BrentOptimizer.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.apache.commons.math3.optimization.univariate;

  19. import org.apache.commons.math3.util.Precision;
  20. import org.apache.commons.math3.util.FastMath;
  21. import org.apache.commons.math3.exception.NumberIsTooSmallException;
  22. import org.apache.commons.math3.exception.NotStrictlyPositiveException;
  23. import org.apache.commons.math3.optimization.ConvergenceChecker;
  24. import org.apache.commons.math3.optimization.GoalType;

  26. /**
  27.  * For a function defined on some interval {@code (lo, hi)}, this class
  28.  * finds an approximation {@code x} to the point at which the function
  29.  * attains its minimum.
  30.  * It implements Richard Brent's algorithm (from his book "Algorithms for
  31.  * Minimization without Derivatives", p. 79) for finding minima of real
  32.  * univariate functions.
  33.  * <br/>
  34.  * This code is an adaptation, partly based on the Python code from SciPy
  35.  * (module "optimize.py" v0.5); the original algorithm is also modified
  36.  * <ul>
  37.  *  <li>to use an initial guess provided by the user,</li>
  38.  *  <li>to ensure that the best point encountered is the one returned.</li>
  39.  * </ul>
  40.  *
  41.  * @deprecated As of 3.1 (to be removed in 4.0).
  42.  * @since 2.0
  43.  */
  44. @Deprecated
  45. public class BrentOptimizer extends BaseAbstractUnivariateOptimizer {
  46.     /**
  47.      * Golden section.
  48.      */
  49.     private static final double GOLDEN_SECTION = 0.5 * (3 - FastMath.sqrt(5));
  50.     /**
  51.      * Minimum relative tolerance.
  52.      */
  53.     private static final double MIN_RELATIVE_TOLERANCE = 2 * FastMath.ulp(1d);
  54.     /**
  55.      * Relative threshold.
  56.      */
  57.     private final double relativeThreshold;
  58.     /**
  59.      * Absolute threshold.
  60.      */
  61.     private final double absoluteThreshold;

  63.     /**
  64.      * The arguments are used implement the original stopping criterion
  65.      * of Brent's algorithm.
  66.      * {@code abs} and {@code rel} define a tolerance
  67.      * {@code tol = rel |x| + abs}. {@code rel} should be no smaller than
  68.      * <em>2 macheps</em> and preferably not much less than <em>sqrt(macheps)</em>,
  69.      * where <em>macheps</em> is the relative machine precision. {@code abs} must
  70.      * be positive.
  71.      *
  72.      * @param rel Relative threshold.
  73.      * @param abs Absolute threshold.
  74.      * @param checker Additional, user-defined, convergence checking
  75.      * procedure.
  76.      * @throws NotStrictlyPositiveException if {@code abs <= 0}.
  77.      * @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
  78.      */
  79.     public BrentOptimizer(double rel,
  80.                           double abs,
  81.                           ConvergenceChecker<UnivariatePointValuePair> checker) {
  82.         super(checker);

  84.         if (rel < MIN_RELATIVE_TOLERANCE) {
  85.             throw new NumberIsTooSmallException(rel, MIN_RELATIVE_TOLERANCE, true);
  86.         }
  87.         if (abs <= 0) {
  88.             throw new NotStrictlyPositiveException(abs);
  89.         }

  91.         relativeThreshold = rel;
  92.         absoluteThreshold = abs;
  93.     }

  95.     /**
  96.      * The arguments are used for implementing the original stopping criterion
  97.      * of Brent's algorithm.
  98.      * {@code abs} and {@code rel} define a tolerance
  99.      * {@code tol = rel |x| + abs}. {@code rel} should be no smaller than
  100.      * <em>2 macheps</em> and preferably not much less than <em>sqrt(macheps)</em>,
  101.      * where <em>macheps</em> is the relative machine precision. {@code abs} must
  102.      * be positive.
  103.      *
  104.      * @param rel Relative threshold.
  105.      * @param abs Absolute threshold.
  106.      * @throws NotStrictlyPositiveException if {@code abs <= 0}.
  107.      * @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
  108.      */
  109.     public BrentOptimizer(double rel,
  110.                           double abs) {
  111.         this(rel, abs, null);
  112.     }

  114.     /** {@inheritDoc} */
  115.     @Override
  116.     protected UnivariatePointValuePair doOptimize() {
  117.         final boolean isMinim = getGoalType() == GoalType.MINIMIZE;
  118.         final double lo = getMin();
  119.         final double mid = getStartValue();
  120.         final double hi = getMax();

  122.         // Optional additional convergence criteria.
  123.         final ConvergenceChecker<UnivariatePointValuePair> checker
  124.             = getConvergenceChecker();

  126.         double a;
  127.         double b;
  128.         if (lo < hi) {
  129.             a = lo;
  130.             b = hi;
  131.         } else {
  132.             a = hi;
  133.             b = lo;
  134.         }

  136.         double x = mid;
  137.         double v = x;
  138.         double w = x;
  139.         double d = 0;
  140.         double e = 0;
  141.         double fx = computeObjectiveValue(x);
  142.         if (!isMinim) {
  143.             fx = -fx;
  144.         }
  145.         double fv = fx;
  146.         double fw = fx;

  148.         UnivariatePointValuePair previous = null;
  149.         UnivariatePointValuePair current
  150.             = new UnivariatePointValuePair(x, isMinim ? fx : -fx);
  151.         // Best point encountered so far (which is the initial guess).
  152.         UnivariatePointValuePair best = current;

  154.         int iter = 0;
  155.         while (true) {
  156.             final double m = 0.5 * (a + b);
  157.             final double tol1 = relativeThreshold * FastMath.abs(x) + absoluteThreshold;
  158.             final double tol2 = 2 * tol1;

  160.             // Default stopping criterion.
  161.             final boolean stop = FastMath.abs(x - m) <= tol2 - 0.5 * (b - a);
  162.             if (!stop) {
  163.                 double p = 0;
  164.                 double q = 0;
  165.                 double r = 0;
  166.                 double u = 0;

  168.                 if (FastMath.abs(e) > tol1) { // Fit parabola.
  169.                     r = (x - w) * (fx - fv);
  170.                     q = (x - v) * (fx - fw);
  171.                     p = (x - v) * q - (x - w) * r;
  172.                     q = 2 * (q - r);

  174.                     if (q > 0) {
  175.                         p = -p;
  176.                     } else {
  177.                         q = -q;
  178.                     }

  180.                     r = e;
  181.                     e = d;

  183.                     if (p > q * (a - x) &&
  184.                         p < q * (b - x) &&
  185.                         FastMath.abs(p) < FastMath.abs(0.5 * q * r)) {
  186.                         // Parabolic interpolation step.
  187.                         d = p / q;
  188.                         u = x + d;

  190.                         // f must not be evaluated too close to a or b.
  191.                         if (u - a < tol2 || b - u < tol2) {
  192.                             if (x <= m) {
  193.                                 d = tol1;
  194.                             } else {
  195.                                 d = -tol1;
  196.                             }
  197.                         }
  198.                     } else {
  199.                         // Golden section step.
  200.                         if (x < m) {
  201.                             e = b - x;
  202.                         } else {
  203.                             e = a - x;
  204.                         }
  205.                         d = GOLDEN_SECTION * e;
  206.                     }
  207.                 } else {
  208.                     // Golden section step.
  209.                     if (x < m) {
  210.                         e = b - x;
  211.                     } else {
  212.                         e = a - x;
  213.                     }
  214.                     d = GOLDEN_SECTION * e;
  215.                 }

  217.                 // Update by at least "tol1".
  218.                 if (FastMath.abs(d) < tol1) {
  219.                     if (d >= 0) {
  220.                         u = x + tol1;
  221.                     } else {
  222.                         u = x - tol1;
  223.                     }
  224.                 } else {
  225.                     u = x + d;
  226.                 }

  228.                 double fu = computeObjectiveValue(u);
  229.                 if (!isMinim) {
  230.                     fu = -fu;
  231.                 }

  233.                 // User-defined convergence checker.
  234.                 previous = current;
  235.                 current = new UnivariatePointValuePair(u, isMinim ? fu : -fu);
  236.                 best = best(best,
  237.                             best(previous,
  238.                                  current,
  239.                                  isMinim),
  240.                             isMinim);

  242.                 if (checker != null && checker.converged(iter, previous, current)) {
  243.                     return best;
  244.                 }

  246.                 // Update a, b, v, w and x.
  247.                 if (fu <= fx) {
  248.                     if (u < x) {
  249.                         b = x;
  250.                     } else {
  251.                         a = x;
  252.                     }
  253.                     v = w;
  254.                     fv = fw;
  255.                     w = x;
  256.                     fw = fx;
  257.                     x = u;
  258.                     fx = fu;
  259.                 } else {
  260.                     if (u < x) {
  261.                         a = u;
  262.                     } else {
  263.                         b = u;
  264.                     }
  265.                     if (fu <= fw ||
  266.                         Precision.equals(w, x)) {
  267.                         v = w;
  268.                         fv = fw;
  269.                         w = u;
  270.                         fw = fu;
  271.                     } else if (fu <= fv ||
  272.                                Precision.equals(v, x) ||
  273.                                Precision.equals(v, w)) {
  274.                         v = u;
  275.                         fv = fu;
  276.                     }
  277.                 }
  278.             } else { // Default termination (Brent's criterion).
  279.                 return best(best,
  280.                             best(previous,
  281.                                  current,
  282.                                  isMinim),
  283.                             isMinim);
  284.             }
  285.             ++iter;
  286.         }
  287.     }

  289.     /**
  290.      * Selects the best of two points.
  291.      *
  292.      * @param a Point and value.
  293.      * @param b Point and value.
  294.      * @param isMinim {@code true} if the selected point must be the one with
  295.      * the lowest value.
  296.      * @return the best point, or {@code null} if {@code a} and {@code b} are
  297.      * both {@code null}. When {@code a} and {@code b} have the same function
  298.      * value, {@code a} is returned.
  299.      */
  300.     private UnivariatePointValuePair best(UnivariatePointValuePair a,
  301.                                           UnivariatePointValuePair b,
  302.                                           boolean isMinim) {
  303.         if (a == null) {
  304.             return b;
  305.         }
  306.         if (b == null) {
  307.             return a;
  308.         }

  310.         if (isMinim) {
  311.             return a.getValue() <= b.getValue() ? a : b;
  312.         } else {
  313.             return a.getValue() >= b.getValue() ? a : b;
  314.         }
  315.     }
  316. }
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