BOBYQAOptimizer.java

  1. // CHECKSTYLE: stop all
  2. /*
  3.  * Licensed to the Apache Software Foundation (ASF) under one or more
  4.  * contributor license agreements.  See the NOTICE file distributed with
  5.  * this work for additional information regarding copyright ownership.
  6.  * The ASF licenses this file to You under the Apache License, Version 2.0
  7.  * (the "License"); you may not use this file except in compliance with
  8.  * the License.  You may obtain a copy of the License at
  9.  *
  10.  *      http://www.apache.org/licenses/LICENSE-2.0
  11.  *
  12.  * Unless required by applicable law or agreed to in writing, software
  13.  * distributed under the License is distributed on an "AS IS" BASIS,
  14.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  15.  * See the License for the specific language governing permissions and
  16.  * limitations under the License.
  17.  */

  19. package org.apache.commons.math3.optimization.direct;

  21. import org.apache.commons.math3.analysis.MultivariateFunction;
  22. import org.apache.commons.math3.exception.MathIllegalStateException;
  23. import org.apache.commons.math3.exception.NumberIsTooSmallException;
  24. import org.apache.commons.math3.exception.OutOfRangeException;
  25. import org.apache.commons.math3.exception.util.LocalizedFormats;
  26. import org.apache.commons.math3.linear.Array2DRowRealMatrix;
  27. import org.apache.commons.math3.linear.ArrayRealVector;
  28. import org.apache.commons.math3.linear.RealVector;
  29. import org.apache.commons.math3.optimization.GoalType;
  30. import org.apache.commons.math3.optimization.PointValuePair;
  31. import org.apache.commons.math3.optimization.MultivariateOptimizer;
  32. import org.apache.commons.math3.util.FastMath;

  34. /**
  35.  * Powell's BOBYQA algorithm. This implementation is translated and
  36.  * adapted from the Fortran version available
  37.  * <a href="http://plato.asu.edu/ftp/other_software/bobyqa.zip">here</a>.
  38.  * See <a href="http://www.optimization-online.org/DB_HTML/2010/05/2616.html">
  39.  * this paper</a> for an introduction.
  40.  * <br/>
  41.  * BOBYQA is particularly well suited for high dimensional problems
  42.  * where derivatives are not available. In most cases it outperforms the
  43.  * {@link PowellOptimizer} significantly. Stochastic algorithms like
  44.  * {@link CMAESOptimizer} succeed more often than BOBYQA, but are more
  45.  * expensive. BOBYQA could also be considered as a replacement of any
  46.  * derivative-based optimizer when the derivatives are approximated by
  47.  * finite differences.
  48.  *
  49.  * @deprecated As of 3.1 (to be removed in 4.0).
  50.  * @since 3.0
  51.  */
  52. @Deprecated
  53. public class BOBYQAOptimizer
  54.     extends BaseAbstractMultivariateSimpleBoundsOptimizer<MultivariateFunction>
  55.     implements MultivariateOptimizer {
  56.     /** Minimum dimension of the problem: {@value} */
  57.     public static final int MINIMUM_PROBLEM_DIMENSION = 2;
  58.     /** Default value for {@link #initialTrustRegionRadius}: {@value} . */
  59.     public static final double DEFAULT_INITIAL_RADIUS = 10.0;
  60.     /** Default value for {@link #stoppingTrustRegionRadius}: {@value} . */
  61.     public static final double DEFAULT_STOPPING_RADIUS = 1E-8;

  63.     private static final double ZERO = 0d;
  64.     private static final double ONE = 1d;
  65.     private static final double TWO = 2d;
  66.     private static final double TEN = 10d;
  67.     private static final double SIXTEEN = 16d;
  68.     private static final double TWO_HUNDRED_FIFTY = 250d;
  69.     private static final double MINUS_ONE = -ONE;
  70.     private static final double HALF = ONE / 2;
  71.     private static final double ONE_OVER_FOUR = ONE / 4;
  72.     private static final double ONE_OVER_EIGHT = ONE / 8;
  73.     private static final double ONE_OVER_TEN = ONE / 10;
  74.     private static final double ONE_OVER_A_THOUSAND = ONE / 1000;

  76.     /**
  77.      * numberOfInterpolationPoints XXX
  78.      */
  79.     private final int numberOfInterpolationPoints;
  80.     /**
  81.      * initialTrustRegionRadius XXX
  82.      */
  83.     private double initialTrustRegionRadius;
  84.     /**
  85.      * stoppingTrustRegionRadius XXX
  86.      */
  87.     private final double stoppingTrustRegionRadius;
  88.     /** Goal type (minimize or maximize). */
  89.     private boolean isMinimize;
  90.     /**
  91.      * Current best values for the variables to be optimized.
  92.      * The vector will be changed in-place to contain the values of the least
  93.      * calculated objective function values.
  94.      */
  95.     private ArrayRealVector currentBest;
  96.     /** Differences between the upper and lower bounds. */
  97.     private double[] boundDifference;
  98.     /**
  99.      * Index of the interpolation point at the trust region center.
  100.      */
  101.     private int trustRegionCenterInterpolationPointIndex;
  102.     /**
  103.      * Last <em>n</em> columns of matrix H (where <em>n</em> is the dimension
  104.      * of the problem).
  105.      * XXX "bmat" in the original code.
  106.      */
  107.     private Array2DRowRealMatrix bMatrix;
  108.     /**
  109.      * Factorization of the leading <em>npt</em> square submatrix of H, this
  110.      * factorization being Z Z<sup>T</sup>, which provides both the correct
  111.      * rank and positive semi-definiteness.
  112.      * XXX "zmat" in the original code.
  113.      */
  114.     private Array2DRowRealMatrix zMatrix;
  115.     /**
  116.      * Coordinates of the interpolation points relative to {@link #originShift}.
  117.      * XXX "xpt" in the original code.
  118.      */
  119.     private Array2DRowRealMatrix interpolationPoints;
  120.     /**
  121.      * Shift of origin that should reduce the contributions from rounding
  122.      * errors to values of the model and Lagrange functions.
  123.      * XXX "xbase" in the original code.
  124.      */
  125.     private ArrayRealVector originShift;
  126.     /**
  127.      * Values of the objective function at the interpolation points.
  128.      * XXX "fval" in the original code.
  129.      */
  130.     private ArrayRealVector fAtInterpolationPoints;
  131.     /**
  132.      * Displacement from {@link #originShift} of the trust region center.
  133.      * XXX "xopt" in the original code.
  134.      */
  135.     private ArrayRealVector trustRegionCenterOffset;
  136.     /**
  137.      * Gradient of the quadratic model at {@link #originShift} +
  138.      * {@link #trustRegionCenterOffset}.
  139.      * XXX "gopt" in the original code.
  140.      */
  141.     private ArrayRealVector gradientAtTrustRegionCenter;
  142.     /**
  143.      * Differences {@link #getLowerBound()} - {@link #originShift}.
  144.      * All the components of every {@link #trustRegionCenterOffset} are going
  145.      * to satisfy the bounds<br/>
  146.      * {@link #getLowerBound() lowerBound}<sub>i</sub> &le;
  147.      * {@link #trustRegionCenterOffset}<sub>i</sub>,<br/>
  148.      * with appropriate equalities when {@link #trustRegionCenterOffset} is
  149.      * on a constraint boundary.
  150.      * XXX "sl" in the original code.
  151.      */
  152.     private ArrayRealVector lowerDifference;
  153.     /**
  154.      * Differences {@link #getUpperBound()} - {@link #originShift}
  155.      * All the components of every {@link #trustRegionCenterOffset} are going
  156.      * to satisfy the bounds<br/>
  157.      *  {@link #trustRegionCenterOffset}<sub>i</sub> &le;
  158.      *  {@link #getUpperBound() upperBound}<sub>i</sub>,<br/>
  159.      * with appropriate equalities when {@link #trustRegionCenterOffset} is
  160.      * on a constraint boundary.
  161.      * XXX "su" in the original code.
  162.      */
  163.     private ArrayRealVector upperDifference;
  164.     /**
  165.      * Parameters of the implicit second derivatives of the quadratic model.
  166.      * XXX "pq" in the original code.
  167.      */
  168.     private ArrayRealVector modelSecondDerivativesParameters;
  169.     /**
  170.      * Point chosen by function {@link #trsbox(double,ArrayRealVector,
  171.      * ArrayRealVector, ArrayRealVector,ArrayRealVector,ArrayRealVector) trsbox}
  172.      * or {@link #altmov(int,double) altmov}.
  173.      * Usually {@link #originShift} + {@link #newPoint} is the vector of
  174.      * variables for the next evaluation of the objective function.
  175.      * It also satisfies the constraints indicated in {@link #lowerDifference}
  176.      * and {@link #upperDifference}.
  177.      * XXX "xnew" in the original code.
  178.      */
  179.     private ArrayRealVector newPoint;
  180.     /**
  181.      * Alternative to {@link #newPoint}, chosen by
  182.      * {@link #altmov(int,double) altmov}.
  183.      * It may replace {@link #newPoint} in order to increase the denominator
  184.      * in the {@link #update(double, double, int) updating procedure}.
  185.      * XXX "xalt" in the original code.
  186.      */
  187.     private ArrayRealVector alternativeNewPoint;
  188.     /**
  189.      * Trial step from {@link #trustRegionCenterOffset} which is usually
  190.      * {@link #newPoint} - {@link #trustRegionCenterOffset}.
  191.      * XXX "d__" in the original code.
  192.      */
  193.     private ArrayRealVector trialStepPoint;
  194.     /**
  195.      * Values of the Lagrange functions at a new point.
  196.      * XXX "vlag" in the original code.
  197.      */
  198.     private ArrayRealVector lagrangeValuesAtNewPoint;
  199.     /**
  200.      * Explicit second derivatives of the quadratic model.
  201.      * XXX "hq" in the original code.
  202.      */
  203.     private ArrayRealVector modelSecondDerivativesValues;

  205.     /**
  206.      * @param numberOfInterpolationPoints Number of interpolation conditions.
  207.      * For a problem of dimension {@code n}, its value must be in the interval
  208.      * {@code [n+2, (n+1)(n+2)/2]}.
  209.      * Choices that exceed {@code 2n+1} are not recommended.
  210.      */
  211.     public BOBYQAOptimizer(int numberOfInterpolationPoints) {
  212.         this(numberOfInterpolationPoints,
  213.              DEFAULT_INITIAL_RADIUS,
  214.              DEFAULT_STOPPING_RADIUS);
  215.     }

  217.     /**
  218.      * @param numberOfInterpolationPoints Number of interpolation conditions.
  219.      * For a problem of dimension {@code n}, its value must be in the interval
  220.      * {@code [n+2, (n+1)(n+2)/2]}.
  221.      * Choices that exceed {@code 2n+1} are not recommended.
  222.      * @param initialTrustRegionRadius Initial trust region radius.
  223.      * @param stoppingTrustRegionRadius Stopping trust region radius.
  224.      */
  225.     public BOBYQAOptimizer(int numberOfInterpolationPoints,
  226.                            double initialTrustRegionRadius,
  227.                            double stoppingTrustRegionRadius) {
  228.         super(null); // No custom convergence criterion.
  229.         this.numberOfInterpolationPoints = numberOfInterpolationPoints;
  230.         this.initialTrustRegionRadius = initialTrustRegionRadius;
  231.         this.stoppingTrustRegionRadius = stoppingTrustRegionRadius;
  232.     }

  234.     /** {@inheritDoc} */
  235.     @Override
  236.     protected PointValuePair doOptimize() {
  237.         final double[] lowerBound = getLowerBound();
  238.         final double[] upperBound = getUpperBound();

  240.         // Validity checks.
  241.         setup(lowerBound, upperBound);

  243.         isMinimize = (getGoalType() == GoalType.MINIMIZE);
  244.         currentBest = new ArrayRealVector(getStartPoint());

  246.         final double value = bobyqa(lowerBound, upperBound);

  248.         return new PointValuePair(currentBest.getDataRef(),
  249.                                       isMinimize ? value : -value);
  250.     }

  252.     /**
  253.      *     This subroutine seeks the least value of a function of many variables,
  254.      *     by applying a trust region method that forms quadratic models by
  255.      *     interpolation. There is usually some freedom in the interpolation
  256.      *     conditions, which is taken up by minimizing the Frobenius norm of
  257.      *     the change to the second derivative of the model, beginning with the
  258.      *     zero matrix. The values of the variables are constrained by upper and
  259.      *     lower bounds. The arguments of the subroutine are as follows.
  260.      *
  261.      *     N must be set to the number of variables and must be at least two.
  262.      *     NPT is the number of interpolation conditions. Its value must be in
  263.      *       the interval [N+2,(N+1)(N+2)/2]. Choices that exceed 2*N+1 are not
  264.      *       recommended.
  265.      *     Initial values of the variables must be set in X(1),X(2),...,X(N). They
  266.      *       will be changed to the values that give the least calculated F.
  267.      *     For I=1,2,...,N, XL(I) and XU(I) must provide the lower and upper
  268.      *       bounds, respectively, on X(I). The construction of quadratic models
  269.      *       requires XL(I) to be strictly less than XU(I) for each I. Further,
  270.      *       the contribution to a model from changes to the I-th variable is
  271.      *       damaged severely by rounding errors if XU(I)-XL(I) is too small.
  272.      *     RHOBEG and RHOEND must be set to the initial and final values of a trust
  273.      *       region radius, so both must be positive with RHOEND no greater than
  274.      *       RHOBEG. Typically, RHOBEG should be about one tenth of the greatest
  275.      *       expected change to a variable, while RHOEND should indicate the
  276.      *       accuracy that is required in the final values of the variables. An
  277.      *       error return occurs if any of the differences XU(I)-XL(I), I=1,...,N,
  278.      *       is less than 2*RHOBEG.
  279.      *     MAXFUN must be set to an upper bound on the number of calls of CALFUN.
  280.      *     The array W will be used for working space. Its length must be at least
  281.      *       (NPT+5)*(NPT+N)+3*N*(N+5)/2.
  282.      *
  283.      * @param lowerBound Lower bounds.
  284.      * @param upperBound Upper bounds.
  285.      * @return the value of the objective at the optimum.
  286.      */
  287.     private double bobyqa(double[] lowerBound,
  288.                           double[] upperBound) {
  289.         printMethod(); // XXX

  291.         final int n = currentBest.getDimension();

  293.         // Return if there is insufficient space between the bounds. Modify the
  294.         // initial X if necessary in order to avoid conflicts between the bounds
  295.         // and the construction of the first quadratic model. The lower and upper
  296.         // bounds on moves from the updated X are set now, in the ISL and ISU
  297.         // partitions of W, in order to provide useful and exact information about
  298.         // components of X that become within distance RHOBEG from their bounds.

  300.         for (int j = 0; j < n; j++) {
  301.             final double boundDiff = boundDifference[j];
  302.             lowerDifference.setEntry(j, lowerBound[j] - currentBest.getEntry(j));
  303.             upperDifference.setEntry(j, upperBound[j] - currentBest.getEntry(j));
  304.             if (lowerDifference.getEntry(j) >= -initialTrustRegionRadius) {
  305.                 if (lowerDifference.getEntry(j) >= ZERO) {
  306.                     currentBest.setEntry(j, lowerBound[j]);
  307.                     lowerDifference.setEntry(j, ZERO);
  308.                     upperDifference.setEntry(j, boundDiff);
  309.                 } else {
  310.                     currentBest.setEntry(j, lowerBound[j] + initialTrustRegionRadius);
  311.                     lowerDifference.setEntry(j, -initialTrustRegionRadius);
  312.                     // Computing MAX
  313.                     final double deltaOne = upperBound[j] - currentBest.getEntry(j);
  314.                     upperDifference.setEntry(j, FastMath.max(deltaOne, initialTrustRegionRadius));
  315.                 }
  316.             } else if (upperDifference.getEntry(j) <= initialTrustRegionRadius) {
  317.                 if (upperDifference.getEntry(j) <= ZERO) {
  318.                     currentBest.setEntry(j, upperBound[j]);
  319.                     lowerDifference.setEntry(j, -boundDiff);
  320.                     upperDifference.setEntry(j, ZERO);
  321.                 } else {
  322.                     currentBest.setEntry(j, upperBound[j] - initialTrustRegionRadius);
  323.                     // Computing MIN
  324.                     final double deltaOne = lowerBound[j] - currentBest.getEntry(j);
  325.                     final double deltaTwo = -initialTrustRegionRadius;
  326.                     lowerDifference.setEntry(j, FastMath.min(deltaOne, deltaTwo));
  327.                     upperDifference.setEntry(j, initialTrustRegionRadius);
  328.                 }
  329.             }
  330.         }

  332.         // Make the call of BOBYQB.

  334.         return bobyqb(lowerBound, upperBound);
  335.     } // bobyqa

  337.     // ----------------------------------------------------------------------------------------

  339.     /**
  340.      *     The arguments N, NPT, X, XL, XU, RHOBEG, RHOEND, IPRINT and MAXFUN
  341.      *       are identical to the corresponding arguments in SUBROUTINE BOBYQA.
  342.      *     XBASE holds a shift of origin that should reduce the contributions
  343.      *       from rounding errors to values of the model and Lagrange functions.
  344.      *     XPT is a two-dimensional array that holds the coordinates of the
  345.      *       interpolation points relative to XBASE.
  346.      *     FVAL holds the values of F at the interpolation points.
  347.      *     XOPT is set to the displacement from XBASE of the trust region centre.
  348.      *     GOPT holds the gradient of the quadratic model at XBASE+XOPT.
  349.      *     HQ holds the explicit second derivatives of the quadratic model.
  350.      *     PQ contains the parameters of the implicit second derivatives of the
  351.      *       quadratic model.
  352.      *     BMAT holds the last N columns of H.
  353.      *     ZMAT holds the factorization of the leading NPT by NPT submatrix of H,
  354.      *       this factorization being ZMAT times ZMAT^T, which provides both the
  355.      *       correct rank and positive semi-definiteness.
  356.      *     NDIM is the first dimension of BMAT and has the value NPT+N.
  357.      *     SL and SU hold the differences XL-XBASE and XU-XBASE, respectively.
  358.      *       All the components of every XOPT are going to satisfy the bounds
  359.      *       SL(I) .LEQ. XOPT(I) .LEQ. SU(I), with appropriate equalities when
  360.      *       XOPT is on a constraint boundary.
  361.      *     XNEW is chosen by SUBROUTINE TRSBOX or ALTMOV. Usually XBASE+XNEW is the
  362.      *       vector of variables for the next call of CALFUN. XNEW also satisfies
  363.      *       the SL and SU constraints in the way that has just been mentioned.
  364.      *     XALT is an alternative to XNEW, chosen by ALTMOV, that may replace XNEW
  365.      *       in order to increase the denominator in the updating of UPDATE.
  366.      *     D is reserved for a trial step from XOPT, which is usually XNEW-XOPT.
  367.      *     VLAG contains the values of the Lagrange functions at a new point X.
  368.      *       They are part of a product that requires VLAG to be of length NDIM.
  369.      *     W is a one-dimensional array that is used for working space. Its length
  370.      *       must be at least 3*NDIM = 3*(NPT+N).
  371.      *
  372.      * @param lowerBound Lower bounds.
  373.      * @param upperBound Upper bounds.
  374.      * @return the value of the objective at the optimum.
  375.      */
  376.     private double bobyqb(double[] lowerBound,
  377.                           double[] upperBound) {
  378.         printMethod(); // XXX

  380.         final int n = currentBest.getDimension();
  381.         final int npt = numberOfInterpolationPoints;
  382.         final int np = n + 1;
  383.         final int nptm = npt - np;
  384.         final int nh = n * np / 2;

  386.         final ArrayRealVector work1 = new ArrayRealVector(n);
  387.         final ArrayRealVector work2 = new ArrayRealVector(npt);
  388.         final ArrayRealVector work3 = new ArrayRealVector(npt);

  390.         double cauchy = Double.NaN;
  391.         double alpha = Double.NaN;
  392.         double dsq = Double.NaN;
  393.         double crvmin = Double.NaN;

  395.         // Set some constants.
  396.         // Parameter adjustments

  398.         // Function Body

  400.         // The call of PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ,
  401.         // BMAT and ZMAT for the first iteration, with the corresponding values of
  402.         // of NF and KOPT, which are the number of calls of CALFUN so far and the
  403.         // index of the interpolation point at the trust region centre. Then the
  404.         // initial XOPT is set too. The branch to label 720 occurs if MAXFUN is
  405.         // less than NPT. GOPT will be updated if KOPT is different from KBASE.

  407.         trustRegionCenterInterpolationPointIndex = 0;

  409.         prelim(lowerBound, upperBound);
  410.         double xoptsq = ZERO;
  411.         for (int i = 0; i < n; i++) {
  412.             trustRegionCenterOffset.setEntry(i, interpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex, i));
  413.             // Computing 2nd power
  414.             final double deltaOne = trustRegionCenterOffset.getEntry(i);
  415.             xoptsq += deltaOne * deltaOne;
  416.         }
  417.         double fsave = fAtInterpolationPoints.getEntry(0);
  418.         final int kbase = 0;

  420.         // Complete the settings that are required for the iterative procedure.

  422.         int ntrits = 0;
  423.         int itest = 0;
  424.         int knew = 0;
  425.         int nfsav = getEvaluations();
  426.         double rho = initialTrustRegionRadius;
  427.         double delta = rho;
  428.         double diffa = ZERO;
  429.         double diffb = ZERO;
  430.         double diffc = ZERO;
  431.         double f = ZERO;
  432.         double beta = ZERO;
  433.         double adelt = ZERO;
  434.         double denom = ZERO;
  435.         double ratio = ZERO;
  436.         double dnorm = ZERO;
  437.         double scaden = ZERO;
  438.         double biglsq = ZERO;
  439.         double distsq = ZERO;

  441.         // Update GOPT if necessary before the first iteration and after each
  442.         // call of RESCUE that makes a call of CALFUN.

  444.         int state = 20;
  445.         for(;;) switch (state) {
  446.         case 20: {
  447.             printState(20); // XXX
  448.             if (trustRegionCenterInterpolationPointIndex != kbase) {
  449.                 int ih = 0;
  450.                 for (int j = 0; j < n; j++) {
  451.                     for (int i = 0; i <= j; i++) {
  452.                         if (i < j) {
  453.                             gradientAtTrustRegionCenter.setEntry(j,  gradientAtTrustRegionCenter.getEntry(j) + modelSecondDerivativesValues.getEntry(ih) * trustRegionCenterOffset.getEntry(i));
  454.                         }
  455.                         gradientAtTrustRegionCenter.setEntry(i,  gradientAtTrustRegionCenter.getEntry(i) + modelSecondDerivativesValues.getEntry(ih) * trustRegionCenterOffset.getEntry(j));
  456.                         ih++;
  457.                     }
  458.                 }
  459.                 if (getEvaluations() > npt) {
  460.                     for (int k = 0; k < npt; k++) {
  461.                         double temp = ZERO;
  462.                         for (int j = 0; j < n; j++) {
  463.                             temp += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j);
  464.                         }
  465.                         temp *= modelSecondDerivativesParameters.getEntry(k);
  466.                         for (int i = 0; i < n; i++) {
  467.                             gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + temp * interpolationPoints.getEntry(k, i));
  468.                         }
  469.                     }
  470.                     // throw new PathIsExploredException(); // XXX
  471.                 }
  472.             }

  474.             // Generate the next point in the trust region that provides a small value
  475.             // of the quadratic model subject to the constraints on the variables.
  476.             // The int NTRITS is set to the number "trust region" iterations that
  477.             // have occurred since the last "alternative" iteration. If the length
  478.             // of XNEW-XOPT is less than HALF*RHO, however, then there is a branch to
  479.             // label 650 or 680 with NTRITS=-1, instead of calculating F at XNEW.

  481.         }
  482.         case 60: {
  483.             printState(60); // XXX
  484.             final ArrayRealVector gnew = new ArrayRealVector(n);
  485.             final ArrayRealVector xbdi = new ArrayRealVector(n);
  486.             final ArrayRealVector s = new ArrayRealVector(n);
  487.             final ArrayRealVector hs = new ArrayRealVector(n);
  488.             final ArrayRealVector hred = new ArrayRealVector(n);

  490.             final double[] dsqCrvmin = trsbox(delta, gnew, xbdi, s,
  491.                                               hs, hred);
  492.             dsq = dsqCrvmin[0];
  493.             crvmin = dsqCrvmin[1];

  495.             // Computing MIN
  496.             double deltaOne = delta;
  497.             double deltaTwo = FastMath.sqrt(dsq);
  498.             dnorm = FastMath.min(deltaOne, deltaTwo);
  499.             if (dnorm < HALF * rho) {
  500.                 ntrits = -1;
  501.                 // Computing 2nd power
  502.                 deltaOne = TEN * rho;
  503.                 distsq = deltaOne * deltaOne;
  504.                 if (getEvaluations() <= nfsav + 2) {
  505.                     state = 650; break;
  506.                 }

  508.                 // The following choice between labels 650 and 680 depends on whether or
  509.                 // not our work with the current RHO seems to be complete. Either RHO is
  510.                 // decreased or termination occurs if the errors in the quadratic model at
  511.                 // the last three interpolation points compare favourably with predictions
  512.                 // of likely improvements to the model within distance HALF*RHO of XOPT.

  514.                 // Computing MAX
  515.                 deltaOne = FastMath.max(diffa, diffb);
  516.                 final double errbig = FastMath.max(deltaOne, diffc);
  517.                 final double frhosq = rho * ONE_OVER_EIGHT * rho;
  518.                 if (crvmin > ZERO &&
  519.                     errbig > frhosq * crvmin) {
  520.                     state = 650; break;
  521.                 }
  522.                 final double bdtol = errbig / rho;
  523.                 for (int j = 0; j < n; j++) {
  524.                     double bdtest = bdtol;
  525.                     if (newPoint.getEntry(j) == lowerDifference.getEntry(j)) {
  526.                         bdtest = work1.getEntry(j);
  527.                     }
  528.                     if (newPoint.getEntry(j) == upperDifference.getEntry(j)) {
  529.                         bdtest = -work1.getEntry(j);
  530.                     }
  531.                     if (bdtest < bdtol) {
  532.                         double curv = modelSecondDerivativesValues.getEntry((j + j * j) / 2);
  533.                         for (int k = 0; k < npt; k++) {
  534.                             // Computing 2nd power
  535.                             final double d1 = interpolationPoints.getEntry(k, j);
  536.                             curv += modelSecondDerivativesParameters.getEntry(k) * (d1 * d1);
  537.                         }
  538.                         bdtest += HALF * curv * rho;
  539.                         if (bdtest < bdtol) {
  540.                             state = 650; break;
  541.                         }
  542.                         // throw new PathIsExploredException(); // XXX
  543.                     }
  544.                 }
  545.                 state = 680; break;
  546.             }
  547.             ++ntrits;

  549.             // Severe cancellation is likely to occur if XOPT is too far from XBASE.
  550.             // If the following test holds, then XBASE is shifted so that XOPT becomes
  551.             // zero. The appropriate changes are made to BMAT and to the second
  552.             // derivatives of the current model, beginning with the changes to BMAT
  553.             // that do not depend on ZMAT. VLAG is used temporarily for working space.

  555.         }
  556.         case 90: {
  557.             printState(90); // XXX
  558.             if (dsq <= xoptsq * ONE_OVER_A_THOUSAND) {
  559.                 final double fracsq = xoptsq * ONE_OVER_FOUR;
  560.                 double sumpq = ZERO;
  561.                 // final RealVector sumVector
  562.                 //     = new ArrayRealVector(npt, -HALF * xoptsq).add(interpolationPoints.operate(trustRegionCenter));
  563.                 for (int k = 0; k < npt; k++) {
  564.                     sumpq += modelSecondDerivativesParameters.getEntry(k);
  565.                     double sum = -HALF * xoptsq;
  566.                     for (int i = 0; i < n; i++) {
  567.                         sum += interpolationPoints.getEntry(k, i) * trustRegionCenterOffset.getEntry(i);
  568.                     }
  569.                     // sum = sumVector.getEntry(k); // XXX "testAckley" and "testDiffPow" fail.
  570.                     work2.setEntry(k, sum);
  571.                     final double temp = fracsq - HALF * sum;
  572.                     for (int i = 0; i < n; i++) {
  573.                         work1.setEntry(i, bMatrix.getEntry(k, i));
  574.                         lagrangeValuesAtNewPoint.setEntry(i, sum * interpolationPoints.getEntry(k, i) + temp * trustRegionCenterOffset.getEntry(i));
  575.                         final int ip = npt + i;
  576.                         for (int j = 0; j <= i; j++) {
  577.                             bMatrix.setEntry(ip, j,
  578.                                           bMatrix.getEntry(ip, j)
  579.                                           + work1.getEntry(i) * lagrangeValuesAtNewPoint.getEntry(j)
  580.                                           + lagrangeValuesAtNewPoint.getEntry(i) * work1.getEntry(j));
  581.                         }
  582.                     }
  583.                 }

  585.                 // Then the revisions of BMAT that depend on ZMAT are calculated.

  587.                 for (int m = 0; m < nptm; m++) {
  588.                     double sumz = ZERO;
  589.                     double sumw = ZERO;
  590.                     for (int k = 0; k < npt; k++) {
  591.                         sumz += zMatrix.getEntry(k, m);
  592.                         lagrangeValuesAtNewPoint.setEntry(k, work2.getEntry(k) * zMatrix.getEntry(k, m));
  593.                         sumw += lagrangeValuesAtNewPoint.getEntry(k);
  594.                     }
  595.                     for (int j = 0; j < n; j++) {
  596.                         double sum = (fracsq * sumz - HALF * sumw) * trustRegionCenterOffset.getEntry(j);
  597.                         for (int k = 0; k < npt; k++) {
  598.                             sum += lagrangeValuesAtNewPoint.getEntry(k) * interpolationPoints.getEntry(k, j);
  599.                         }
  600.                         work1.setEntry(j, sum);
  601.                         for (int k = 0; k < npt; k++) {
  602.                             bMatrix.setEntry(k, j,
  603.                                           bMatrix.getEntry(k, j)
  604.                                           + sum * zMatrix.getEntry(k, m));
  605.                         }
  606.                     }
  607.                     for (int i = 0; i < n; i++) {
  608.                         final int ip = i + npt;
  609.                         final double temp = work1.getEntry(i);
  610.                         for (int j = 0; j <= i; j++) {
  611.                             bMatrix.setEntry(ip, j,
  612.                                           bMatrix.getEntry(ip, j)
  613.                                           + temp * work1.getEntry(j));
  614.                         }
  615.                     }
  616.                 }

  618.                 // The following instructions complete the shift, including the changes
  619.                 // to the second derivative parameters of the quadratic model.

  621.                 int ih = 0;
  622.                 for (int j = 0; j < n; j++) {
  623.                     work1.setEntry(j, -HALF * sumpq * trustRegionCenterOffset.getEntry(j));
  624.                     for (int k = 0; k < npt; k++) {
  625.                         work1.setEntry(j, work1.getEntry(j) + modelSecondDerivativesParameters.getEntry(k) * interpolationPoints.getEntry(k, j));
  626.                         interpolationPoints.setEntry(k, j, interpolationPoints.getEntry(k, j) - trustRegionCenterOffset.getEntry(j));
  627.                     }
  628.                     for (int i = 0; i <= j; i++) {
  629.                          modelSecondDerivativesValues.setEntry(ih,
  630.                                     modelSecondDerivativesValues.getEntry(ih)
  631.                                     + work1.getEntry(i) * trustRegionCenterOffset.getEntry(j)
  632.                                     + trustRegionCenterOffset.getEntry(i) * work1.getEntry(j));
  633.                         bMatrix.setEntry(npt + i, j, bMatrix.getEntry(npt + j, i));
  634.                         ih++;
  635.                     }
  636.                 }
  637.                 for (int i = 0; i < n; i++) {
  638.                     originShift.setEntry(i, originShift.getEntry(i) + trustRegionCenterOffset.getEntry(i));
  639.                     newPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  640.                     lowerDifference.setEntry(i, lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  641.                     upperDifference.setEntry(i, upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  642.                     trustRegionCenterOffset.setEntry(i, ZERO);
  643.                 }
  644.                 xoptsq = ZERO;
  645.             }
  646.             if (ntrits == 0) {
  647.                 state = 210; break;
  648.             }
  649.             state = 230; break;

  651.             // XBASE is also moved to XOPT by a call of RESCUE. This calculation is
  652.             // more expensive than the previous shift, because new matrices BMAT and
  653.             // ZMAT are generated from scratch, which may include the replacement of
  654.             // interpolation points whose positions seem to be causing near linear
  655.             // dependence in the interpolation conditions. Therefore RESCUE is called
  656.             // only if rounding errors have reduced by at least a factor of two the
  657.             // denominator of the formula for updating the H matrix. It provides a
  658.             // useful safeguard, but is not invoked in most applications of BOBYQA.

  660.         }
  661.         case 210: {
  662.             printState(210); // XXX
  663.             // Pick two alternative vectors of variables, relative to XBASE, that
  664.             // are suitable as new positions of the KNEW-th interpolation point.
  665.             // Firstly, XNEW is set to the point on a line through XOPT and another
  666.             // interpolation point that minimizes the predicted value of the next
  667.             // denominator, subject to ||XNEW - XOPT|| .LEQ. ADELT and to the SL
  668.             // and SU bounds. Secondly, XALT is set to the best feasible point on
  669.             // a constrained version of the Cauchy step of the KNEW-th Lagrange
  670.             // function, the corresponding value of the square of this function
  671.             // being returned in CAUCHY. The choice between these alternatives is
  672.             // going to be made when the denominator is calculated.

  674.             final double[] alphaCauchy = altmov(knew, adelt);
  675.             alpha = alphaCauchy[0];
  676.             cauchy = alphaCauchy[1];

  678.             for (int i = 0; i < n; i++) {
  679.                 trialStepPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  680.             }

  682.             // Calculate VLAG and BETA for the current choice of D. The scalar
  683.             // product of D with XPT(K,.) is going to be held in W(NPT+K) for
  684.             // use when VQUAD is calculated.

  686.         }
  687.         case 230: {
  688.             printState(230); // XXX
  689.             for (int k = 0; k < npt; k++) {
  690.                 double suma = ZERO;
  691.                 double sumb = ZERO;
  692.                 double sum = ZERO;
  693.                 for (int j = 0; j < n; j++) {
  694.                     suma += interpolationPoints.getEntry(k, j) * trialStepPoint.getEntry(j);
  695.                     sumb += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j);
  696.                     sum += bMatrix.getEntry(k, j) * trialStepPoint.getEntry(j);
  697.                 }
  698.                 work3.setEntry(k, suma * (HALF * suma + sumb));
  699.                 lagrangeValuesAtNewPoint.setEntry(k, sum);
  700.                 work2.setEntry(k, suma);
  701.             }
  702.             beta = ZERO;
  703.             for (int m = 0; m < nptm; m++) {
  704.                 double sum = ZERO;
  705.                 for (int k = 0; k < npt; k++) {
  706.                     sum += zMatrix.getEntry(k, m) * work3.getEntry(k);
  707.                 }
  708.                 beta -= sum * sum;
  709.                 for (int k = 0; k < npt; k++) {
  710.                     lagrangeValuesAtNewPoint.setEntry(k, lagrangeValuesAtNewPoint.getEntry(k) + sum * zMatrix.getEntry(k, m));
  711.                 }
  712.             }
  713.             dsq = ZERO;
  714.             double bsum = ZERO;
  715.             double dx = ZERO;
  716.             for (int j = 0; j < n; j++) {
  717.                 // Computing 2nd power
  718.                 final double d1 = trialStepPoint.getEntry(j);
  719.                 dsq += d1 * d1;
  720.                 double sum = ZERO;
  721.                 for (int k = 0; k < npt; k++) {
  722.                     sum += work3.getEntry(k) * bMatrix.getEntry(k, j);
  723.                 }
  724.                 bsum += sum * trialStepPoint.getEntry(j);
  725.                 final int jp = npt + j;
  726.                 for (int i = 0; i < n; i++) {
  727.                     sum += bMatrix.getEntry(jp, i) * trialStepPoint.getEntry(i);
  728.                 }
  729.                 lagrangeValuesAtNewPoint.setEntry(jp, sum);
  730.                 bsum += sum * trialStepPoint.getEntry(j);
  731.                 dx += trialStepPoint.getEntry(j) * trustRegionCenterOffset.getEntry(j);
  732.             }

  734.             beta = dx * dx + dsq * (xoptsq + dx + dx + HALF * dsq) + beta - bsum; // Original
  735.             // beta += dx * dx + dsq * (xoptsq + dx + dx + HALF * dsq) - bsum; // XXX "testAckley" and "testDiffPow" fail.
  736.             // beta = dx * dx + dsq * (xoptsq + 2 * dx + HALF * dsq) + beta - bsum; // XXX "testDiffPow" fails.

  738.             lagrangeValuesAtNewPoint.setEntry(trustRegionCenterInterpolationPointIndex,
  739.                           lagrangeValuesAtNewPoint.getEntry(trustRegionCenterInterpolationPointIndex) + ONE);

  741.             // If NTRITS is zero, the denominator may be increased by replacing
  742.             // the step D of ALTMOV by a Cauchy step. Then RESCUE may be called if
  743.             // rounding errors have damaged the chosen denominator.

  745.             if (ntrits == 0) {
  746.                 // Computing 2nd power
  747.                 final double d1 = lagrangeValuesAtNewPoint.getEntry(knew);
  748.                 denom = d1 * d1 + alpha * beta;
  749.                 if (denom < cauchy && cauchy > ZERO) {
  750.                     for (int i = 0; i < n; i++) {
  751.                         newPoint.setEntry(i, alternativeNewPoint.getEntry(i));
  752.                         trialStepPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  753.                     }
  754.                     cauchy = ZERO; // XXX Useful statement?
  755.                     state = 230; break;
  756.                 }
  757.                 // Alternatively, if NTRITS is positive, then set KNEW to the index of
  758.                 // the next interpolation point to be deleted to make room for a trust
  759.                 // region step. Again RESCUE may be called if rounding errors have damaged_
  760.                 // the chosen denominator, which is the reason for attempting to select
  761.                 // KNEW before calculating the next value of the objective function.

  763.             } else {
  764.                 final double delsq = delta * delta;
  765.                 scaden = ZERO;
  766.                 biglsq = ZERO;
  767.                 knew = 0;
  768.                 for (int k = 0; k < npt; k++) {
  769.                     if (k == trustRegionCenterInterpolationPointIndex) {
  770.                         continue;
  771.                     }
  772.                     double hdiag = ZERO;
  773.                     for (int m = 0; m < nptm; m++) {
  774.                         // Computing 2nd power
  775.                         final double d1 = zMatrix.getEntry(k, m);
  776.                         hdiag += d1 * d1;
  777.                     }
  778.                     // Computing 2nd power
  779.                     final double d2 = lagrangeValuesAtNewPoint.getEntry(k);
  780.                     final double den = beta * hdiag + d2 * d2;
  781.                     distsq = ZERO;
  782.                     for (int j = 0; j < n; j++) {
  783.                         // Computing 2nd power
  784.                         final double d3 = interpolationPoints.getEntry(k, j) - trustRegionCenterOffset.getEntry(j);
  785.                         distsq += d3 * d3;
  786.                     }
  787.                     // Computing MAX
  788.                     // Computing 2nd power
  789.                     final double d4 = distsq / delsq;
  790.                     final double temp = FastMath.max(ONE, d4 * d4);
  791.                     if (temp * den > scaden) {
  792.                         scaden = temp * den;
  793.                         knew = k;
  794.                         denom = den;
  795.                     }
  796.                     // Computing MAX
  797.                     // Computing 2nd power
  798.                     final double d5 = lagrangeValuesAtNewPoint.getEntry(k);
  799.                     biglsq = FastMath.max(biglsq, temp * (d5 * d5));
  800.                 }
  801.             }

  803.             // Put the variables for the next calculation of the objective function
  804.             //   in XNEW, with any adjustments for the bounds.

  806.             // Calculate the value of the objective function at XBASE+XNEW, unless
  807.             //   the limit on the number of calculations of F has been reached.

  809.         }
  810.         case 360: {
  811.             printState(360); // XXX
  812.             for (int i = 0; i < n; i++) {
  813.                 // Computing MIN
  814.                 // Computing MAX
  815.                 final double d3 = lowerBound[i];
  816.                 final double d4 = originShift.getEntry(i) + newPoint.getEntry(i);
  817.                 final double d1 = FastMath.max(d3, d4);
  818.                 final double d2 = upperBound[i];
  819.                 currentBest.setEntry(i, FastMath.min(d1, d2));
  820.                 if (newPoint.getEntry(i) == lowerDifference.getEntry(i)) {
  821.                     currentBest.setEntry(i, lowerBound[i]);
  822.                 }
  823.                 if (newPoint.getEntry(i) == upperDifference.getEntry(i)) {
  824.                     currentBest.setEntry(i, upperBound[i]);
  825.                 }
  826.             }

  828.             f = computeObjectiveValue(currentBest.toArray());

  830.             if (!isMinimize)
  831.                 f = -f;
  832.             if (ntrits == -1) {
  833.                 fsave = f;
  834.                 state = 720; break;
  835.             }

  837.             // Use the quadratic model to predict the change in F due to the step D,
  838.             //   and set DIFF to the error of this prediction.

  840.             final double fopt = fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex);
  841.             double vquad = ZERO;
  842.             int ih = 0;
  843.             for (int j = 0; j < n; j++) {
  844.                 vquad += trialStepPoint.getEntry(j) * gradientAtTrustRegionCenter.getEntry(j);
  845.                 for (int i = 0; i <= j; i++) {
  846.                     double temp = trialStepPoint.getEntry(i) * trialStepPoint.getEntry(j);
  847.                     if (i == j) {
  848.                         temp *= HALF;
  849.                     }
  850.                     vquad += modelSecondDerivativesValues.getEntry(ih) * temp;
  851.                     ih++;
  852.                }
  853.             }
  854.             for (int k = 0; k < npt; k++) {
  855.                 // Computing 2nd power
  856.                 final double d1 = work2.getEntry(k);
  857.                 final double d2 = d1 * d1; // "d1" must be squared first to prevent test failures.
  858.                 vquad += HALF * modelSecondDerivativesParameters.getEntry(k) * d2;
  859.             }
  860.             final double diff = f - fopt - vquad;
  861.             diffc = diffb;
  862.             diffb = diffa;
  863.             diffa = FastMath.abs(diff);
  864.             if (dnorm > rho) {
  865.                 nfsav = getEvaluations();
  866.             }

  868.             // Pick the next value of DELTA after a trust region step.

  870.             if (ntrits > 0) {
  871.                 if (vquad >= ZERO) {
  872.                     throw new MathIllegalStateException(LocalizedFormats.TRUST_REGION_STEP_FAILED, vquad);
  873.                 }
  874.                 ratio = (f - fopt) / vquad;
  875.                 final double hDelta = HALF * delta;
  876.                 if (ratio <= ONE_OVER_TEN) {
  877.                     // Computing MIN
  878.                     delta = FastMath.min(hDelta, dnorm);
  879.                 } else if (ratio <= .7) {
  880.                     // Computing MAX
  881.                     delta = FastMath.max(hDelta, dnorm);
  882.                 } else {
  883.                     // Computing MAX
  884.                     delta = FastMath.max(hDelta, 2 * dnorm);
  885.                 }
  886.                 if (delta <= rho * 1.5) {
  887.                     delta = rho;
  888.                 }

  890.                 // Recalculate KNEW and DENOM if the new F is less than FOPT.

  892.                 if (f < fopt) {
  893.                     final int ksav = knew;
  894.                     final double densav = denom;
  895.                     final double delsq = delta * delta;
  896.                     scaden = ZERO;
  897.                     biglsq = ZERO;
  898.                     knew = 0;
  899.                     for (int k = 0; k < npt; k++) {
  900.                         double hdiag = ZERO;
  901.                         for (int m = 0; m < nptm; m++) {
  902.                             // Computing 2nd power
  903.                             final double d1 = zMatrix.getEntry(k, m);
  904.                             hdiag += d1 * d1;
  905.                         }
  906.                         // Computing 2nd power
  907.                         final double d1 = lagrangeValuesAtNewPoint.getEntry(k);
  908.                         final double den = beta * hdiag + d1 * d1;
  909.                         distsq = ZERO;
  910.                         for (int j = 0; j < n; j++) {
  911.                             // Computing 2nd power
  912.                             final double d2 = interpolationPoints.getEntry(k, j) - newPoint.getEntry(j);
  913.                             distsq += d2 * d2;
  914.                         }
  915.                         // Computing MAX
  916.                         // Computing 2nd power
  917.                         final double d3 = distsq / delsq;
  918.                         final double temp = FastMath.max(ONE, d3 * d3);
  919.                         if (temp * den > scaden) {
  920.                             scaden = temp * den;
  921.                             knew = k;
  922.                             denom = den;
  923.                         }
  924.                         // Computing MAX
  925.                         // Computing 2nd power
  926.                         final double d4 = lagrangeValuesAtNewPoint.getEntry(k);
  927.                         final double d5 = temp * (d4 * d4);
  928.                         biglsq = FastMath.max(biglsq, d5);
  929.                     }
  930.                     if (scaden <= HALF * biglsq) {
  931.                         knew = ksav;
  932.                         denom = densav;
  933.                     }
  934.                 }
  935.             }

  937.             // Update BMAT and ZMAT, so that the KNEW-th interpolation point can be
  938.             // moved. Also update the second derivative terms of the model.

  940.             update(beta, denom, knew);

  942.             ih = 0;
  943.             final double pqold = modelSecondDerivativesParameters.getEntry(knew);
  944.             modelSecondDerivativesParameters.setEntry(knew, ZERO);
  945.             for (int i = 0; i < n; i++) {
  946.                 final double temp = pqold * interpolationPoints.getEntry(knew, i);
  947.                 for (int j = 0; j <= i; j++) {
  948.                     modelSecondDerivativesValues.setEntry(ih, modelSecondDerivativesValues.getEntry(ih) + temp * interpolationPoints.getEntry(knew, j));
  949.                     ih++;
  950.                 }
  951.             }
  952.             for (int m = 0; m < nptm; m++) {
  953.                 final double temp = diff * zMatrix.getEntry(knew, m);
  954.                 for (int k = 0; k < npt; k++) {
  955.                     modelSecondDerivativesParameters.setEntry(k, modelSecondDerivativesParameters.getEntry(k) + temp * zMatrix.getEntry(k, m));
  956.                 }
  957.             }

  959.             // Include the new interpolation point, and make the changes to GOPT at
  960.             // the old XOPT that are caused by the updating of the quadratic model.

  962.             fAtInterpolationPoints.setEntry(knew,  f);
  963.             for (int i = 0; i < n; i++) {
  964.                 interpolationPoints.setEntry(knew, i, newPoint.getEntry(i));
  965.                 work1.setEntry(i, bMatrix.getEntry(knew, i));
  966.             }
  967.             for (int k = 0; k < npt; k++) {
  968.                 double suma = ZERO;
  969.                 for (int m = 0; m < nptm; m++) {
  970.                     suma += zMatrix.getEntry(knew, m) * zMatrix.getEntry(k, m);
  971.                 }
  972.                 double sumb = ZERO;
  973.                 for (int j = 0; j < n; j++) {
  974.                     sumb += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j);
  975.                 }
  976.                 final double temp = suma * sumb;
  977.                 for (int i = 0; i < n; i++) {
  978.                     work1.setEntry(i, work1.getEntry(i) + temp * interpolationPoints.getEntry(k, i));
  979.                 }
  980.             }
  981.             for (int i = 0; i < n; i++) {
  982.                 gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + diff * work1.getEntry(i));
  983.             }

  985.             // Update XOPT, GOPT and KOPT if the new calculated F is less than FOPT.

  987.             if (f < fopt) {
  988.                 trustRegionCenterInterpolationPointIndex = knew;
  989.                 xoptsq = ZERO;
  990.                 ih = 0;
  991.                 for (int j = 0; j < n; j++) {
  992.                     trustRegionCenterOffset.setEntry(j, newPoint.getEntry(j));
  993.                     // Computing 2nd power
  994.                     final double d1 = trustRegionCenterOffset.getEntry(j);
  995.                     xoptsq += d1 * d1;
  996.                     for (int i = 0; i <= j; i++) {
  997.                         if (i < j) {
  998.                             gradientAtTrustRegionCenter.setEntry(j, gradientAtTrustRegionCenter.getEntry(j) + modelSecondDerivativesValues.getEntry(ih) * trialStepPoint.getEntry(i));
  999.                         }
  1000.                         gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + modelSecondDerivativesValues.getEntry(ih) * trialStepPoint.getEntry(j));
  1001.                         ih++;
  1002.                     }
  1003.                 }
  1004.                 for (int k = 0; k < npt; k++) {
  1005.                     double temp = ZERO;
  1006.                     for (int j = 0; j < n; j++) {
  1007.                         temp += interpolationPoints.getEntry(k, j) * trialStepPoint.getEntry(j);
  1008.                     }
  1009.                     temp *= modelSecondDerivativesParameters.getEntry(k);
  1010.                     for (int i = 0; i < n; i++) {
  1011.                         gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + temp * interpolationPoints.getEntry(k, i));
  1012.                     }
  1013.                 }
  1014.             }

  1016.             // Calculate the parameters of the least Frobenius norm interpolant to
  1017.             // the current data, the gradient of this interpolant at XOPT being put
  1018.             // into VLAG(NPT+I), I=1,2,...,N.

  1020.             if (ntrits > 0) {
  1021.                 for (int k = 0; k < npt; k++) {
  1022.                     lagrangeValuesAtNewPoint.setEntry(k, fAtInterpolationPoints.getEntry(k) - fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex));
  1023.                     work3.setEntry(k, ZERO);
  1024.                 }
  1025.                 for (int j = 0; j < nptm; j++) {
  1026.                     double sum = ZERO;
  1027.                     for (int k = 0; k < npt; k++) {
  1028.                         sum += zMatrix.getEntry(k, j) * lagrangeValuesAtNewPoint.getEntry(k);
  1029.                     }
  1030.                     for (int k = 0; k < npt; k++) {
  1031.                         work3.setEntry(k, work3.getEntry(k) + sum * zMatrix.getEntry(k, j));
  1032.                     }
  1033.                 }
  1034.                 for (int k = 0; k < npt; k++) {
  1035.                     double sum = ZERO;
  1036.                     for (int j = 0; j < n; j++) {
  1037.                         sum += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j);
  1038.                     }
  1039.                     work2.setEntry(k, work3.getEntry(k));
  1040.                     work3.setEntry(k, sum * work3.getEntry(k));
  1041.                 }
  1042.                 double gqsq = ZERO;
  1043.                 double gisq = ZERO;
  1044.                 for (int i = 0; i < n; i++) {
  1045.                     double sum = ZERO;
  1046.                     for (int k = 0; k < npt; k++) {
  1047.                         sum += bMatrix.getEntry(k, i) *
  1048.                             lagrangeValuesAtNewPoint.getEntry(k) + interpolationPoints.getEntry(k, i) * work3.getEntry(k);
  1049.                     }
  1050.                     if (trustRegionCenterOffset.getEntry(i) == lowerDifference.getEntry(i)) {
  1051.                         // Computing MIN
  1052.                         // Computing 2nd power
  1053.                         final double d1 = FastMath.min(ZERO, gradientAtTrustRegionCenter.getEntry(i));
  1054.                         gqsq += d1 * d1;
  1055.                         // Computing 2nd power
  1056.                         final double d2 = FastMath.min(ZERO, sum);
  1057.                         gisq += d2 * d2;
  1058.                     } else if (trustRegionCenterOffset.getEntry(i) == upperDifference.getEntry(i)) {
  1059.                         // Computing MAX
  1060.                         // Computing 2nd power
  1061.                         final double d1 = FastMath.max(ZERO, gradientAtTrustRegionCenter.getEntry(i));
  1062.                         gqsq += d1 * d1;
  1063.                         // Computing 2nd power
  1064.                         final double d2 = FastMath.max(ZERO, sum);
  1065.                         gisq += d2 * d2;
  1066.                     } else {
  1067.                         // Computing 2nd power
  1068.                         final double d1 = gradientAtTrustRegionCenter.getEntry(i);
  1069.                         gqsq += d1 * d1;
  1070.                         gisq += sum * sum;
  1071.                     }
  1072.                     lagrangeValuesAtNewPoint.setEntry(npt + i, sum);
  1073.                 }

  1075.                 // Test whether to replace the new quadratic model by the least Frobenius
  1076.                 // norm interpolant, making the replacement if the test is satisfied.

  1078.                 ++itest;
  1079.                 if (gqsq < TEN * gisq) {
  1080.                     itest = 0;
  1081.                 }
  1082.                 if (itest >= 3) {
  1083.                     for (int i = 0, max = FastMath.max(npt, nh); i < max; i++) {
  1084.                         if (i < n) {
  1085.                             gradientAtTrustRegionCenter.setEntry(i, lagrangeValuesAtNewPoint.getEntry(npt + i));
  1086.                         }
  1087.                         if (i < npt) {
  1088.                             modelSecondDerivativesParameters.setEntry(i, work2.getEntry(i));
  1089.                         }
  1090.                         if (i < nh) {
  1091.                             modelSecondDerivativesValues.setEntry(i, ZERO);
  1092.                         }
  1093.                         itest = 0;
  1094.                     }
  1095.                 }
  1096.             }

  1098.             // If a trust region step has provided a sufficient decrease in F, then
  1099.             // branch for another trust region calculation. The case NTRITS=0 occurs
  1100.             // when the new interpolation point was reached by an alternative step.

  1102.             if (ntrits == 0) {
  1103.                 state = 60; break;
  1104.             }
  1105.             if (f <= fopt + ONE_OVER_TEN * vquad) {
  1106.                 state = 60; break;
  1107.             }

  1109.             // Alternatively, find out if the interpolation points are close enough
  1110.             //   to the best point so far.

  1112.             // Computing MAX
  1113.             // Computing 2nd power
  1114.             final double d1 = TWO * delta;
  1115.             // Computing 2nd power
  1116.             final double d2 = TEN * rho;
  1117.             distsq = FastMath.max(d1 * d1, d2 * d2);
  1118.         }
  1119.         case 650: {
  1120.             printState(650); // XXX
  1121.             knew = -1;
  1122.             for (int k = 0; k < npt; k++) {
  1123.                 double sum = ZERO;
  1124.                 for (int j = 0; j < n; j++) {
  1125.                     // Computing 2nd power
  1126.                     final double d1 = interpolationPoints.getEntry(k, j) - trustRegionCenterOffset.getEntry(j);
  1127.                     sum += d1 * d1;
  1128.                 }
  1129.                 if (sum > distsq) {
  1130.                     knew = k;
  1131.                     distsq = sum;
  1132.                 }
  1133.             }

  1135.             // If KNEW is positive, then ALTMOV finds alternative new positions for
  1136.             // the KNEW-th interpolation point within distance ADELT of XOPT. It is
  1137.             // reached via label 90. Otherwise, there is a branch to label 60 for
  1138.             // another trust region iteration, unless the calculations with the
  1139.             // current RHO are complete.

  1141.             if (knew >= 0) {
  1142.                 final double dist = FastMath.sqrt(distsq);
  1143.                 if (ntrits == -1) {
  1144.                     // Computing MIN
  1145.                     delta = FastMath.min(ONE_OVER_TEN * delta, HALF * dist);
  1146.                     if (delta <= rho * 1.5) {
  1147.                         delta = rho;
  1148.                     }
  1149.                 }
  1150.                 ntrits = 0;
  1151.                 // Computing MAX
  1152.                 // Computing MIN
  1153.                 final double d1 = FastMath.min(ONE_OVER_TEN * dist, delta);
  1154.                 adelt = FastMath.max(d1, rho);
  1155.                 dsq = adelt * adelt;
  1156.                 state = 90; break;
  1157.             }
  1158.             if (ntrits == -1) {
  1159.                 state = 680; break;
  1160.             }
  1161.             if (ratio > ZERO) {
  1162.                 state = 60; break;
  1163.             }
  1164.             if (FastMath.max(delta, dnorm) > rho) {
  1165.                 state = 60; break;
  1166.             }

  1168.             // The calculations with the current value of RHO are complete. Pick the
  1169.             //   next values of RHO and DELTA.
  1170.         }
  1171.         case 680: {
  1172.             printState(680); // XXX
  1173.             if (rho > stoppingTrustRegionRadius) {
  1174.                 delta = HALF * rho;
  1175.                 ratio = rho / stoppingTrustRegionRadius;
  1176.                 if (ratio <= SIXTEEN) {
  1177.                     rho = stoppingTrustRegionRadius;
  1178.                 } else if (ratio <= TWO_HUNDRED_FIFTY) {
  1179.                     rho = FastMath.sqrt(ratio) * stoppingTrustRegionRadius;
  1180.                 } else {
  1181.                     rho *= ONE_OVER_TEN;
  1182.                 }
  1183.                 delta = FastMath.max(delta, rho);
  1184.                 ntrits = 0;
  1185.                 nfsav = getEvaluations();
  1186.                 state = 60; break;
  1187.             }

  1189.             // Return from the calculation, after another Newton-Raphson step, if
  1190.             //   it is too short to have been tried before.

  1192.             if (ntrits == -1) {
  1193.                 state = 360; break;
  1194.             }
  1195.         }
  1196.         case 720: {
  1197.             printState(720); // XXX
  1198.             if (fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex) <= fsave) {
  1199.                 for (int i = 0; i < n; i++) {
  1200.                     // Computing MIN
  1201.                     // Computing MAX
  1202.                     final double d3 = lowerBound[i];
  1203.                     final double d4 = originShift.getEntry(i) + trustRegionCenterOffset.getEntry(i);
  1204.                     final double d1 = FastMath.max(d3, d4);
  1205.                     final double d2 = upperBound[i];
  1206.                     currentBest.setEntry(i, FastMath.min(d1, d2));
  1207.                     if (trustRegionCenterOffset.getEntry(i) == lowerDifference.getEntry(i)) {
  1208.                         currentBest.setEntry(i, lowerBound[i]);
  1209.                     }
  1210.                     if (trustRegionCenterOffset.getEntry(i) == upperDifference.getEntry(i)) {
  1211.                         currentBest.setEntry(i, upperBound[i]);
  1212.                     }
  1213.                 }
  1214.                 f = fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex);
  1215.             }
  1216.             return f;
  1217.         }
  1218.         default: {
  1219.             throw new MathIllegalStateException(LocalizedFormats.SIMPLE_MESSAGE, "bobyqb");
  1220.         }}
  1221.     } // bobyqb

  1223.     // ----------------------------------------------------------------------------------------

  1225.     /**
  1226.      *     The arguments N, NPT, XPT, XOPT, BMAT, ZMAT, NDIM, SL and SU all have
  1227.      *       the same meanings as the corresponding arguments of BOBYQB.
  1228.      *     KOPT is the index of the optimal interpolation point.
  1229.      *     KNEW is the index of the interpolation point that is going to be moved.
  1230.      *     ADELT is the current trust region bound.
  1231.      *     XNEW will be set to a suitable new position for the interpolation point
  1232.      *       XPT(KNEW,.). Specifically, it satisfies the SL, SU and trust region
  1233.      *       bounds and it should provide a large denominator in the next call of
  1234.      *       UPDATE. The step XNEW-XOPT from XOPT is restricted to moves along the
  1235.      *       straight lines through XOPT and another interpolation point.
  1236.      *     XALT also provides a large value of the modulus of the KNEW-th Lagrange
  1237.      *       function subject to the constraints that have been mentioned, its main
  1238.      *       difference from XNEW being that XALT-XOPT is a constrained version of
  1239.      *       the Cauchy step within the trust region. An exception is that XALT is
  1240.      *       not calculated if all components of GLAG (see below) are zero.
  1241.      *     ALPHA will be set to the KNEW-th diagonal element of the H matrix.
  1242.      *     CAUCHY will be set to the square of the KNEW-th Lagrange function at
  1243.      *       the step XALT-XOPT from XOPT for the vector XALT that is returned,
  1244.      *       except that CAUCHY is set to zero if XALT is not calculated.
  1245.      *     GLAG is a working space vector of length N for the gradient of the
  1246.      *       KNEW-th Lagrange function at XOPT.
  1247.      *     HCOL is a working space vector of length NPT for the second derivative
  1248.      *       coefficients of the KNEW-th Lagrange function.
  1249.      *     W is a working space vector of length 2N that is going to hold the
  1250.      *       constrained Cauchy step from XOPT of the Lagrange function, followed
  1251.      *       by the downhill version of XALT when the uphill step is calculated.
  1252.      *
  1253.      *     Set the first NPT components of W to the leading elements of the
  1254.      *     KNEW-th column of the H matrix.
  1255.      * @param knew
  1256.      * @param adelt
  1257.      */
  1258.     private double[] altmov(
  1259.             int knew,
  1260.             double adelt
  1261.     ) {
  1262.         printMethod(); // XXX

  1264.         final int n = currentBest.getDimension();
  1265.         final int npt = numberOfInterpolationPoints;

  1267.         final ArrayRealVector glag = new ArrayRealVector(n);
  1268.         final ArrayRealVector hcol = new ArrayRealVector(npt);

  1270.         final ArrayRealVector work1 = new ArrayRealVector(n);
  1271.         final ArrayRealVector work2 = new ArrayRealVector(n);

  1273.         for (int k = 0; k < npt; k++) {
  1274.             hcol.setEntry(k, ZERO);
  1275.         }
  1276.         for (int j = 0, max = npt - n - 1; j < max; j++) {
  1277.             final double tmp = zMatrix.getEntry(knew, j);
  1278.             for (int k = 0; k < npt; k++) {
  1279.                 hcol.setEntry(k, hcol.getEntry(k) + tmp * zMatrix.getEntry(k, j));
  1280.             }
  1281.         }
  1282.         final double alpha = hcol.getEntry(knew);
  1283.         final double ha = HALF * alpha;

  1285.         // Calculate the gradient of the KNEW-th Lagrange function at XOPT.

  1287.         for (int i = 0; i < n; i++) {
  1288.             glag.setEntry(i, bMatrix.getEntry(knew, i));
  1289.         }
  1290.         for (int k = 0; k < npt; k++) {
  1291.             double tmp = ZERO;
  1292.             for (int j = 0; j < n; j++) {
  1293.                 tmp += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j);
  1294.             }
  1295.             tmp *= hcol.getEntry(k);
  1296.             for (int i = 0; i < n; i++) {
  1297.                 glag.setEntry(i, glag.getEntry(i) + tmp * interpolationPoints.getEntry(k, i));
  1298.             }
  1299.         }

  1301.         // Search for a large denominator along the straight lines through XOPT
  1302.         // and another interpolation point. SLBD and SUBD will be lower and upper
  1303.         // bounds on the step along each of these lines in turn. PREDSQ will be
  1304.         // set to the square of the predicted denominator for each line. PRESAV
  1305.         // will be set to the largest admissible value of PREDSQ that occurs.

  1307.         double presav = ZERO;
  1308.         double step = Double.NaN;
  1309.         int ksav = 0;
  1310.         int ibdsav = 0;
  1311.         double stpsav = 0;
  1312.         for (int k = 0; k < npt; k++) {
  1313.             if (k == trustRegionCenterInterpolationPointIndex) {
  1314.                 continue;
  1315.             }
  1316.             double dderiv = ZERO;
  1317.             double distsq = ZERO;
  1318.             for (int i = 0; i < n; i++) {
  1319.                 final double tmp = interpolationPoints.getEntry(k, i) - trustRegionCenterOffset.getEntry(i);
  1320.                 dderiv += glag.getEntry(i) * tmp;
  1321.                 distsq += tmp * tmp;
  1322.             }
  1323.             double subd = adelt / FastMath.sqrt(distsq);
  1324.             double slbd = -subd;
  1325.             int ilbd = 0;
  1326.             int iubd = 0;
  1327.             final double sumin = FastMath.min(ONE, subd);

  1329.             // Revise SLBD and SUBD if necessary because of the bounds in SL and SU.

  1331.             for (int i = 0; i < n; i++) {
  1332.                 final double tmp = interpolationPoints.getEntry(k, i) - trustRegionCenterOffset.getEntry(i);
  1333.                 if (tmp > ZERO) {
  1334.                     if (slbd * tmp < lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) {
  1335.                         slbd = (lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp;
  1336.                         ilbd = -i - 1;
  1337.                     }
  1338.                     if (subd * tmp > upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) {
  1339.                         // Computing MAX
  1340.                         subd = FastMath.max(sumin,
  1341.                                             (upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp);
  1342.                         iubd = i + 1;
  1343.                     }
  1344.                 } else if (tmp < ZERO) {
  1345.                     if (slbd * tmp > upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) {
  1346.                         slbd = (upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp;
  1347.                         ilbd = i + 1;
  1348.                     }
  1349.                     if (subd * tmp < lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) {
  1350.                         // Computing MAX
  1351.                         subd = FastMath.max(sumin,
  1352.                                             (lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp);
  1353.                         iubd = -i - 1;
  1354.                     }
  1355.                 }
  1356.             }

  1358.             // Seek a large modulus of the KNEW-th Lagrange function when the index
  1359.             // of the other interpolation point on the line through XOPT is KNEW.

  1361.             step = slbd;
  1362.             int isbd = ilbd;
  1363.             double vlag = Double.NaN;
  1364.             if (k == knew) {
  1365.                 final double diff = dderiv - ONE;
  1366.                 vlag = slbd * (dderiv - slbd * diff);
  1367.                 final double d1 = subd * (dderiv - subd * diff);
  1368.                 if (FastMath.abs(d1) > FastMath.abs(vlag)) {
  1369.                     step = subd;
  1370.                     vlag = d1;
  1371.                     isbd = iubd;
  1372.                 }
  1373.                 final double d2 = HALF * dderiv;
  1374.                 final double d3 = d2 - diff * slbd;
  1375.                 final double d4 = d2 - diff * subd;
  1376.                 if (d3 * d4 < ZERO) {
  1377.                     final double d5 = d2 * d2 / diff;
  1378.                     if (FastMath.abs(d5) > FastMath.abs(vlag)) {
  1379.                         step = d2 / diff;
  1380.                         vlag = d5;
  1381.                         isbd = 0;
  1382.                     }
  1383.                 }

  1385.                 // Search along each of the other lines through XOPT and another point.

  1387.             } else {
  1388.                 vlag = slbd * (ONE - slbd);
  1389.                 final double tmp = subd * (ONE - subd);
  1390.                 if (FastMath.abs(tmp) > FastMath.abs(vlag)) {
  1391.                     step = subd;
  1392.                     vlag = tmp;
  1393.                     isbd = iubd;
  1394.                 }
  1395.                 if (subd > HALF && FastMath.abs(vlag) < ONE_OVER_FOUR) {
  1396.                     step = HALF;
  1397.                     vlag = ONE_OVER_FOUR;
  1398.                     isbd = 0;
  1399.                 }
  1400.                 vlag *= dderiv;
  1401.             }

  1403.             // Calculate PREDSQ for the current line search and maintain PRESAV.

  1405.             final double tmp = step * (ONE - step) * distsq;
  1406.             final double predsq = vlag * vlag * (vlag * vlag + ha * tmp * tmp);
  1407.             if (predsq > presav) {
  1408.                 presav = predsq;
  1409.                 ksav = k;
  1410.                 stpsav = step;
  1411.                 ibdsav = isbd;
  1412.             }
  1413.         }

  1415.         // Construct XNEW in a way that satisfies the bound constraints exactly.

  1417.         for (int i = 0; i < n; i++) {
  1418.             final double tmp = trustRegionCenterOffset.getEntry(i) + stpsav * (interpolationPoints.getEntry(ksav, i) - trustRegionCenterOffset.getEntry(i));
  1419.             newPoint.setEntry(i, FastMath.max(lowerDifference.getEntry(i),
  1420.                                               FastMath.min(upperDifference.getEntry(i), tmp)));
  1421.         }
  1422.         if (ibdsav < 0) {
  1423.             newPoint.setEntry(-ibdsav - 1, lowerDifference.getEntry(-ibdsav - 1));
  1424.         }
  1425.         if (ibdsav > 0) {
  1426.             newPoint.setEntry(ibdsav - 1, upperDifference.getEntry(ibdsav - 1));
  1427.         }

  1429.         // Prepare for the iterative method that assembles the constrained Cauchy
  1430.         // step in W. The sum of squares of the fixed components of W is formed in
  1431.         // WFIXSQ, and the free components of W are set to BIGSTP.

  1433.         final double bigstp = adelt + adelt;
  1434.         int iflag = 0;
  1435.         double cauchy = Double.NaN;
  1436.         double csave = ZERO;
  1437.         while (true) {
  1438.             double wfixsq = ZERO;
  1439.             double ggfree = ZERO;
  1440.             for (int i = 0; i < n; i++) {
  1441.                 final double glagValue = glag.getEntry(i);
  1442.                 work1.setEntry(i, ZERO);
  1443.                 if (FastMath.min(trustRegionCenterOffset.getEntry(i) - lowerDifference.getEntry(i), glagValue) > ZERO ||
  1444.                     FastMath.max(trustRegionCenterOffset.getEntry(i) - upperDifference.getEntry(i), glagValue) < ZERO) {
  1445.                     work1.setEntry(i, bigstp);
  1446.                     // Computing 2nd power
  1447.                     ggfree += glagValue * glagValue;
  1448.                 }
  1449.             }
  1450.             if (ggfree == ZERO) {
  1451.                 return new double[] { alpha, ZERO };
  1452.             }

  1454.             // Investigate whether more components of W can be fixed.
  1455.             final double tmp1 = adelt * adelt - wfixsq;
  1456.             if (tmp1 > ZERO) {
  1457.                 step = FastMath.sqrt(tmp1 / ggfree);
  1458.                 ggfree = ZERO;
  1459.                 for (int i = 0; i < n; i++) {
  1460.                     if (work1.getEntry(i) == bigstp) {
  1461.                         final double tmp2 = trustRegionCenterOffset.getEntry(i) - step * glag.getEntry(i);
  1462.                         if (tmp2 <= lowerDifference.getEntry(i)) {
  1463.                             work1.setEntry(i, lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  1464.                             // Computing 2nd power
  1465.                             final double d1 = work1.getEntry(i);
  1466.                             wfixsq += d1 * d1;
  1467.                         } else if (tmp2 >= upperDifference.getEntry(i)) {
  1468.                             work1.setEntry(i, upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  1469.                             // Computing 2nd power
  1470.                             final double d1 = work1.getEntry(i);
  1471.                             wfixsq += d1 * d1;
  1472.                         } else {
  1473.                             // Computing 2nd power
  1474.                             final double d1 = glag.getEntry(i);
  1475.                             ggfree += d1 * d1;
  1476.                         }
  1477.                     }
  1478.                 }
  1479.             }

  1481.             // Set the remaining free components of W and all components of XALT,
  1482.             // except that W may be scaled later.

  1484.             double gw = ZERO;
  1485.             for (int i = 0; i < n; i++) {
  1486.                 final double glagValue = glag.getEntry(i);
  1487.                 if (work1.getEntry(i) == bigstp) {
  1488.                     work1.setEntry(i, -step * glagValue);
  1489.                     final double min = FastMath.min(upperDifference.getEntry(i),
  1490.                                                     trustRegionCenterOffset.getEntry(i) + work1.getEntry(i));
  1491.                     alternativeNewPoint.setEntry(i, FastMath.max(lowerDifference.getEntry(i), min));
  1492.                 } else if (work1.getEntry(i) == ZERO) {
  1493.                     alternativeNewPoint.setEntry(i, trustRegionCenterOffset.getEntry(i));
  1494.                 } else if (glagValue > ZERO) {
  1495.                     alternativeNewPoint.setEntry(i, lowerDifference.getEntry(i));
  1496.                 } else {
  1497.                     alternativeNewPoint.setEntry(i, upperDifference.getEntry(i));
  1498.                 }
  1499.                 gw += glagValue * work1.getEntry(i);
  1500.             }

  1502.             // Set CURV to the curvature of the KNEW-th Lagrange function along W.
  1503.             // Scale W by a factor less than one if that can reduce the modulus of
  1504.             // the Lagrange function at XOPT+W. Set CAUCHY to the final value of
  1505.             // the square of this function.

  1507.             double curv = ZERO;
  1508.             for (int k = 0; k < npt; k++) {
  1509.                 double tmp = ZERO;
  1510.                 for (int j = 0; j < n; j++) {
  1511.                     tmp += interpolationPoints.getEntry(k, j) * work1.getEntry(j);
  1512.                 }
  1513.                 curv += hcol.getEntry(k) * tmp * tmp;
  1514.             }
  1515.             if (iflag == 1) {
  1516.                 curv = -curv;
  1517.             }
  1518.             if (curv > -gw &&
  1519.                 curv < -gw * (ONE + FastMath.sqrt(TWO))) {
  1520.                 final double scale = -gw / curv;
  1521.                 for (int i = 0; i < n; i++) {
  1522.                     final double tmp = trustRegionCenterOffset.getEntry(i) + scale * work1.getEntry(i);
  1523.                     alternativeNewPoint.setEntry(i, FastMath.max(lowerDifference.getEntry(i),
  1524.                                                     FastMath.min(upperDifference.getEntry(i), tmp)));
  1525.                 }
  1526.                 // Computing 2nd power
  1527.                 final double d1 = HALF * gw * scale;
  1528.                 cauchy = d1 * d1;
  1529.             } else {
  1530.                 // Computing 2nd power
  1531.                 final double d1 = gw + HALF * curv;
  1532.                 cauchy = d1 * d1;
  1533.             }

  1535.             // If IFLAG is zero, then XALT is calculated as before after reversing
  1536.             // the sign of GLAG. Thus two XALT vectors become available. The one that
  1537.             // is chosen is the one that gives the larger value of CAUCHY.

  1539.             if (iflag == 0) {
  1540.                 for (int i = 0; i < n; i++) {
  1541.                     glag.setEntry(i, -glag.getEntry(i));
  1542.                     work2.setEntry(i, alternativeNewPoint.getEntry(i));
  1543.                 }
  1544.                 csave = cauchy;
  1545.                 iflag = 1;
  1546.             } else {
  1547.                 break;
  1548.             }
  1549.         }
  1550.         if (csave > cauchy) {
  1551.             for (int i = 0; i < n; i++) {
  1552.                 alternativeNewPoint.setEntry(i, work2.getEntry(i));
  1553.             }
  1554.             cauchy = csave;
  1555.         }

  1557.         return new double[] { alpha, cauchy };
  1558.     } // altmov

  1560.     // ----------------------------------------------------------------------------------------

  1562.     /**
  1563.      *     SUBROUTINE PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ,
  1564.      *     BMAT and ZMAT for the first iteration, and it maintains the values of
  1565.      *     NF and KOPT. The vector X is also changed by PRELIM.
  1566.      *
  1567.      *     The arguments N, NPT, X, XL, XU, RHOBEG, IPRINT and MAXFUN are the
  1568.      *       same as the corresponding arguments in SUBROUTINE BOBYQA.
  1569.      *     The arguments XBASE, XPT, FVAL, HQ, PQ, BMAT, ZMAT, NDIM, SL and SU
  1570.      *       are the same as the corresponding arguments in BOBYQB, the elements
  1571.      *       of SL and SU being set in BOBYQA.
  1572.      *     GOPT is usually the gradient of the quadratic model at XOPT+XBASE, but
  1573.      *       it is set by PRELIM to the gradient of the quadratic model at XBASE.
  1574.      *       If XOPT is nonzero, BOBYQB will change it to its usual value later.
  1575.      *     NF is maintaned as the number of calls of CALFUN so far.
  1576.      *     KOPT will be such that the least calculated value of F so far is at
  1577.      *       the point XPT(KOPT,.)+XBASE in the space of the variables.
  1578.      *
  1579.      * @param lowerBound Lower bounds.
  1580.      * @param upperBound Upper bounds.
  1581.      */
  1582.     private void prelim(double[] lowerBound,
  1583.                         double[] upperBound) {
  1584.         printMethod(); // XXX

  1586.         final int n = currentBest.getDimension();
  1587.         final int npt = numberOfInterpolationPoints;
  1588.         final int ndim = bMatrix.getRowDimension();

  1590.         final double rhosq = initialTrustRegionRadius * initialTrustRegionRadius;
  1591.         final double recip = 1d / rhosq;
  1592.         final int np = n + 1;

  1594.         // Set XBASE to the initial vector of variables, and set the initial
  1595.         // elements of XPT, BMAT, HQ, PQ and ZMAT to zero.

  1597.         for (int j = 0; j < n; j++) {
  1598.             originShift.setEntry(j, currentBest.getEntry(j));
  1599.             for (int k = 0; k < npt; k++) {
  1600.                 interpolationPoints.setEntry(k, j, ZERO);
  1601.             }
  1602.             for (int i = 0; i < ndim; i++) {
  1603.                 bMatrix.setEntry(i, j, ZERO);
  1604.             }
  1605.         }
  1606.         for (int i = 0, max = n * np / 2; i < max; i++) {
  1607.             modelSecondDerivativesValues.setEntry(i, ZERO);
  1608.         }
  1609.         for (int k = 0; k < npt; k++) {
  1610.             modelSecondDerivativesParameters.setEntry(k, ZERO);
  1611.             for (int j = 0, max = npt - np; j < max; j++) {
  1612.                 zMatrix.setEntry(k, j, ZERO);
  1613.             }
  1614.         }

  1616.         // Begin the initialization procedure. NF becomes one more than the number
  1617.         // of function values so far. The coordinates of the displacement of the
  1618.         // next initial interpolation point from XBASE are set in XPT(NF+1,.).

  1620.         int ipt = 0;
  1621.         int jpt = 0;
  1622.         double fbeg = Double.NaN;
  1623.         do {
  1624.             final int nfm = getEvaluations();
  1625.             final int nfx = nfm - n;
  1626.             final int nfmm = nfm - 1;
  1627.             final int nfxm = nfx - 1;
  1628.             double stepa = 0;
  1629.             double stepb = 0;
  1630.             if (nfm <= 2 * n) {
  1631.                 if (nfm >= 1 &&
  1632.                     nfm <= n) {
  1633.                     stepa = initialTrustRegionRadius;
  1634.                     if (upperDifference.getEntry(nfmm) == ZERO) {
  1635.                         stepa = -stepa;
  1636.                         // throw new PathIsExploredException(); // XXX
  1637.                     }
  1638.                     interpolationPoints.setEntry(nfm, nfmm, stepa);
  1639.                 } else if (nfm > n) {
  1640.                     stepa = interpolationPoints.getEntry(nfx, nfxm);
  1641.                     stepb = -initialTrustRegionRadius;
  1642.                     if (lowerDifference.getEntry(nfxm) == ZERO) {
  1643.                         stepb = FastMath.min(TWO * initialTrustRegionRadius, upperDifference.getEntry(nfxm));
  1644.                         // throw new PathIsExploredException(); // XXX
  1645.                     }
  1646.                     if (upperDifference.getEntry(nfxm) == ZERO) {
  1647.                         stepb = FastMath.max(-TWO * initialTrustRegionRadius, lowerDifference.getEntry(nfxm));
  1648.                         // throw new PathIsExploredException(); // XXX
  1649.                     }
  1650.                     interpolationPoints.setEntry(nfm, nfxm, stepb);
  1651.                 }
  1652.             } else {
  1653.                 final int tmp1 = (nfm - np) / n;
  1654.                 jpt = nfm - tmp1 * n - n;
  1655.                 ipt = jpt + tmp1;
  1656.                 if (ipt > n) {
  1657.                     final int tmp2 = jpt;
  1658.                     jpt = ipt - n;
  1659.                     ipt = tmp2;
  1660. //                     throw new PathIsExploredException(); // XXX
  1661.                 }
  1662.                 final int iptMinus1 = ipt - 1;
  1663.                 final int jptMinus1 = jpt - 1;
  1664.                 interpolationPoints.setEntry(nfm, iptMinus1, interpolationPoints.getEntry(ipt, iptMinus1));
  1665.                 interpolationPoints.setEntry(nfm, jptMinus1, interpolationPoints.getEntry(jpt, jptMinus1));
  1666.             }

  1668.             // Calculate the next value of F. The least function value so far and
  1669.             // its index are required.

  1671.             for (int j = 0; j < n; j++) {
  1672.                 currentBest.setEntry(j, FastMath.min(FastMath.max(lowerBound[j],
  1673.                                                                   originShift.getEntry(j) + interpolationPoints.getEntry(nfm, j)),
  1674.                                                      upperBound[j]));
  1675.                 if (interpolationPoints.getEntry(nfm, j) == lowerDifference.getEntry(j)) {
  1676.                     currentBest.setEntry(j, lowerBound[j]);
  1677.                 }
  1678.                 if (interpolationPoints.getEntry(nfm, j) == upperDifference.getEntry(j)) {
  1679.                     currentBest.setEntry(j, upperBound[j]);
  1680.                 }
  1681.             }

  1683.             final double objectiveValue = computeObjectiveValue(currentBest.toArray());
  1684.             final double f = isMinimize ? objectiveValue : -objectiveValue;
  1685.             final int numEval = getEvaluations(); // nfm + 1
  1686.             fAtInterpolationPoints.setEntry(nfm, f);

  1688.             if (numEval == 1) {
  1689.                 fbeg = f;
  1690.                 trustRegionCenterInterpolationPointIndex = 0;
  1691.             } else if (f < fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex)) {
  1692.                 trustRegionCenterInterpolationPointIndex = nfm;
  1693.             }

  1695.             // Set the nonzero initial elements of BMAT and the quadratic model in the
  1696.             // cases when NF is at most 2*N+1. If NF exceeds N+1, then the positions
  1697.             // of the NF-th and (NF-N)-th interpolation points may be switched, in
  1698.             // order that the function value at the first of them contributes to the
  1699.             // off-diagonal second derivative terms of the initial quadratic model.

  1701.             if (numEval <= 2 * n + 1) {
  1702.                 if (numEval >= 2 &&
  1703.                     numEval <= n + 1) {
  1704.                     gradientAtTrustRegionCenter.setEntry(nfmm, (f - fbeg) / stepa);
  1705.                     if (npt < numEval + n) {
  1706.                         final double oneOverStepA = ONE / stepa;
  1707.                         bMatrix.setEntry(0, nfmm, -oneOverStepA);
  1708.                         bMatrix.setEntry(nfm, nfmm, oneOverStepA);
  1709.                         bMatrix.setEntry(npt + nfmm, nfmm, -HALF * rhosq);
  1710.                         // throw new PathIsExploredException(); // XXX
  1711.                     }
  1712.                 } else if (numEval >= n + 2) {
  1713.                     final int ih = nfx * (nfx + 1) / 2 - 1;
  1714.                     final double tmp = (f - fbeg) / stepb;
  1715.                     final double diff = stepb - stepa;
  1716.                     modelSecondDerivativesValues.setEntry(ih, TWO * (tmp - gradientAtTrustRegionCenter.getEntry(nfxm)) / diff);
  1717.                     gradientAtTrustRegionCenter.setEntry(nfxm, (gradientAtTrustRegionCenter.getEntry(nfxm) * stepb - tmp * stepa) / diff);
  1718.                     if (stepa * stepb < ZERO && f < fAtInterpolationPoints.getEntry(nfm - n)) {
  1719.                         fAtInterpolationPoints.setEntry(nfm, fAtInterpolationPoints.getEntry(nfm - n));
  1720.                         fAtInterpolationPoints.setEntry(nfm - n, f);
  1721.                         if (trustRegionCenterInterpolationPointIndex == nfm) {
  1722.                             trustRegionCenterInterpolationPointIndex = nfm - n;
  1723.                         }
  1724.                         interpolationPoints.setEntry(nfm - n, nfxm, stepb);
  1725.                         interpolationPoints.setEntry(nfm, nfxm, stepa);
  1726.                     }
  1727.                     bMatrix.setEntry(0, nfxm, -(stepa + stepb) / (stepa * stepb));
  1728.                     bMatrix.setEntry(nfm, nfxm, -HALF / interpolationPoints.getEntry(nfm - n, nfxm));
  1729.                     bMatrix.setEntry(nfm - n, nfxm,
  1730.                                   -bMatrix.getEntry(0, nfxm) - bMatrix.getEntry(nfm, nfxm));
  1731.                     zMatrix.setEntry(0, nfxm, FastMath.sqrt(TWO) / (stepa * stepb));
  1732.                     zMatrix.setEntry(nfm, nfxm, FastMath.sqrt(HALF) / rhosq);
  1733.                     // zMatrix.setEntry(nfm, nfxm, FastMath.sqrt(HALF) * recip); // XXX "testAckley" and "testDiffPow" fail.
  1734.                     zMatrix.setEntry(nfm - n, nfxm,
  1735.                                   -zMatrix.getEntry(0, nfxm) - zMatrix.getEntry(nfm, nfxm));
  1736.                 }

  1738.                 // Set the off-diagonal second derivatives of the Lagrange functions and
  1739.                 // the initial quadratic model.

  1741.             } else {
  1742.                 zMatrix.setEntry(0, nfxm, recip);
  1743.                 zMatrix.setEntry(nfm, nfxm, recip);
  1744.                 zMatrix.setEntry(ipt, nfxm, -recip);
  1745.                 zMatrix.setEntry(jpt, nfxm, -recip);

  1747.                 final int ih = ipt * (ipt - 1) / 2 + jpt - 1;
  1748.                 final double tmp = interpolationPoints.getEntry(nfm, ipt - 1) * interpolationPoints.getEntry(nfm, jpt - 1);
  1749.                 modelSecondDerivativesValues.setEntry(ih, (fbeg - fAtInterpolationPoints.getEntry(ipt) - fAtInterpolationPoints.getEntry(jpt) + f) / tmp);
  1750. //                 throw new PathIsExploredException(); // XXX
  1751.             }
  1752.         } while (getEvaluations() < npt);
  1753.     } // prelim


  1756.     // ----------------------------------------------------------------------------------------

  1758.     /**
  1759.      *     A version of the truncated conjugate gradient is applied. If a line
  1760.      *     search is restricted by a constraint, then the procedure is restarted,
  1761.      *     the values of the variables that are at their bounds being fixed. If
  1762.      *     the trust region boundary is reached, then further changes may be made
  1763.      *     to D, each one being in the two dimensional space that is spanned
  1764.      *     by the current D and the gradient of Q at XOPT+D, staying on the trust
  1765.      *     region boundary. Termination occurs when the reduction in Q seems to
  1766.      *     be close to the greatest reduction that can be achieved.
  1767.      *     The arguments N, NPT, XPT, XOPT, GOPT, HQ, PQ, SL and SU have the same
  1768.      *       meanings as the corresponding arguments of BOBYQB.
  1769.      *     DELTA is the trust region radius for the present calculation, which
  1770.      *       seeks a small value of the quadratic model within distance DELTA of
  1771.      *       XOPT subject to the bounds on the variables.
  1772.      *     XNEW will be set to a new vector of variables that is approximately
  1773.      *       the one that minimizes the quadratic model within the trust region
  1774.      *       subject to the SL and SU constraints on the variables. It satisfies
  1775.      *       as equations the bounds that become active during the calculation.
  1776.      *     D is the calculated trial step from XOPT, generated iteratively from an
  1777.      *       initial value of zero. Thus XNEW is XOPT+D after the final iteration.
  1778.      *     GNEW holds the gradient of the quadratic model at XOPT+D. It is updated
  1779.      *       when D is updated.
  1780.      *     xbdi.get( is a working space vector. For I=1,2,...,N, the element xbdi.get((I) is
  1781.      *       set to -1.0, 0.0, or 1.0, the value being nonzero if and only if the
  1782.      *       I-th variable has become fixed at a bound, the bound being SL(I) or
  1783.      *       SU(I) in the case xbdi.get((I)=-1.0 or xbdi.get((I)=1.0, respectively. This
  1784.      *       information is accumulated during the construction of XNEW.
  1785.      *     The arrays S, HS and HRED are also used for working space. They hold the
  1786.      *       current search direction, and the changes in the gradient of Q along S
  1787.      *       and the reduced D, respectively, where the reduced D is the same as D,
  1788.      *       except that the components of the fixed variables are zero.
  1789.      *     DSQ will be set to the square of the length of XNEW-XOPT.
  1790.      *     CRVMIN is set to zero if D reaches the trust region boundary. Otherwise
  1791.      *       it is set to the least curvature of H that occurs in the conjugate
  1792.      *       gradient searches that are not restricted by any constraints. The
  1793.      *       value CRVMIN=-1.0D0 is set, however, if all of these searches are
  1794.      *       constrained.
  1795.      * @param delta
  1796.      * @param gnew
  1797.      * @param xbdi
  1798.      * @param s
  1799.      * @param hs
  1800.      * @param hred
  1801.      */
  1802.     private double[] trsbox(
  1803.             double delta,
  1804.             ArrayRealVector gnew,
  1805.             ArrayRealVector xbdi,
  1806.             ArrayRealVector s,
  1807.             ArrayRealVector hs,
  1808.             ArrayRealVector hred
  1809.     ) {
  1810.         printMethod(); // XXX

  1812.         final int n = currentBest.getDimension();
  1813.         final int npt = numberOfInterpolationPoints;

  1815.         double dsq = Double.NaN;
  1816.         double crvmin = Double.NaN;

  1818.         // Local variables
  1819.         double ds;
  1820.         int iu;
  1821.         double dhd, dhs, cth, shs, sth, ssq, beta=0, sdec, blen;
  1822.         int iact = -1;
  1823.         int nact = 0;
  1824.         double angt = 0, qred;
  1825.         int isav;
  1826.         double temp = 0, xsav = 0, xsum = 0, angbd = 0, dredg = 0, sredg = 0;
  1827.         int iterc;
  1828.         double resid = 0, delsq = 0, ggsav = 0, tempa = 0, tempb = 0,
  1829.         redmax = 0, dredsq = 0, redsav = 0, gredsq = 0, rednew = 0;
  1830.         int itcsav = 0;
  1831.         double rdprev = 0, rdnext = 0, stplen = 0, stepsq = 0;
  1832.         int itermax = 0;

  1834.         // Set some constants.

  1836.         // Function Body

  1838.         // The sign of GOPT(I) gives the sign of the change to the I-th variable
  1839.         // that will reduce Q from its value at XOPT. Thus xbdi.get((I) shows whether
  1840.         // or not to fix the I-th variable at one of its bounds initially, with
  1841.         // NACT being set to the number of fixed variables. D and GNEW are also
  1842.         // set for the first iteration. DELSQ is the upper bound on the sum of
  1843.         // squares of the free variables. QRED is the reduction in Q so far.

  1845.         iterc = 0;
  1846.         nact = 0;
  1847.         for (int i = 0; i < n; i++) {
  1848.             xbdi.setEntry(i, ZERO);
  1849.             if (trustRegionCenterOffset.getEntry(i) <= lowerDifference.getEntry(i)) {
  1850.                 if (gradientAtTrustRegionCenter.getEntry(i) >= ZERO) {
  1851.                     xbdi.setEntry(i, MINUS_ONE);
  1852.                 }
  1853.             } else if (trustRegionCenterOffset.getEntry(i) >= upperDifference.getEntry(i) &&
  1854.                        gradientAtTrustRegionCenter.getEntry(i) <= ZERO) {
  1855.                 xbdi.setEntry(i, ONE);
  1856.             }
  1857.             if (xbdi.getEntry(i) != ZERO) {
  1858.                 ++nact;
  1859.             }
  1860.             trialStepPoint.setEntry(i, ZERO);
  1861.             gnew.setEntry(i, gradientAtTrustRegionCenter.getEntry(i));
  1862.         }
  1863.         delsq = delta * delta;
  1864.         qred = ZERO;
  1865.         crvmin = MINUS_ONE;

  1867.         // Set the next search direction of the conjugate gradient method. It is
  1868.         // the steepest descent direction initially and when the iterations are
  1869.         // restarted because a variable has just been fixed by a bound, and of
  1870.         // course the components of the fixed variables are zero. ITERMAX is an
  1871.         // upper bound on the indices of the conjugate gradient iterations.

  1873.         int state = 20;
  1874.         for(;;) {
  1875.             switch (state) {
  1876.         case 20: {
  1877.             printState(20); // XXX
  1878.             beta = ZERO;
  1879.         }
  1880.         case 30: {
  1881.             printState(30); // XXX
  1882.             stepsq = ZERO;
  1883.             for (int i = 0; i < n; i++) {
  1884.                 if (xbdi.getEntry(i) != ZERO) {
  1885.                     s.setEntry(i, ZERO);
  1886.                 } else if (beta == ZERO) {
  1887.                     s.setEntry(i, -gnew.getEntry(i));
  1888.                 } else {
  1889.                     s.setEntry(i, beta * s.getEntry(i) - gnew.getEntry(i));
  1890.                 }
  1891.                 // Computing 2nd power
  1892.                 final double d1 = s.getEntry(i);
  1893.                 stepsq += d1 * d1;
  1894.             }
  1895.             if (stepsq == ZERO) {
  1896.                 state = 190; break;
  1897.             }
  1898.             if (beta == ZERO) {
  1899.                 gredsq = stepsq;
  1900.                 itermax = iterc + n - nact;
  1901.             }
  1902.             if (gredsq * delsq <= qred * 1e-4 * qred) {
  1903.                 state = 190; break;
  1904.             }

  1906.             // Multiply the search direction by the second derivative matrix of Q and
  1907.             // calculate some scalars for the choice of steplength. Then set BLEN to
  1908.             // the length of the the step to the trust region boundary and STPLEN to
  1909.             // the steplength, ignoring the simple bounds.

  1911.             state = 210; break;
  1912.         }
  1913.         case 50: {
  1914.             printState(50); // XXX
  1915.             resid = delsq;
  1916.             ds = ZERO;
  1917.             shs = ZERO;
  1918.             for (int i = 0; i < n; i++) {
  1919.                 if (xbdi.getEntry(i) == ZERO) {
  1920.                     // Computing 2nd power
  1921.                     final double d1 = trialStepPoint.getEntry(i);
  1922.                     resid -= d1 * d1;
  1923.                     ds += s.getEntry(i) * trialStepPoint.getEntry(i);
  1924.                     shs += s.getEntry(i) * hs.getEntry(i);
  1925.                 }
  1926.             }
  1927.             if (resid <= ZERO) {
  1928.                 state = 90; break;
  1929.             }
  1930.             temp = FastMath.sqrt(stepsq * resid + ds * ds);
  1931.             if (ds < ZERO) {
  1932.                 blen = (temp - ds) / stepsq;
  1933.             } else {
  1934.                 blen = resid / (temp + ds);
  1935.             }
  1936.             stplen = blen;
  1937.             if (shs > ZERO) {
  1938.                 // Computing MIN
  1939.                 stplen = FastMath.min(blen, gredsq / shs);
  1940.             }

  1942.             // Reduce STPLEN if necessary in order to preserve the simple bounds,
  1943.             // letting IACT be the index of the new constrained variable.

  1945.             iact = -1;
  1946.             for (int i = 0; i < n; i++) {
  1947.                 if (s.getEntry(i) != ZERO) {
  1948.                     xsum = trustRegionCenterOffset.getEntry(i) + trialStepPoint.getEntry(i);
  1949.                     if (s.getEntry(i) > ZERO) {
  1950.                         temp = (upperDifference.getEntry(i) - xsum) / s.getEntry(i);
  1951.                     } else {
  1952.                         temp = (lowerDifference.getEntry(i) - xsum) / s.getEntry(i);
  1953.                     }
  1954.                     if (temp < stplen) {
  1955.                         stplen = temp;
  1956.                         iact = i;
  1957.                     }
  1958.                 }
  1959.             }

  1961.             // Update CRVMIN, GNEW and D. Set SDEC to the decrease that occurs in Q.

  1963.             sdec = ZERO;
  1964.             if (stplen > ZERO) {
  1965.                 ++iterc;
  1966.                 temp = shs / stepsq;
  1967.                 if (iact == -1 && temp > ZERO) {
  1968.                     crvmin = FastMath.min(crvmin,temp);
  1969.                     if (crvmin == MINUS_ONE) {
  1970.                         crvmin = temp;
  1971.                     }
  1972.                 }
  1973.                 ggsav = gredsq;
  1974.                 gredsq = ZERO;
  1975.                 for (int i = 0; i < n; i++) {
  1976.                     gnew.setEntry(i, gnew.getEntry(i) + stplen * hs.getEntry(i));
  1977.                     if (xbdi.getEntry(i) == ZERO) {
  1978.                         // Computing 2nd power
  1979.                         final double d1 = gnew.getEntry(i);
  1980.                         gredsq += d1 * d1;
  1981.                     }
  1982.                     trialStepPoint.setEntry(i, trialStepPoint.getEntry(i) + stplen * s.getEntry(i));
  1983.                 }
  1984.                 // Computing MAX
  1985.                 final double d1 = stplen * (ggsav - HALF * stplen * shs);
  1986.                 sdec = FastMath.max(d1, ZERO);
  1987.                 qred += sdec;
  1988.             }

  1990.             // Restart the conjugate gradient method if it has hit a new bound.

  1992.             if (iact >= 0) {
  1993.                 ++nact;
  1994.                 xbdi.setEntry(iact, ONE);
  1995.                 if (s.getEntry(iact) < ZERO) {
  1996.                     xbdi.setEntry(iact, MINUS_ONE);
  1997.                 }
  1998.                 // Computing 2nd power
  1999.                 final double d1 = trialStepPoint.getEntry(iact);
  2000.                 delsq -= d1 * d1;
  2001.                 if (delsq <= ZERO) {
  2002.                     state = 190; break;
  2003.                 }
  2004.                 state = 20; break;
  2005.             }

  2007.             // If STPLEN is less than BLEN, then either apply another conjugate
  2008.             // gradient iteration or RETURN.

  2010.             if (stplen < blen) {
  2011.                 if (iterc == itermax) {
  2012.                     state = 190; break;
  2013.                 }
  2014.                 if (sdec <= qred * .01) {
  2015.                     state = 190; break;
  2016.                 }
  2017.                 beta = gredsq / ggsav;
  2018.                 state = 30; break;
  2019.             }
  2020.         }
  2021.         case 90: {
  2022.             printState(90); // XXX
  2023.             crvmin = ZERO;

  2025.             // Prepare for the alternative iteration by calculating some scalars
  2026.             // and by multiplying the reduced D by the second derivative matrix of
  2027.             // Q, where S holds the reduced D in the call of GGMULT.

  2029.         }
  2030.         case 100: {
  2031.             printState(100); // XXX
  2032.             if (nact >= n - 1) {
  2033.                 state = 190; break;
  2034.             }
  2035.             dredsq = ZERO;
  2036.             dredg = ZERO;
  2037.             gredsq = ZERO;
  2038.             for (int i = 0; i < n; i++) {
  2039.                 if (xbdi.getEntry(i) == ZERO) {
  2040.                     // Computing 2nd power
  2041.                     double d1 = trialStepPoint.getEntry(i);
  2042.                     dredsq += d1 * d1;
  2043.                     dredg += trialStepPoint.getEntry(i) * gnew.getEntry(i);
  2044.                     // Computing 2nd power
  2045.                     d1 = gnew.getEntry(i);
  2046.                     gredsq += d1 * d1;
  2047.                     s.setEntry(i, trialStepPoint.getEntry(i));
  2048.                 } else {
  2049.                     s.setEntry(i, ZERO);
  2050.                 }
  2051.             }
  2052.             itcsav = iterc;
  2053.             state = 210; break;
  2054.             // Let the search direction S be a linear combination of the reduced D
  2055.             // and the reduced G that is orthogonal to the reduced D.
  2056.         }
  2057.         case 120: {
  2058.             printState(120); // XXX
  2059.             ++iterc;
  2060.             temp = gredsq * dredsq - dredg * dredg;
  2061.             if (temp <= qred * 1e-4 * qred) {
  2062.                 state = 190; break;
  2063.             }
  2064.             temp = FastMath.sqrt(temp);
  2065.             for (int i = 0; i < n; i++) {
  2066.                 if (xbdi.getEntry(i) == ZERO) {
  2067.                     s.setEntry(i, (dredg * trialStepPoint.getEntry(i) - dredsq * gnew.getEntry(i)) / temp);
  2068.                 } else {
  2069.                     s.setEntry(i, ZERO);
  2070.                 }
  2071.             }
  2072.             sredg = -temp;

  2074.             // By considering the simple bounds on the variables, calculate an upper
  2075.             // bound on the tangent of half the angle of the alternative iteration,
  2076.             // namely ANGBD, except that, if already a free variable has reached a
  2077.             // bound, there is a branch back to label 100 after fixing that variable.

  2079.             angbd = ONE;
  2080.             iact = -1;
  2081.             for (int i = 0; i < n; i++) {
  2082.                 if (xbdi.getEntry(i) == ZERO) {
  2083.                     tempa = trustRegionCenterOffset.getEntry(i) + trialStepPoint.getEntry(i) - lowerDifference.getEntry(i);
  2084.                     tempb = upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i) - trialStepPoint.getEntry(i);
  2085.                     if (tempa <= ZERO) {
  2086.                         ++nact;
  2087.                         xbdi.setEntry(i, MINUS_ONE);
  2088.                         state = 100; break;
  2089.                     } else if (tempb <= ZERO) {
  2090.                         ++nact;
  2091.                         xbdi.setEntry(i, ONE);
  2092.                         state = 100; break;
  2093.                     }
  2094.                     // Computing 2nd power
  2095.                     double d1 = trialStepPoint.getEntry(i);
  2096.                     // Computing 2nd power
  2097.                     double d2 = s.getEntry(i);
  2098.                     ssq = d1 * d1 + d2 * d2;
  2099.                     // Computing 2nd power
  2100.                     d1 = trustRegionCenterOffset.getEntry(i) - lowerDifference.getEntry(i);
  2101.                     temp = ssq - d1 * d1;
  2102.                     if (temp > ZERO) {
  2103.                         temp = FastMath.sqrt(temp) - s.getEntry(i);
  2104.                         if (angbd * temp > tempa) {
  2105.                             angbd = tempa / temp;
  2106.                             iact = i;
  2107.                             xsav = MINUS_ONE;
  2108.                         }
  2109.                     }
  2110.                     // Computing 2nd power
  2111.                     d1 = upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i);
  2112.                     temp = ssq - d1 * d1;
  2113.                     if (temp > ZERO) {
  2114.                         temp = FastMath.sqrt(temp) + s.getEntry(i);
  2115.                         if (angbd * temp > tempb) {
  2116.                             angbd = tempb / temp;
  2117.                             iact = i;
  2118.                             xsav = ONE;
  2119.                         }
  2120.                     }
  2121.                 }
  2122.             }

  2124.             // Calculate HHD and some curvatures for the alternative iteration.

  2126.             state = 210; break;
  2127.         }
  2128.         case 150: {
  2129.             printState(150); // XXX
  2130.             shs = ZERO;
  2131.             dhs = ZERO;
  2132.             dhd = ZERO;
  2133.             for (int i = 0; i < n; i++) {
  2134.                 if (xbdi.getEntry(i) == ZERO) {
  2135.                     shs += s.getEntry(i) * hs.getEntry(i);
  2136.                     dhs += trialStepPoint.getEntry(i) * hs.getEntry(i);
  2137.                     dhd += trialStepPoint.getEntry(i) * hred.getEntry(i);
  2138.                 }
  2139.             }

  2141.             // Seek the greatest reduction in Q for a range of equally spaced values
  2142.             // of ANGT in [0,ANGBD], where ANGT is the tangent of half the angle of
  2143.             // the alternative iteration.

  2145.             redmax = ZERO;
  2146.             isav = -1;
  2147.             redsav = ZERO;
  2148.             iu = (int) (angbd * 17. + 3.1);
  2149.             for (int i = 0; i < iu; i++) {
  2150.                 angt = angbd * i / iu;
  2151.                 sth = (angt + angt) / (ONE + angt * angt);
  2152.                 temp = shs + angt * (angt * dhd - dhs - dhs);
  2153.                 rednew = sth * (angt * dredg - sredg - HALF * sth * temp);
  2154.                 if (rednew > redmax) {
  2155.                     redmax = rednew;
  2156.                     isav = i;
  2157.                     rdprev = redsav;
  2158.                 } else if (i == isav + 1) {
  2159.                     rdnext = rednew;
  2160.                 }
  2161.                 redsav = rednew;
  2162.             }

  2164.             // Return if the reduction is zero. Otherwise, set the sine and cosine
  2165.             // of the angle of the alternative iteration, and calculate SDEC.

  2167.             if (isav < 0) {
  2168.                 state = 190; break;
  2169.             }
  2170.             if (isav < iu) {
  2171.                 temp = (rdnext - rdprev) / (redmax + redmax - rdprev - rdnext);
  2172.                 angt = angbd * (isav + HALF * temp) / iu;
  2173.             }
  2174.             cth = (ONE - angt * angt) / (ONE + angt * angt);
  2175.             sth = (angt + angt) / (ONE + angt * angt);
  2176.             temp = shs + angt * (angt * dhd - dhs - dhs);
  2177.             sdec = sth * (angt * dredg - sredg - HALF * sth * temp);
  2178.             if (sdec <= ZERO) {
  2179.                 state = 190; break;
  2180.             }

  2182.             // Update GNEW, D and HRED. If the angle of the alternative iteration
  2183.             // is restricted by a bound on a free variable, that variable is fixed
  2184.             // at the bound.

  2186.             dredg = ZERO;
  2187.             gredsq = ZERO;
  2188.             for (int i = 0; i < n; i++) {
  2189.                 gnew.setEntry(i, gnew.getEntry(i) + (cth - ONE) * hred.getEntry(i) + sth * hs.getEntry(i));
  2190.                 if (xbdi.getEntry(i) == ZERO) {
  2191.                     trialStepPoint.setEntry(i, cth * trialStepPoint.getEntry(i) + sth * s.getEntry(i));
  2192.                     dredg += trialStepPoint.getEntry(i) * gnew.getEntry(i);
  2193.                     // Computing 2nd power
  2194.                     final double d1 = gnew.getEntry(i);
  2195.                     gredsq += d1 * d1;
  2196.                 }
  2197.                 hred.setEntry(i, cth * hred.getEntry(i) + sth * hs.getEntry(i));
  2198.             }
  2199.             qred += sdec;
  2200.             if (iact >= 0 && isav == iu) {
  2201.                 ++nact;
  2202.                 xbdi.setEntry(iact, xsav);
  2203.                 state = 100; break;
  2204.             }

  2206.             // If SDEC is sufficiently small, then RETURN after setting XNEW to
  2207.             // XOPT+D, giving careful attention to the bounds.

  2209.             if (sdec > qred * .01) {
  2210.                 state = 120; break;
  2211.             }
  2212.         }
  2213.         case 190: {
  2214.             printState(190); // XXX
  2215.             dsq = ZERO;
  2216.             for (int i = 0; i < n; i++) {
  2217.                 // Computing MAX
  2218.                 // Computing MIN
  2219.                 final double min = FastMath.min(trustRegionCenterOffset.getEntry(i) + trialStepPoint.getEntry(i),
  2220.                                                 upperDifference.getEntry(i));
  2221.                 newPoint.setEntry(i, FastMath.max(min, lowerDifference.getEntry(i)));
  2222.                 if (xbdi.getEntry(i) == MINUS_ONE) {
  2223.                     newPoint.setEntry(i, lowerDifference.getEntry(i));
  2224.                 }
  2225.                 if (xbdi.getEntry(i) == ONE) {
  2226.                     newPoint.setEntry(i, upperDifference.getEntry(i));
  2227.                 }
  2228.                 trialStepPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i));
  2229.                 // Computing 2nd power
  2230.                 final double d1 = trialStepPoint.getEntry(i);
  2231.                 dsq += d1 * d1;
  2232.             }
  2233.             return new double[] { dsq, crvmin };
  2234.             // The following instructions multiply the current S-vector by the second
  2235.             // derivative matrix of the quadratic model, putting the product in HS.
  2236.             // They are reached from three different parts of the software above and
  2237.             // they can be regarded as an external subroutine.
  2238.         }
  2239.         case 210: {
  2240.             printState(210); // XXX
  2241.             int ih = 0;
  2242.             for (int j = 0; j < n; j++) {
  2243.                 hs.setEntry(j, ZERO);
  2244.                 for (int i = 0; i <= j; i++) {
  2245.                     if (i < j) {
  2246.                         hs.setEntry(j, hs.getEntry(j) + modelSecondDerivativesValues.getEntry(ih) * s.getEntry(i));
  2247.                     }
  2248.                     hs.setEntry(i, hs.getEntry(i) + modelSecondDerivativesValues.getEntry(ih) * s.getEntry(j));
  2249.                     ih++;
  2250.                 }
  2251.             }
  2252.             final RealVector tmp = interpolationPoints.operate(s).ebeMultiply(modelSecondDerivativesParameters);
  2253.             for (int k = 0; k < npt; k++) {
  2254.                 if (modelSecondDerivativesParameters.getEntry(k) != ZERO) {
  2255.                     for (int i = 0; i < n; i++) {
  2256.                         hs.setEntry(i, hs.getEntry(i) + tmp.getEntry(k) * interpolationPoints.getEntry(k, i));
  2257.                     }
  2258.                 }
  2259.             }
  2260.             if (crvmin != ZERO) {
  2261.                 state = 50; break;
  2262.             }
  2263.             if (iterc > itcsav) {
  2264.                 state = 150; break;
  2265.             }
  2266.             for (int i = 0; i < n; i++) {
  2267.                 hred.setEntry(i, hs.getEntry(i));
  2268.             }
  2269.             state = 120; break;
  2270.         }
  2271.         default: {
  2272.             throw new MathIllegalStateException(LocalizedFormats.SIMPLE_MESSAGE, "trsbox");
  2273.         }}
  2274.         }
  2275.     } // trsbox

  2277.     // ----------------------------------------------------------------------------------------

  2279.     /**
  2280.      *     The arrays BMAT and ZMAT are updated, as required by the new position
  2281.      *     of the interpolation point that has the index KNEW. The vector VLAG has
  2282.      *     N+NPT components, set on entry to the first NPT and last N components
  2283.      *     of the product Hw in equation (4.11) of the Powell (2006) paper on
  2284.      *     NEWUOA. Further, BETA is set on entry to the value of the parameter
  2285.      *     with that name, and DENOM is set to the denominator of the updating
  2286.      *     formula. Elements of ZMAT may be treated as zero if their moduli are
  2287.      *     at most ZTEST. The first NDIM elements of W are used for working space.
  2288.      * @param beta
  2289.      * @param denom
  2290.      * @param knew
  2291.      */
  2292.     private void update(
  2293.             double beta,
  2294.             double denom,
  2295.             int knew
  2296.     ) {
  2297.         printMethod(); // XXX

  2299.         final int n = currentBest.getDimension();
  2300.         final int npt = numberOfInterpolationPoints;
  2301.         final int nptm = npt - n - 1;

  2303.         // XXX Should probably be split into two arrays.
  2304.         final ArrayRealVector work = new ArrayRealVector(npt + n);

  2306.         double ztest = ZERO;
  2307.         for (int k = 0; k < npt; k++) {
  2308.             for (int j = 0; j < nptm; j++) {
  2309.                 // Computing MAX
  2310.                 ztest = FastMath.max(ztest, FastMath.abs(zMatrix.getEntry(k, j)));
  2311.             }
  2312.         }
  2313.         ztest *= 1e-20;

  2315.         // Apply the rotations that put zeros in the KNEW-th row of ZMAT.

  2317.         for (int j = 1; j < nptm; j++) {
  2318.             final double d1 = zMatrix.getEntry(knew, j);
  2319.             if (FastMath.abs(d1) > ztest) {
  2320.                 // Computing 2nd power
  2321.                 final double d2 = zMatrix.getEntry(knew, 0);
  2322.                 // Computing 2nd power
  2323.                 final double d3 = zMatrix.getEntry(knew, j);
  2324.                 final double d4 = FastMath.sqrt(d2 * d2 + d3 * d3);
  2325.                 final double d5 = zMatrix.getEntry(knew, 0) / d4;
  2326.                 final double d6 = zMatrix.getEntry(knew, j) / d4;
  2327.                 for (int i = 0; i < npt; i++) {
  2328.                     final double d7 = d5 * zMatrix.getEntry(i, 0) + d6 * zMatrix.getEntry(i, j);
  2329.                     zMatrix.setEntry(i, j, d5 * zMatrix.getEntry(i, j) - d6 * zMatrix.getEntry(i, 0));
  2330.                     zMatrix.setEntry(i, 0, d7);
  2331.                 }
  2332.             }
  2333.             zMatrix.setEntry(knew, j, ZERO);
  2334.         }

  2336.         // Put the first NPT components of the KNEW-th column of HLAG into W,
  2337.         // and calculate the parameters of the updating formula.

  2339.         for (int i = 0; i < npt; i++) {
  2340.             work.setEntry(i, zMatrix.getEntry(knew, 0) * zMatrix.getEntry(i, 0));
  2341.         }
  2342.         final double alpha = work.getEntry(knew);
  2343.         final double tau = lagrangeValuesAtNewPoint.getEntry(knew);
  2344.         lagrangeValuesAtNewPoint.setEntry(knew, lagrangeValuesAtNewPoint.getEntry(knew) - ONE);

  2346.         // Complete the updating of ZMAT.

  2348.         final double sqrtDenom = FastMath.sqrt(denom);
  2349.         final double d1 = tau / sqrtDenom;
  2350.         final double d2 = zMatrix.getEntry(knew, 0) / sqrtDenom;
  2351.         for (int i = 0; i < npt; i++) {
  2352.             zMatrix.setEntry(i, 0,
  2353.                           d1 * zMatrix.getEntry(i, 0) - d2 * lagrangeValuesAtNewPoint.getEntry(i));
  2354.         }

  2356.         // Finally, update the matrix BMAT.

  2358.         for (int j = 0; j < n; j++) {
  2359.             final int jp = npt + j;
  2360.             work.setEntry(jp, bMatrix.getEntry(knew, j));
  2361.             final double d3 = (alpha * lagrangeValuesAtNewPoint.getEntry(jp) - tau * work.getEntry(jp)) / denom;
  2362.             final double d4 = (-beta * work.getEntry(jp) - tau * lagrangeValuesAtNewPoint.getEntry(jp)) / denom;
  2363.             for (int i = 0; i <= jp; i++) {
  2364.                 bMatrix.setEntry(i, j,
  2365.                               bMatrix.getEntry(i, j) + d3 * lagrangeValuesAtNewPoint.getEntry(i) + d4 * work.getEntry(i));
  2366.                 if (i >= npt) {
  2367.                     bMatrix.setEntry(jp, (i - npt), bMatrix.getEntry(i, j));
  2368.                 }
  2369.             }
  2370.         }
  2371.     } // update

  2373.     /**
  2374.      * Performs validity checks.
  2375.      *
  2376.      * @param lowerBound Lower bounds (constraints) of the objective variables.
  2377.      * @param upperBound Upperer bounds (constraints) of the objective variables.
  2378.      */
  2379.     private void setup(double[] lowerBound,
  2380.                        double[] upperBound) {
  2381.         printMethod(); // XXX

  2383.         double[] init = getStartPoint();
  2384.         final int dimension = init.length;

  2386.         // Check problem dimension.
  2387.         if (dimension < MINIMUM_PROBLEM_DIMENSION) {
  2388.             throw new NumberIsTooSmallException(dimension, MINIMUM_PROBLEM_DIMENSION, true);
  2389.         }
  2390.         // Check number of interpolation points.
  2391.         final int[] nPointsInterval = { dimension + 2, (dimension + 2) * (dimension + 1) / 2 };
  2392.         if (numberOfInterpolationPoints < nPointsInterval[0] ||
  2393.             numberOfInterpolationPoints > nPointsInterval[1]) {
  2394.             throw new OutOfRangeException(LocalizedFormats.NUMBER_OF_INTERPOLATION_POINTS,
  2395.                                           numberOfInterpolationPoints,
  2396.                                           nPointsInterval[0],
  2397.                                           nPointsInterval[1]);
  2398.         }

  2400.         // Initialize bound differences.
  2401.         boundDifference = new double[dimension];

  2403.         double requiredMinDiff = 2 * initialTrustRegionRadius;
  2404.         double minDiff = Double.POSITIVE_INFINITY;
  2405.         for (int i = 0; i < dimension; i++) {
  2406.             boundDifference[i] = upperBound[i] - lowerBound[i];
  2407.             minDiff = FastMath.min(minDiff, boundDifference[i]);
  2408.         }
  2409.         if (minDiff < requiredMinDiff) {
  2410.             initialTrustRegionRadius = minDiff / 3.0;
  2411.         }

  2413.         // Initialize the data structures used by the "bobyqa" method.
  2414.         bMatrix = new Array2DRowRealMatrix(dimension + numberOfInterpolationPoints,
  2415.                                            dimension);
  2416.         zMatrix = new Array2DRowRealMatrix(numberOfInterpolationPoints,
  2417.                                            numberOfInterpolationPoints - dimension - 1);
  2418.         interpolationPoints = new Array2DRowRealMatrix(numberOfInterpolationPoints,
  2419.                                                        dimension);
  2420.         originShift = new ArrayRealVector(dimension);
  2421.         fAtInterpolationPoints = new ArrayRealVector(numberOfInterpolationPoints);
  2422.         trustRegionCenterOffset = new ArrayRealVector(dimension);
  2423.         gradientAtTrustRegionCenter = new ArrayRealVector(dimension);
  2424.         lowerDifference = new ArrayRealVector(dimension);
  2425.         upperDifference = new ArrayRealVector(dimension);
  2426.         modelSecondDerivativesParameters = new ArrayRealVector(numberOfInterpolationPoints);
  2427.         newPoint = new ArrayRealVector(dimension);
  2428.         alternativeNewPoint = new ArrayRealVector(dimension);
  2429.         trialStepPoint = new ArrayRealVector(dimension);
  2430.         lagrangeValuesAtNewPoint = new ArrayRealVector(dimension + numberOfInterpolationPoints);
  2431.         modelSecondDerivativesValues = new ArrayRealVector(dimension * (dimension + 1) / 2);
  2432.     }

  2434.     // XXX utility for figuring out call sequence.
  2435.     private static String caller(int n) {
  2436.         final Throwable t = new Throwable();
  2437.         final StackTraceElement[] elements = t.getStackTrace();
  2438.         final StackTraceElement e = elements[n];
  2439.         return e.getMethodName() + " (at line " + e.getLineNumber() + ")";
  2440.     }
  2441.     // XXX utility for figuring out call sequence.
  2442.     private static void printState(int s) {
  2443.         //        System.out.println(caller(2) + ": state " + s);
  2444.     }
  2445.     // XXX utility for figuring out call sequence.
  2446.     private static void printMethod() {
  2447.         //        System.out.println(caller(2));
  2448.     }

  2450.     /**
  2451.      * Marker for code paths that are not explored with the current unit tests.
  2452.      * If the path becomes explored, it should just be removed from the code.
  2453.      */
  2454.     private static class PathIsExploredException extends RuntimeException {
  2455.         private static final long serialVersionUID = 745350979634801853L;

  2457.         private static final String PATH_IS_EXPLORED
  2458.             = "If this exception is thrown, just remove it from the code";

  2460.         PathIsExploredException() {
  2461.             super(PATH_IS_EXPLORED + " " + BOBYQAOptimizer.caller(3));
  2462.         }
  2463.     }
  2464. }
  2465. //CHECKSTYLE: resume all
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