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Java example source code file (BOBYQAOptimizer.java)

This example Java source code file (BOBYQAOptimizer.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

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Java - Java tags/keywords

array2drowrealmatrix, arrayrealvector, bobyqaoptimizer, half, minimum_problem_dimension, minus_one, one, one_over_a_thousand, one_over_four, one_over_ten, sixteen, two, xxx, zero

The BOBYQAOptimizer.java Java example source code

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

// CHECKSTYLE: stop all
package org.apache.commons.math3.optimization.direct;

import org.apache.commons.math3.analysis.MultivariateFunction;
import org.apache.commons.math3.exception.MathIllegalStateException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.ArrayRealVector;
import org.apache.commons.math3.linear.RealVector;
import org.apache.commons.math3.optimization.GoalType;
import org.apache.commons.math3.optimization.PointValuePair;
import org.apache.commons.math3.optimization.MultivariateOptimizer;
import org.apache.commons.math3.util.FastMath;

/**
 * Powell's BOBYQA algorithm. This implementation is translated and
 * adapted from the Fortran version available
 * <a href="http://plato.asu.edu/ftp/other_software/bobyqa.zip">here.
 * See <a href="http://www.optimization-online.org/DB_HTML/2010/05/2616.html">
 * this paper</a> for an introduction.
 * <br/>
 * BOBYQA is particularly well suited for high dimensional problems
 * where derivatives are not available. In most cases it outperforms the
 * {@link PowellOptimizer} significantly. Stochastic algorithms like
 * {@link CMAESOptimizer} succeed more often than BOBYQA, but are more
 * expensive. BOBYQA could also be considered as a replacement of any
 * derivative-based optimizer when the derivatives are approximated by
 * finite differences.
 *
 * @deprecated As of 3.1 (to be removed in 4.0).
 * @since 3.0
 */
@Deprecated
public class BOBYQAOptimizer
    extends BaseAbstractMultivariateSimpleBoundsOptimizer<MultivariateFunction>
    implements MultivariateOptimizer {
    /** Minimum dimension of the problem: {@value} */
    public static final int MINIMUM_PROBLEM_DIMENSION = 2;
    /** Default value for {@link #initialTrustRegionRadius}: {@value} . */
    public static final double DEFAULT_INITIAL_RADIUS = 10.0;
    /** Default value for {@link #stoppingTrustRegionRadius}: {@value} . */
    public static final double DEFAULT_STOPPING_RADIUS = 1E-8;
    /** Constant 0. */
    private static final double ZERO = 0d;
    /** Constant 1. */
    private static final double ONE = 1d;
    /** Constant 2. */
    private static final double TWO = 2d;
    /** Constant 10. */
    private static final double TEN = 10d;
    /** Constant 16. */
    private static final double SIXTEEN = 16d;
    /** Constant 250. */
    private static final double TWO_HUNDRED_FIFTY = 250d;
    /** Constant -1. */
    private static final double MINUS_ONE = -ONE;
    /** Constant 1/2. */
    private static final double HALF = ONE / 2;
    /** Constant 1/4. */
    private static final double ONE_OVER_FOUR = ONE / 4;
    /** Constant 1/8. */
    private static final double ONE_OVER_EIGHT = ONE / 8;
    /** Constant 1/10. */
    private static final double ONE_OVER_TEN = ONE / 10;
    /** Constant 1/1000. */
    private static final double ONE_OVER_A_THOUSAND = ONE / 1000;

    /**
     * numberOfInterpolationPoints XXX
     */
    private final int numberOfInterpolationPoints;
    /**
     * initialTrustRegionRadius XXX
     */
    private double initialTrustRegionRadius;
    /**
     * stoppingTrustRegionRadius XXX
     */
    private final double stoppingTrustRegionRadius;
    /** Goal type (minimize or maximize). */
    private boolean isMinimize;
    /**
     * Current best values for the variables to be optimized.
     * The vector will be changed in-place to contain the values of the least
     * calculated objective function values.
     */
    private ArrayRealVector currentBest;
    /** Differences between the upper and lower bounds. */
    private double[] boundDifference;
    /**
     * Index of the interpolation point at the trust region center.
     */
    private int trustRegionCenterInterpolationPointIndex;
    /**
     * Last <em>n columns of matrix H (where n is the dimension
     * of the problem).
     * XXX "bmat" in the original code.
     */
    private Array2DRowRealMatrix bMatrix;
    /**
     * Factorization of the leading <em>npt square submatrix of H, this
     * factorization being Z Z<sup>T, which provides both the correct
     * rank and positive semi-definiteness.
     * XXX "zmat" in the original code.
     */
    private Array2DRowRealMatrix zMatrix;
    /**
     * Coordinates of the interpolation points relative to {@link #originShift}.
     * XXX "xpt" in the original code.
     */
    private Array2DRowRealMatrix interpolationPoints;
    /**
     * Shift of origin that should reduce the contributions from rounding
     * errors to values of the model and Lagrange functions.
     * XXX "xbase" in the original code.
     */
    private ArrayRealVector originShift;
    /**
     * Values of the objective function at the interpolation points.
     * XXX "fval" in the original code.
     */
    private ArrayRealVector fAtInterpolationPoints;
    /**
     * Displacement from {@link #originShift} of the trust region center.
     * XXX "xopt" in the original code.
     */
    private ArrayRealVector trustRegionCenterOffset;
    /**
     * Gradient of the quadratic model at {@link #originShift} +
     * {@link #trustRegionCenterOffset}.
     * XXX "gopt" in the original code.
     */
    private ArrayRealVector gradientAtTrustRegionCenter;
    /**
     * Differences {@link #getLowerBound()} - {@link #originShift}.
     * All the components of every {@link #trustRegionCenterOffset} are going
     * to satisfy the bounds<br/>
     * {@link #getLowerBound() lowerBound}<sub>i ≤
     * {@link #trustRegionCenterOffset}<sub>i,
* with appropriate equalities when {@link #trustRegionCenterOffset} is * on a constraint boundary. * XXX "sl" in the original code. */ private ArrayRealVector lowerDifference; /** * Differences {@link #getUpperBound()} - {@link #originShift} * All the components of every {@link #trustRegionCenterOffset} are going * to satisfy the bounds<br/> * {@link #trustRegionCenterOffset}<sub>i ≤ * {@link #getUpperBound() upperBound}<sub>i,
* with appropriate equalities when {@link #trustRegionCenterOffset} is * on a constraint boundary. * XXX "su" in the original code. */ private ArrayRealVector upperDifference; /** * Parameters of the implicit second derivatives of the quadratic model. * XXX "pq" in the original code. */ private ArrayRealVector modelSecondDerivativesParameters; /** * Point chosen by function {@link #trsbox(double,ArrayRealVector, * ArrayRealVector, ArrayRealVector,ArrayRealVector,ArrayRealVector) trsbox} * or {@link #altmov(int,double) altmov}. * Usually {@link #originShift} + {@link #newPoint} is the vector of * variables for the next evaluation of the objective function. * It also satisfies the constraints indicated in {@link #lowerDifference} * and {@link #upperDifference}. * XXX "xnew" in the original code. */ private ArrayRealVector newPoint; /** * Alternative to {@link #newPoint}, chosen by * {@link #altmov(int,double) altmov}. * It may replace {@link #newPoint} in order to increase the denominator * in the {@link #update(double, double, int) updating procedure}. * XXX "xalt" in the original code. */ private ArrayRealVector alternativeNewPoint; /** * Trial step from {@link #trustRegionCenterOffset} which is usually * {@link #newPoint} - {@link #trustRegionCenterOffset}. * XXX "d__" in the original code. */ private ArrayRealVector trialStepPoint; /** * Values of the Lagrange functions at a new point. * XXX "vlag" in the original code. */ private ArrayRealVector lagrangeValuesAtNewPoint; /** * Explicit second derivatives of the quadratic model. * XXX "hq" in the original code. */ private ArrayRealVector modelSecondDerivativesValues; /** * @param numberOfInterpolationPoints Number of interpolation conditions. * For a problem of dimension {@code n}, its value must be in the interval * {@code [n+2, (n+1)(n+2)/2]}. * Choices that exceed {@code 2n+1} are not recommended. */ public BOBYQAOptimizer(int numberOfInterpolationPoints) { this(numberOfInterpolationPoints, DEFAULT_INITIAL_RADIUS, DEFAULT_STOPPING_RADIUS); } /** * @param numberOfInterpolationPoints Number of interpolation conditions. * For a problem of dimension {@code n}, its value must be in the interval * {@code [n+2, (n+1)(n+2)/2]}. * Choices that exceed {@code 2n+1} are not recommended. * @param initialTrustRegionRadius Initial trust region radius. * @param stoppingTrustRegionRadius Stopping trust region radius. */ public BOBYQAOptimizer(int numberOfInterpolationPoints, double initialTrustRegionRadius, double stoppingTrustRegionRadius) { super(null); // No custom convergence criterion. this.numberOfInterpolationPoints = numberOfInterpolationPoints; this.initialTrustRegionRadius = initialTrustRegionRadius; this.stoppingTrustRegionRadius = stoppingTrustRegionRadius; } /** {@inheritDoc} */ @Override protected PointValuePair doOptimize() { final double[] lowerBound = getLowerBound(); final double[] upperBound = getUpperBound(); // Validity checks. setup(lowerBound, upperBound); isMinimize = (getGoalType() == GoalType.MINIMIZE); currentBest = new ArrayRealVector(getStartPoint()); final double value = bobyqa(lowerBound, upperBound); return new PointValuePair(currentBest.getDataRef(), isMinimize ? value : -value); } /** * This subroutine seeks the least value of a function of many variables, * by applying a trust region method that forms quadratic models by * interpolation. There is usually some freedom in the interpolation * conditions, which is taken up by minimizing the Frobenius norm of * the change to the second derivative of the model, beginning with the * zero matrix. The values of the variables are constrained by upper and * lower bounds. The arguments of the subroutine are as follows. * * N must be set to the number of variables and must be at least two. * NPT is the number of interpolation conditions. Its value must be in * the interval [N+2,(N+1)(N+2)/2]. Choices that exceed 2*N+1 are not * recommended. * Initial values of the variables must be set in X(1),X(2),...,X(N). They * will be changed to the values that give the least calculated F. * For I=1,2,...,N, XL(I) and XU(I) must provide the lower and upper * bounds, respectively, on X(I). The construction of quadratic models * requires XL(I) to be strictly less than XU(I) for each I. Further, * the contribution to a model from changes to the I-th variable is * damaged severely by rounding errors if XU(I)-XL(I) is too small. * RHOBEG and RHOEND must be set to the initial and final values of a trust * region radius, so both must be positive with RHOEND no greater than * RHOBEG. Typically, RHOBEG should be about one tenth of the greatest * expected change to a variable, while RHOEND should indicate the * accuracy that is required in the final values of the variables. An * error return occurs if any of the differences XU(I)-XL(I), I=1,...,N, * is less than 2*RHOBEG. * MAXFUN must be set to an upper bound on the number of calls of CALFUN. * The array W will be used for working space. Its length must be at least * (NPT+5)*(NPT+N)+3*N*(N+5)/2. * * @param lowerBound Lower bounds. * @param upperBound Upper bounds. * @return the value of the objective at the optimum. */ private double bobyqa(double[] lowerBound, double[] upperBound) { printMethod(); // XXX final int n = currentBest.getDimension(); // Return if there is insufficient space between the bounds. Modify the // initial X if necessary in order to avoid conflicts between the bounds // and the construction of the first quadratic model. The lower and upper // bounds on moves from the updated X are set now, in the ISL and ISU // partitions of W, in order to provide useful and exact information about // components of X that become within distance RHOBEG from their bounds. for (int j = 0; j < n; j++) { final double boundDiff = boundDifference[j]; lowerDifference.setEntry(j, lowerBound[j] - currentBest.getEntry(j)); upperDifference.setEntry(j, upperBound[j] - currentBest.getEntry(j)); if (lowerDifference.getEntry(j) >= -initialTrustRegionRadius) { if (lowerDifference.getEntry(j) >= ZERO) { currentBest.setEntry(j, lowerBound[j]); lowerDifference.setEntry(j, ZERO); upperDifference.setEntry(j, boundDiff); } else { currentBest.setEntry(j, lowerBound[j] + initialTrustRegionRadius); lowerDifference.setEntry(j, -initialTrustRegionRadius); // Computing MAX final double deltaOne = upperBound[j] - currentBest.getEntry(j); upperDifference.setEntry(j, FastMath.max(deltaOne, initialTrustRegionRadius)); } } else if (upperDifference.getEntry(j) <= initialTrustRegionRadius) { if (upperDifference.getEntry(j) <= ZERO) { currentBest.setEntry(j, upperBound[j]); lowerDifference.setEntry(j, -boundDiff); upperDifference.setEntry(j, ZERO); } else { currentBest.setEntry(j, upperBound[j] - initialTrustRegionRadius); // Computing MIN final double deltaOne = lowerBound[j] - currentBest.getEntry(j); final double deltaTwo = -initialTrustRegionRadius; lowerDifference.setEntry(j, FastMath.min(deltaOne, deltaTwo)); upperDifference.setEntry(j, initialTrustRegionRadius); } } } // Make the call of BOBYQB. return bobyqb(lowerBound, upperBound); } // bobyqa // ---------------------------------------------------------------------------------------- /** * The arguments N, NPT, X, XL, XU, RHOBEG, RHOEND, IPRINT and MAXFUN * are identical to the corresponding arguments in SUBROUTINE BOBYQA. * XBASE holds a shift of origin that should reduce the contributions * from rounding errors to values of the model and Lagrange functions. * XPT is a two-dimensional array that holds the coordinates of the * interpolation points relative to XBASE. * FVAL holds the values of F at the interpolation points. * XOPT is set to the displacement from XBASE of the trust region centre. * GOPT holds the gradient of the quadratic model at XBASE+XOPT. * HQ holds the explicit second derivatives of the quadratic model. * PQ contains the parameters of the implicit second derivatives of the * quadratic model. * BMAT holds the last N columns of H. * ZMAT holds the factorization of the leading NPT by NPT submatrix of H, * this factorization being ZMAT times ZMAT^T, which provides both the * correct rank and positive semi-definiteness. * NDIM is the first dimension of BMAT and has the value NPT+N. * SL and SU hold the differences XL-XBASE and XU-XBASE, respectively. * All the components of every XOPT are going to satisfy the bounds * SL(I) .LEQ. XOPT(I) .LEQ. SU(I), with appropriate equalities when * XOPT is on a constraint boundary. * XNEW is chosen by SUBROUTINE TRSBOX or ALTMOV. Usually XBASE+XNEW is the * vector of variables for the next call of CALFUN. XNEW also satisfies * the SL and SU constraints in the way that has just been mentioned. * XALT is an alternative to XNEW, chosen by ALTMOV, that may replace XNEW * in order to increase the denominator in the updating of UPDATE. * D is reserved for a trial step from XOPT, which is usually XNEW-XOPT. * VLAG contains the values of the Lagrange functions at a new point X. * They are part of a product that requires VLAG to be of length NDIM. * W is a one-dimensional array that is used for working space. Its length * must be at least 3*NDIM = 3*(NPT+N). * * @param lowerBound Lower bounds. * @param upperBound Upper bounds. * @return the value of the objective at the optimum. */ private double bobyqb(double[] lowerBound, double[] upperBound) { printMethod(); // XXX final int n = currentBest.getDimension(); final int npt = numberOfInterpolationPoints; final int np = n + 1; final int nptm = npt - np; final int nh = n * np / 2; final ArrayRealVector work1 = new ArrayRealVector(n); final ArrayRealVector work2 = new ArrayRealVector(npt); final ArrayRealVector work3 = new ArrayRealVector(npt); double cauchy = Double.NaN; double alpha = Double.NaN; double dsq = Double.NaN; double crvmin = Double.NaN; // Set some constants. // Parameter adjustments // Function Body // The call of PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ, // BMAT and ZMAT for the first iteration, with the corresponding values of // of NF and KOPT, which are the number of calls of CALFUN so far and the // index of the interpolation point at the trust region centre. Then the // initial XOPT is set too. The branch to label 720 occurs if MAXFUN is // less than NPT. GOPT will be updated if KOPT is different from KBASE. trustRegionCenterInterpolationPointIndex = 0; prelim(lowerBound, upperBound); double xoptsq = ZERO; for (int i = 0; i < n; i++) { trustRegionCenterOffset.setEntry(i, interpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex, i)); // Computing 2nd power final double deltaOne = trustRegionCenterOffset.getEntry(i); xoptsq += deltaOne * deltaOne; } double fsave = fAtInterpolationPoints.getEntry(0); final int kbase = 0; // Complete the settings that are required for the iterative procedure. int ntrits = 0; int itest = 0; int knew = 0; int nfsav = getEvaluations(); double rho = initialTrustRegionRadius; double delta = rho; double diffa = ZERO; double diffb = ZERO; double diffc = ZERO; double f = ZERO; double beta = ZERO; double adelt = ZERO; double denom = ZERO; double ratio = ZERO; double dnorm = ZERO; double scaden = ZERO; double biglsq = ZERO; double distsq = ZERO; // Update GOPT if necessary before the first iteration and after each // call of RESCUE that makes a call of CALFUN. int state = 20; for(;;) { switch (state) { case 20: { printState(20); // XXX if (trustRegionCenterInterpolationPointIndex != kbase) { int ih = 0; for (int j = 0; j < n; j++) { for (int i = 0; i <= j; i++) { if (i < j) { gradientAtTrustRegionCenter.setEntry(j, gradientAtTrustRegionCenter.getEntry(j) + modelSecondDerivativesValues.getEntry(ih) * trustRegionCenterOffset.getEntry(i)); } gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + modelSecondDerivativesValues.getEntry(ih) * trustRegionCenterOffset.getEntry(j)); ih++; } } if (getEvaluations() > npt) { for (int k = 0; k < npt; k++) { double temp = ZERO; for (int j = 0; j < n; j++) { temp += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j); } temp *= modelSecondDerivativesParameters.getEntry(k); for (int i = 0; i < n; i++) { gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + temp * interpolationPoints.getEntry(k, i)); } } // throw new PathIsExploredException(); // XXX } } // Generate the next point in the trust region that provides a small value // of the quadratic model subject to the constraints on the variables. // The int NTRITS is set to the number "trust region" iterations that // have occurred since the last "alternative" iteration. If the length // of XNEW-XOPT is less than HALF*RHO, however, then there is a branch to // label 650 or 680 with NTRITS=-1, instead of calculating F at XNEW. } case 60: { printState(60); // XXX final ArrayRealVector gnew = new ArrayRealVector(n); final ArrayRealVector xbdi = new ArrayRealVector(n); final ArrayRealVector s = new ArrayRealVector(n); final ArrayRealVector hs = new ArrayRealVector(n); final ArrayRealVector hred = new ArrayRealVector(n); final double[] dsqCrvmin = trsbox(delta, gnew, xbdi, s, hs, hred); dsq = dsqCrvmin[0]; crvmin = dsqCrvmin[1]; // Computing MIN double deltaOne = delta; double deltaTwo = FastMath.sqrt(dsq); dnorm = FastMath.min(deltaOne, deltaTwo); if (dnorm < HALF * rho) { ntrits = -1; // Computing 2nd power deltaOne = TEN * rho; distsq = deltaOne * deltaOne; if (getEvaluations() <= nfsav + 2) { state = 650; break; } // The following choice between labels 650 and 680 depends on whether or // not our work with the current RHO seems to be complete. Either RHO is // decreased or termination occurs if the errors in the quadratic model at // the last three interpolation points compare favourably with predictions // of likely improvements to the model within distance HALF*RHO of XOPT. // Computing MAX deltaOne = FastMath.max(diffa, diffb); final double errbig = FastMath.max(deltaOne, diffc); final double frhosq = rho * ONE_OVER_EIGHT * rho; if (crvmin > ZERO && errbig > frhosq * crvmin) { state = 650; break; } final double bdtol = errbig / rho; for (int j = 0; j < n; j++) { double bdtest = bdtol; if (newPoint.getEntry(j) == lowerDifference.getEntry(j)) { bdtest = work1.getEntry(j); } if (newPoint.getEntry(j) == upperDifference.getEntry(j)) { bdtest = -work1.getEntry(j); } if (bdtest < bdtol) { double curv = modelSecondDerivativesValues.getEntry((j + j * j) / 2); for (int k = 0; k < npt; k++) { // Computing 2nd power final double d1 = interpolationPoints.getEntry(k, j); curv += modelSecondDerivativesParameters.getEntry(k) * (d1 * d1); } bdtest += HALF * curv * rho; if (bdtest < bdtol) { state = 650; break; } // throw new PathIsExploredException(); // XXX } } state = 680; break; } ++ntrits; // Severe cancellation is likely to occur if XOPT is too far from XBASE. // If the following test holds, then XBASE is shifted so that XOPT becomes // zero. The appropriate changes are made to BMAT and to the second // derivatives of the current model, beginning with the changes to BMAT // that do not depend on ZMAT. VLAG is used temporarily for working space. } case 90: { printState(90); // XXX if (dsq <= xoptsq * ONE_OVER_A_THOUSAND) { final double fracsq = xoptsq * ONE_OVER_FOUR; double sumpq = ZERO; // final RealVector sumVector // = new ArrayRealVector(npt, -HALF * xoptsq).add(interpolationPoints.operate(trustRegionCenter)); for (int k = 0; k < npt; k++) { sumpq += modelSecondDerivativesParameters.getEntry(k); double sum = -HALF * xoptsq; for (int i = 0; i < n; i++) { sum += interpolationPoints.getEntry(k, i) * trustRegionCenterOffset.getEntry(i); } // sum = sumVector.getEntry(k); // XXX "testAckley" and "testDiffPow" fail. work2.setEntry(k, sum); final double temp = fracsq - HALF * sum; for (int i = 0; i < n; i++) { work1.setEntry(i, bMatrix.getEntry(k, i)); lagrangeValuesAtNewPoint.setEntry(i, sum * interpolationPoints.getEntry(k, i) + temp * trustRegionCenterOffset.getEntry(i)); final int ip = npt + i; for (int j = 0; j <= i; j++) { bMatrix.setEntry(ip, j, bMatrix.getEntry(ip, j) + work1.getEntry(i) * lagrangeValuesAtNewPoint.getEntry(j) + lagrangeValuesAtNewPoint.getEntry(i) * work1.getEntry(j)); } } } // Then the revisions of BMAT that depend on ZMAT are calculated. for (int m = 0; m < nptm; m++) { double sumz = ZERO; double sumw = ZERO; for (int k = 0; k < npt; k++) { sumz += zMatrix.getEntry(k, m); lagrangeValuesAtNewPoint.setEntry(k, work2.getEntry(k) * zMatrix.getEntry(k, m)); sumw += lagrangeValuesAtNewPoint.getEntry(k); } for (int j = 0; j < n; j++) { double sum = (fracsq * sumz - HALF * sumw) * trustRegionCenterOffset.getEntry(j); for (int k = 0; k < npt; k++) { sum += lagrangeValuesAtNewPoint.getEntry(k) * interpolationPoints.getEntry(k, j); } work1.setEntry(j, sum); for (int k = 0; k < npt; k++) { bMatrix.setEntry(k, j, bMatrix.getEntry(k, j) + sum * zMatrix.getEntry(k, m)); } } for (int i = 0; i < n; i++) { final int ip = i + npt; final double temp = work1.getEntry(i); for (int j = 0; j <= i; j++) { bMatrix.setEntry(ip, j, bMatrix.getEntry(ip, j) + temp * work1.getEntry(j)); } } } // The following instructions complete the shift, including the changes // to the second derivative parameters of the quadratic model. int ih = 0; for (int j = 0; j < n; j++) { work1.setEntry(j, -HALF * sumpq * trustRegionCenterOffset.getEntry(j)); for (int k = 0; k < npt; k++) { work1.setEntry(j, work1.getEntry(j) + modelSecondDerivativesParameters.getEntry(k) * interpolationPoints.getEntry(k, j)); interpolationPoints.setEntry(k, j, interpolationPoints.getEntry(k, j) - trustRegionCenterOffset.getEntry(j)); } for (int i = 0; i <= j; i++) { modelSecondDerivativesValues.setEntry(ih, modelSecondDerivativesValues.getEntry(ih) + work1.getEntry(i) * trustRegionCenterOffset.getEntry(j) + trustRegionCenterOffset.getEntry(i) * work1.getEntry(j)); bMatrix.setEntry(npt + i, j, bMatrix.getEntry(npt + j, i)); ih++; } } for (int i = 0; i < n; i++) { originShift.setEntry(i, originShift.getEntry(i) + trustRegionCenterOffset.getEntry(i)); newPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i)); lowerDifference.setEntry(i, lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)); upperDifference.setEntry(i, upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)); trustRegionCenterOffset.setEntry(i, ZERO); } xoptsq = ZERO; } if (ntrits == 0) { state = 210; break; } state = 230; break; // XBASE is also moved to XOPT by a call of RESCUE. This calculation is // more expensive than the previous shift, because new matrices BMAT and // ZMAT are generated from scratch, which may include the replacement of // interpolation points whose positions seem to be causing near linear // dependence in the interpolation conditions. Therefore RESCUE is called // only if rounding errors have reduced by at least a factor of two the // denominator of the formula for updating the H matrix. It provides a // useful safeguard, but is not invoked in most applications of BOBYQA. } case 210: { printState(210); // XXX // Pick two alternative vectors of variables, relative to XBASE, that // are suitable as new positions of the KNEW-th interpolation point. // Firstly, XNEW is set to the point on a line through XOPT and another // interpolation point that minimizes the predicted value of the next // denominator, subject to ||XNEW - XOPT|| .LEQ. ADELT and to the SL // and SU bounds. Secondly, XALT is set to the best feasible point on // a constrained version of the Cauchy step of the KNEW-th Lagrange // function, the corresponding value of the square of this function // being returned in CAUCHY. The choice between these alternatives is // going to be made when the denominator is calculated. final double[] alphaCauchy = altmov(knew, adelt); alpha = alphaCauchy[0]; cauchy = alphaCauchy[1]; for (int i = 0; i < n; i++) { trialStepPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i)); } // Calculate VLAG and BETA for the current choice of D. The scalar // product of D with XPT(K,.) is going to be held in W(NPT+K) for // use when VQUAD is calculated. } case 230: { printState(230); // XXX for (int k = 0; k < npt; k++) { double suma = ZERO; double sumb = ZERO; double sum = ZERO; for (int j = 0; j < n; j++) { suma += interpolationPoints.getEntry(k, j) * trialStepPoint.getEntry(j); sumb += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j); sum += bMatrix.getEntry(k, j) * trialStepPoint.getEntry(j); } work3.setEntry(k, suma * (HALF * suma + sumb)); lagrangeValuesAtNewPoint.setEntry(k, sum); work2.setEntry(k, suma); } beta = ZERO; for (int m = 0; m < nptm; m++) { double sum = ZERO; for (int k = 0; k < npt; k++) { sum += zMatrix.getEntry(k, m) * work3.getEntry(k); } beta -= sum * sum; for (int k = 0; k < npt; k++) { lagrangeValuesAtNewPoint.setEntry(k, lagrangeValuesAtNewPoint.getEntry(k) + sum * zMatrix.getEntry(k, m)); } } dsq = ZERO; double bsum = ZERO; double dx = ZERO; for (int j = 0; j < n; j++) { // Computing 2nd power final double d1 = trialStepPoint.getEntry(j); dsq += d1 * d1; double sum = ZERO; for (int k = 0; k < npt; k++) { sum += work3.getEntry(k) * bMatrix.getEntry(k, j); } bsum += sum * trialStepPoint.getEntry(j); final int jp = npt + j; for (int i = 0; i < n; i++) { sum += bMatrix.getEntry(jp, i) * trialStepPoint.getEntry(i); } lagrangeValuesAtNewPoint.setEntry(jp, sum); bsum += sum * trialStepPoint.getEntry(j); dx += trialStepPoint.getEntry(j) * trustRegionCenterOffset.getEntry(j); } beta = dx * dx + dsq * (xoptsq + dx + dx + HALF * dsq) + beta - bsum; // Original // beta += dx * dx + dsq * (xoptsq + dx + dx + HALF * dsq) - bsum; // XXX "testAckley" and "testDiffPow" fail. // beta = dx * dx + dsq * (xoptsq + 2 * dx + HALF * dsq) + beta - bsum; // XXX "testDiffPow" fails. lagrangeValuesAtNewPoint.setEntry(trustRegionCenterInterpolationPointIndex, lagrangeValuesAtNewPoint.getEntry(trustRegionCenterInterpolationPointIndex) + ONE); // If NTRITS is zero, the denominator may be increased by replacing // the step D of ALTMOV by a Cauchy step. Then RESCUE may be called if // rounding errors have damaged the chosen denominator. if (ntrits == 0) { // Computing 2nd power final double d1 = lagrangeValuesAtNewPoint.getEntry(knew); denom = d1 * d1 + alpha * beta; if (denom < cauchy && cauchy > ZERO) { for (int i = 0; i < n; i++) { newPoint.setEntry(i, alternativeNewPoint.getEntry(i)); trialStepPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i)); } cauchy = ZERO; // XXX Useful statement? state = 230; break; } // Alternatively, if NTRITS is positive, then set KNEW to the index of // the next interpolation point to be deleted to make room for a trust // region step. Again RESCUE may be called if rounding errors have damaged_ // the chosen denominator, which is the reason for attempting to select // KNEW before calculating the next value of the objective function. } else { final double delsq = delta * delta; scaden = ZERO; biglsq = ZERO; knew = 0; for (int k = 0; k < npt; k++) { if (k == trustRegionCenterInterpolationPointIndex) { continue; } double hdiag = ZERO; for (int m = 0; m < nptm; m++) { // Computing 2nd power final double d1 = zMatrix.getEntry(k, m); hdiag += d1 * d1; } // Computing 2nd power final double d2 = lagrangeValuesAtNewPoint.getEntry(k); final double den = beta * hdiag + d2 * d2; distsq = ZERO; for (int j = 0; j < n; j++) { // Computing 2nd power final double d3 = interpolationPoints.getEntry(k, j) - trustRegionCenterOffset.getEntry(j); distsq += d3 * d3; } // Computing MAX // Computing 2nd power final double d4 = distsq / delsq; final double temp = FastMath.max(ONE, d4 * d4); if (temp * den > scaden) { scaden = temp * den; knew = k; denom = den; } // Computing MAX // Computing 2nd power final double d5 = lagrangeValuesAtNewPoint.getEntry(k); biglsq = FastMath.max(biglsq, temp * (d5 * d5)); } } // Put the variables for the next calculation of the objective function // in XNEW, with any adjustments for the bounds. // Calculate the value of the objective function at XBASE+XNEW, unless // the limit on the number of calculations of F has been reached. } case 360: { printState(360); // XXX for (int i = 0; i < n; i++) { // Computing MIN // Computing MAX final double d3 = lowerBound[i]; final double d4 = originShift.getEntry(i) + newPoint.getEntry(i); final double d1 = FastMath.max(d3, d4); final double d2 = upperBound[i]; currentBest.setEntry(i, FastMath.min(d1, d2)); if (newPoint.getEntry(i) == lowerDifference.getEntry(i)) { currentBest.setEntry(i, lowerBound[i]); } if (newPoint.getEntry(i) == upperDifference.getEntry(i)) { currentBest.setEntry(i, upperBound[i]); } } f = computeObjectiveValue(currentBest.toArray()); if (!isMinimize) { f = -f; } if (ntrits == -1) { fsave = f; state = 720; break; } // Use the quadratic model to predict the change in F due to the step D, // and set DIFF to the error of this prediction. final double fopt = fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex); double vquad = ZERO; int ih = 0; for (int j = 0; j < n; j++) { vquad += trialStepPoint.getEntry(j) * gradientAtTrustRegionCenter.getEntry(j); for (int i = 0; i <= j; i++) { double temp = trialStepPoint.getEntry(i) * trialStepPoint.getEntry(j); if (i == j) { temp *= HALF; } vquad += modelSecondDerivativesValues.getEntry(ih) * temp; ih++; } } for (int k = 0; k < npt; k++) { // Computing 2nd power final double d1 = work2.getEntry(k); final double d2 = d1 * d1; // "d1" must be squared first to prevent test failures. vquad += HALF * modelSecondDerivativesParameters.getEntry(k) * d2; } final double diff = f - fopt - vquad; diffc = diffb; diffb = diffa; diffa = FastMath.abs(diff); if (dnorm > rho) { nfsav = getEvaluations(); } // Pick the next value of DELTA after a trust region step. if (ntrits > 0) { if (vquad >= ZERO) { throw new MathIllegalStateException(LocalizedFormats.TRUST_REGION_STEP_FAILED, vquad); } ratio = (f - fopt) / vquad; final double hDelta = HALF * delta; if (ratio <= ONE_OVER_TEN) { // Computing MIN delta = FastMath.min(hDelta, dnorm); } else if (ratio <= .7) { // Computing MAX delta = FastMath.max(hDelta, dnorm); } else { // Computing MAX delta = FastMath.max(hDelta, 2 * dnorm); } if (delta <= rho * 1.5) { delta = rho; } // Recalculate KNEW and DENOM if the new F is less than FOPT. if (f < fopt) { final int ksav = knew; final double densav = denom; final double delsq = delta * delta; scaden = ZERO; biglsq = ZERO; knew = 0; for (int k = 0; k < npt; k++) { double hdiag = ZERO; for (int m = 0; m < nptm; m++) { // Computing 2nd power final double d1 = zMatrix.getEntry(k, m); hdiag += d1 * d1; } // Computing 2nd power final double d1 = lagrangeValuesAtNewPoint.getEntry(k); final double den = beta * hdiag + d1 * d1; distsq = ZERO; for (int j = 0; j < n; j++) { // Computing 2nd power final double d2 = interpolationPoints.getEntry(k, j) - newPoint.getEntry(j); distsq += d2 * d2; } // Computing MAX // Computing 2nd power final double d3 = distsq / delsq; final double temp = FastMath.max(ONE, d3 * d3); if (temp * den > scaden) { scaden = temp * den; knew = k; denom = den; } // Computing MAX // Computing 2nd power final double d4 = lagrangeValuesAtNewPoint.getEntry(k); final double d5 = temp * (d4 * d4); biglsq = FastMath.max(biglsq, d5); } if (scaden <= HALF * biglsq) { knew = ksav; denom = densav; } } } // Update BMAT and ZMAT, so that the KNEW-th interpolation point can be // moved. Also update the second derivative terms of the model. update(beta, denom, knew); ih = 0; final double pqold = modelSecondDerivativesParameters.getEntry(knew); modelSecondDerivativesParameters.setEntry(knew, ZERO); for (int i = 0; i < n; i++) { final double temp = pqold * interpolationPoints.getEntry(knew, i); for (int j = 0; j <= i; j++) { modelSecondDerivativesValues.setEntry(ih, modelSecondDerivativesValues.getEntry(ih) + temp * interpolationPoints.getEntry(knew, j)); ih++; } } for (int m = 0; m < nptm; m++) { final double temp = diff * zMatrix.getEntry(knew, m); for (int k = 0; k < npt; k++) { modelSecondDerivativesParameters.setEntry(k, modelSecondDerivativesParameters.getEntry(k) + temp * zMatrix.getEntry(k, m)); } } // Include the new interpolation point, and make the changes to GOPT at // the old XOPT that are caused by the updating of the quadratic model. fAtInterpolationPoints.setEntry(knew, f); for (int i = 0; i < n; i++) { interpolationPoints.setEntry(knew, i, newPoint.getEntry(i)); work1.setEntry(i, bMatrix.getEntry(knew, i)); } for (int k = 0; k < npt; k++) { double suma = ZERO; for (int m = 0; m < nptm; m++) { suma += zMatrix.getEntry(knew, m) * zMatrix.getEntry(k, m); } double sumb = ZERO; for (int j = 0; j < n; j++) { sumb += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j); } final double temp = suma * sumb; for (int i = 0; i < n; i++) { work1.setEntry(i, work1.getEntry(i) + temp * interpolationPoints.getEntry(k, i)); } } for (int i = 0; i < n; i++) { gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + diff * work1.getEntry(i)); } // Update XOPT, GOPT and KOPT if the new calculated F is less than FOPT. if (f < fopt) { trustRegionCenterInterpolationPointIndex = knew; xoptsq = ZERO; ih = 0; for (int j = 0; j < n; j++) { trustRegionCenterOffset.setEntry(j, newPoint.getEntry(j)); // Computing 2nd power final double d1 = trustRegionCenterOffset.getEntry(j); xoptsq += d1 * d1; for (int i = 0; i <= j; i++) { if (i < j) { gradientAtTrustRegionCenter.setEntry(j, gradientAtTrustRegionCenter.getEntry(j) + modelSecondDerivativesValues.getEntry(ih) * trialStepPoint.getEntry(i)); } gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + modelSecondDerivativesValues.getEntry(ih) * trialStepPoint.getEntry(j)); ih++; } } for (int k = 0; k < npt; k++) { double temp = ZERO; for (int j = 0; j < n; j++) { temp += interpolationPoints.getEntry(k, j) * trialStepPoint.getEntry(j); } temp *= modelSecondDerivativesParameters.getEntry(k); for (int i = 0; i < n; i++) { gradientAtTrustRegionCenter.setEntry(i, gradientAtTrustRegionCenter.getEntry(i) + temp * interpolationPoints.getEntry(k, i)); } } } // Calculate the parameters of the least Frobenius norm interpolant to // the current data, the gradient of this interpolant at XOPT being put // into VLAG(NPT+I), I=1,2,...,N. if (ntrits > 0) { for (int k = 0; k < npt; k++) { lagrangeValuesAtNewPoint.setEntry(k, fAtInterpolationPoints.getEntry(k) - fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex)); work3.setEntry(k, ZERO); } for (int j = 0; j < nptm; j++) { double sum = ZERO; for (int k = 0; k < npt; k++) { sum += zMatrix.getEntry(k, j) * lagrangeValuesAtNewPoint.getEntry(k); } for (int k = 0; k < npt; k++) { work3.setEntry(k, work3.getEntry(k) + sum * zMatrix.getEntry(k, j)); } } for (int k = 0; k < npt; k++) { double sum = ZERO; for (int j = 0; j < n; j++) { sum += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j); } work2.setEntry(k, work3.getEntry(k)); work3.setEntry(k, sum * work3.getEntry(k)); } double gqsq = ZERO; double gisq = ZERO; for (int i = 0; i < n; i++) { double sum = ZERO; for (int k = 0; k < npt; k++) { sum += bMatrix.getEntry(k, i) * lagrangeValuesAtNewPoint.getEntry(k) + interpolationPoints.getEntry(k, i) * work3.getEntry(k); } if (trustRegionCenterOffset.getEntry(i) == lowerDifference.getEntry(i)) { // Computing MIN // Computing 2nd power final double d1 = FastMath.min(ZERO, gradientAtTrustRegionCenter.getEntry(i)); gqsq += d1 * d1; // Computing 2nd power final double d2 = FastMath.min(ZERO, sum); gisq += d2 * d2; } else if (trustRegionCenterOffset.getEntry(i) == upperDifference.getEntry(i)) { // Computing MAX // Computing 2nd power final double d1 = FastMath.max(ZERO, gradientAtTrustRegionCenter.getEntry(i)); gqsq += d1 * d1; // Computing 2nd power final double d2 = FastMath.max(ZERO, sum); gisq += d2 * d2; } else { // Computing 2nd power final double d1 = gradientAtTrustRegionCenter.getEntry(i); gqsq += d1 * d1; gisq += sum * sum; } lagrangeValuesAtNewPoint.setEntry(npt + i, sum); } // Test whether to replace the new quadratic model by the least Frobenius // norm interpolant, making the replacement if the test is satisfied. ++itest; if (gqsq < TEN * gisq) { itest = 0; } if (itest >= 3) { for (int i = 0, max = FastMath.max(npt, nh); i < max; i++) { if (i < n) { gradientAtTrustRegionCenter.setEntry(i, lagrangeValuesAtNewPoint.getEntry(npt + i)); } if (i < npt) { modelSecondDerivativesParameters.setEntry(i, work2.getEntry(i)); } if (i < nh) { modelSecondDerivativesValues.setEntry(i, ZERO); } itest = 0; } } } // If a trust region step has provided a sufficient decrease in F, then // branch for another trust region calculation. The case NTRITS=0 occurs // when the new interpolation point was reached by an alternative step. if (ntrits == 0) { state = 60; break; } if (f <= fopt + ONE_OVER_TEN * vquad) { state = 60; break; } // Alternatively, find out if the interpolation points are close enough // to the best point so far. // Computing MAX // Computing 2nd power final double d1 = TWO * delta; // Computing 2nd power final double d2 = TEN * rho; distsq = FastMath.max(d1 * d1, d2 * d2); } case 650: { printState(650); // XXX knew = -1; for (int k = 0; k < npt; k++) { double sum = ZERO; for (int j = 0; j < n; j++) { // Computing 2nd power final double d1 = interpolationPoints.getEntry(k, j) - trustRegionCenterOffset.getEntry(j); sum += d1 * d1; } if (sum > distsq) { knew = k; distsq = sum; } } // If KNEW is positive, then ALTMOV finds alternative new positions for // the KNEW-th interpolation point within distance ADELT of XOPT. It is // reached via label 90. Otherwise, there is a branch to label 60 for // another trust region iteration, unless the calculations with the // current RHO are complete. if (knew >= 0) { final double dist = FastMath.sqrt(distsq); if (ntrits == -1) { // Computing MIN delta = FastMath.min(ONE_OVER_TEN * delta, HALF * dist); if (delta <= rho * 1.5) { delta = rho; } } ntrits = 0; // Computing MAX // Computing MIN final double d1 = FastMath.min(ONE_OVER_TEN * dist, delta); adelt = FastMath.max(d1, rho); dsq = adelt * adelt; state = 90; break; } if (ntrits == -1) { state = 680; break; } if (ratio > ZERO) { state = 60; break; } if (FastMath.max(delta, dnorm) > rho) { state = 60; break; } // The calculations with the current value of RHO are complete. Pick the // next values of RHO and DELTA. } case 680: { printState(680); // XXX if (rho > stoppingTrustRegionRadius) { delta = HALF * rho; ratio = rho / stoppingTrustRegionRadius; if (ratio <= SIXTEEN) { rho = stoppingTrustRegionRadius; } else if (ratio <= TWO_HUNDRED_FIFTY) { rho = FastMath.sqrt(ratio) * stoppingTrustRegionRadius; } else { rho *= ONE_OVER_TEN; } delta = FastMath.max(delta, rho); ntrits = 0; nfsav = getEvaluations(); state = 60; break; } // Return from the calculation, after another Newton-Raphson step, if // it is too short to have been tried before. if (ntrits == -1) { state = 360; break; } } case 720: { printState(720); // XXX if (fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex) <= fsave) { for (int i = 0; i < n; i++) { // Computing MIN // Computing MAX final double d3 = lowerBound[i]; final double d4 = originShift.getEntry(i) + trustRegionCenterOffset.getEntry(i); final double d1 = FastMath.max(d3, d4); final double d2 = upperBound[i]; currentBest.setEntry(i, FastMath.min(d1, d2)); if (trustRegionCenterOffset.getEntry(i) == lowerDifference.getEntry(i)) { currentBest.setEntry(i, lowerBound[i]); } if (trustRegionCenterOffset.getEntry(i) == upperDifference.getEntry(i)) { currentBest.setEntry(i, upperBound[i]); } } f = fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex); } return f; } default: { throw new MathIllegalStateException(LocalizedFormats.SIMPLE_MESSAGE, "bobyqb"); }}} } // bobyqb // ---------------------------------------------------------------------------------------- /** * The arguments N, NPT, XPT, XOPT, BMAT, ZMAT, NDIM, SL and SU all have * the same meanings as the corresponding arguments of BOBYQB. * KOPT is the index of the optimal interpolation point. * KNEW is the index of the interpolation point that is going to be moved. * ADELT is the current trust region bound. * XNEW will be set to a suitable new position for the interpolation point * XPT(KNEW,.). Specifically, it satisfies the SL, SU and trust region * bounds and it should provide a large denominator in the next call of * UPDATE. The step XNEW-XOPT from XOPT is restricted to moves along the * straight lines through XOPT and another interpolation point. * XALT also provides a large value of the modulus of the KNEW-th Lagrange * function subject to the constraints that have been mentioned, its main * difference from XNEW being that XALT-XOPT is a constrained version of * the Cauchy step within the trust region. An exception is that XALT is * not calculated if all components of GLAG (see below) are zero. * ALPHA will be set to the KNEW-th diagonal element of the H matrix. * CAUCHY will be set to the square of the KNEW-th Lagrange function at * the step XALT-XOPT from XOPT for the vector XALT that is returned, * except that CAUCHY is set to zero if XALT is not calculated. * GLAG is a working space vector of length N for the gradient of the * KNEW-th Lagrange function at XOPT. * HCOL is a working space vector of length NPT for the second derivative * coefficients of the KNEW-th Lagrange function. * W is a working space vector of length 2N that is going to hold the * constrained Cauchy step from XOPT of the Lagrange function, followed * by the downhill version of XALT when the uphill step is calculated. * * Set the first NPT components of W to the leading elements of the * KNEW-th column of the H matrix. * @param knew * @param adelt */ private double[] altmov( int knew, double adelt ) { printMethod(); // XXX final int n = currentBest.getDimension(); final int npt = numberOfInterpolationPoints; final ArrayRealVector glag = new ArrayRealVector(n); final ArrayRealVector hcol = new ArrayRealVector(npt); final ArrayRealVector work1 = new ArrayRealVector(n); final ArrayRealVector work2 = new ArrayRealVector(n); for (int k = 0; k < npt; k++) { hcol.setEntry(k, ZERO); } for (int j = 0, max = npt - n - 1; j < max; j++) { final double tmp = zMatrix.getEntry(knew, j); for (int k = 0; k < npt; k++) { hcol.setEntry(k, hcol.getEntry(k) + tmp * zMatrix.getEntry(k, j)); } } final double alpha = hcol.getEntry(knew); final double ha = HALF * alpha; // Calculate the gradient of the KNEW-th Lagrange function at XOPT. for (int i = 0; i < n; i++) { glag.setEntry(i, bMatrix.getEntry(knew, i)); } for (int k = 0; k < npt; k++) { double tmp = ZERO; for (int j = 0; j < n; j++) { tmp += interpolationPoints.getEntry(k, j) * trustRegionCenterOffset.getEntry(j); } tmp *= hcol.getEntry(k); for (int i = 0; i < n; i++) { glag.setEntry(i, glag.getEntry(i) + tmp * interpolationPoints.getEntry(k, i)); } } // Search for a large denominator along the straight lines through XOPT // and another interpolation point. SLBD and SUBD will be lower and upper // bounds on the step along each of these lines in turn. PREDSQ will be // set to the square of the predicted denominator for each line. PRESAV // will be set to the largest admissible value of PREDSQ that occurs. double presav = ZERO; double step = Double.NaN; int ksav = 0; int ibdsav = 0; double stpsav = 0; for (int k = 0; k < npt; k++) { if (k == trustRegionCenterInterpolationPointIndex) { continue; } double dderiv = ZERO; double distsq = ZERO; for (int i = 0; i < n; i++) { final double tmp = interpolationPoints.getEntry(k, i) - trustRegionCenterOffset.getEntry(i); dderiv += glag.getEntry(i) * tmp; distsq += tmp * tmp; } double subd = adelt / FastMath.sqrt(distsq); double slbd = -subd; int ilbd = 0; int iubd = 0; final double sumin = FastMath.min(ONE, subd); // Revise SLBD and SUBD if necessary because of the bounds in SL and SU. for (int i = 0; i < n; i++) { final double tmp = interpolationPoints.getEntry(k, i) - trustRegionCenterOffset.getEntry(i); if (tmp > ZERO) { if (slbd * tmp < lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) { slbd = (lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp; ilbd = -i - 1; } if (subd * tmp > upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) { // Computing MAX subd = FastMath.max(sumin, (upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp); iubd = i + 1; } } else if (tmp < ZERO) { if (slbd * tmp > upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) { slbd = (upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp; ilbd = i + 1; } if (subd * tmp < lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) { // Computing MAX subd = FastMath.max(sumin, (lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)) / tmp); iubd = -i - 1; } } } // Seek a large modulus of the KNEW-th Lagrange function when the index // of the other interpolation point on the line through XOPT is KNEW. step = slbd; int isbd = ilbd; double vlag = Double.NaN; if (k == knew) { final double diff = dderiv - ONE; vlag = slbd * (dderiv - slbd * diff); final double d1 = subd * (dderiv - subd * diff); if (FastMath.abs(d1) > FastMath.abs(vlag)) { step = subd; vlag = d1; isbd = iubd; } final double d2 = HALF * dderiv; final double d3 = d2 - diff * slbd; final double d4 = d2 - diff * subd; if (d3 * d4 < ZERO) { final double d5 = d2 * d2 / diff; if (FastMath.abs(d5) > FastMath.abs(vlag)) { step = d2 / diff; vlag = d5; isbd = 0; } } // Search along each of the other lines through XOPT and another point. } else { vlag = slbd * (ONE - slbd); final double tmp = subd * (ONE - subd); if (FastMath.abs(tmp) > FastMath.abs(vlag)) { step = subd; vlag = tmp; isbd = iubd; } if (subd > HALF && FastMath.abs(vlag) < ONE_OVER_FOUR) { step = HALF; vlag = ONE_OVER_FOUR; isbd = 0; } vlag *= dderiv; } // Calculate PREDSQ for the current line search and maintain PRESAV. final double tmp = step * (ONE - step) * distsq; final double predsq = vlag * vlag * (vlag * vlag + ha * tmp * tmp); if (predsq > presav) { presav = predsq; ksav = k; stpsav = step; ibdsav = isbd; } } // Construct XNEW in a way that satisfies the bound constraints exactly. for (int i = 0; i < n; i++) { final double tmp = trustRegionCenterOffset.getEntry(i) + stpsav * (interpolationPoints.getEntry(ksav, i) - trustRegionCenterOffset.getEntry(i)); newPoint.setEntry(i, FastMath.max(lowerDifference.getEntry(i), FastMath.min(upperDifference.getEntry(i), tmp))); } if (ibdsav < 0) { newPoint.setEntry(-ibdsav - 1, lowerDifference.getEntry(-ibdsav - 1)); } if (ibdsav > 0) { newPoint.setEntry(ibdsav - 1, upperDifference.getEntry(ibdsav - 1)); } // Prepare for the iterative method that assembles the constrained Cauchy // step in W. The sum of squares of the fixed components of W is formed in // WFIXSQ, and the free components of W are set to BIGSTP. final double bigstp = adelt + adelt; int iflag = 0; double cauchy = Double.NaN; double csave = ZERO; while (true) { double wfixsq = ZERO; double ggfree = ZERO; for (int i = 0; i < n; i++) { final double glagValue = glag.getEntry(i); work1.setEntry(i, ZERO); if (FastMath.min(trustRegionCenterOffset.getEntry(i) - lowerDifference.getEntry(i), glagValue) > ZERO || FastMath.max(trustRegionCenterOffset.getEntry(i) - upperDifference.getEntry(i), glagValue) < ZERO) { work1.setEntry(i, bigstp); // Computing 2nd power ggfree += glagValue * glagValue; } } if (ggfree == ZERO) { return new double[] { alpha, ZERO }; } // Investigate whether more components of W can be fixed. final double tmp1 = adelt * adelt - wfixsq; if (tmp1 > ZERO) { step = FastMath.sqrt(tmp1 / ggfree); ggfree = ZERO; for (int i = 0; i < n; i++) { if (work1.getEntry(i) == bigstp) { final double tmp2 = trustRegionCenterOffset.getEntry(i) - step * glag.getEntry(i); if (tmp2 <= lowerDifference.getEntry(i)) { work1.setEntry(i, lowerDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)); // Computing 2nd power final double d1 = work1.getEntry(i); wfixsq += d1 * d1; } else if (tmp2 >= upperDifference.getEntry(i)) { work1.setEntry(i, upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i)); // Computing 2nd power final double d1 = work1.getEntry(i); wfixsq += d1 * d1; } else { // Computing 2nd power final double d1 = glag.getEntry(i); ggfree += d1 * d1; } } } } // Set the remaining free components of W and all components of XALT, // except that W may be scaled later. double gw = ZERO; for (int i = 0; i < n; i++) { final double glagValue = glag.getEntry(i); if (work1.getEntry(i) == bigstp) { work1.setEntry(i, -step * glagValue); final double min = FastMath.min(upperDifference.getEntry(i), trustRegionCenterOffset.getEntry(i) + work1.getEntry(i)); alternativeNewPoint.setEntry(i, FastMath.max(lowerDifference.getEntry(i), min)); } else if (work1.getEntry(i) == ZERO) { alternativeNewPoint.setEntry(i, trustRegionCenterOffset.getEntry(i)); } else if (glagValue > ZERO) { alternativeNewPoint.setEntry(i, lowerDifference.getEntry(i)); } else { alternativeNewPoint.setEntry(i, upperDifference.getEntry(i)); } gw += glagValue * work1.getEntry(i); } // Set CURV to the curvature of the KNEW-th Lagrange function along W. // Scale W by a factor less than one if that can reduce the modulus of // the Lagrange function at XOPT+W. Set CAUCHY to the final value of // the square of this function. double curv = ZERO; for (int k = 0; k < npt; k++) { double tmp = ZERO; for (int j = 0; j < n; j++) { tmp += interpolationPoints.getEntry(k, j) * work1.getEntry(j); } curv += hcol.getEntry(k) * tmp * tmp; } if (iflag == 1) { curv = -curv; } if (curv > -gw && curv < -gw * (ONE + FastMath.sqrt(TWO))) { final double scale = -gw / curv; for (int i = 0; i < n; i++) { final double tmp = trustRegionCenterOffset.getEntry(i) + scale * work1.getEntry(i); alternativeNewPoint.setEntry(i, FastMath.max(lowerDifference.getEntry(i), FastMath.min(upperDifference.getEntry(i), tmp))); } // Computing 2nd power final double d1 = HALF * gw * scale; cauchy = d1 * d1; } else { // Computing 2nd power final double d1 = gw + HALF * curv; cauchy = d1 * d1; } // If IFLAG is zero, then XALT is calculated as before after reversing // the sign of GLAG. Thus two XALT vectors become available. The one that // is chosen is the one that gives the larger value of CAUCHY. if (iflag == 0) { for (int i = 0; i < n; i++) { glag.setEntry(i, -glag.getEntry(i)); work2.setEntry(i, alternativeNewPoint.getEntry(i)); } csave = cauchy; iflag = 1; } else { break; } } if (csave > cauchy) { for (int i = 0; i < n; i++) { alternativeNewPoint.setEntry(i, work2.getEntry(i)); } cauchy = csave; } return new double[] { alpha, cauchy }; } // altmov // ---------------------------------------------------------------------------------------- /** * SUBROUTINE PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ, * BMAT and ZMAT for the first iteration, and it maintains the values of * NF and KOPT. The vector X is also changed by PRELIM. * * The arguments N, NPT, X, XL, XU, RHOBEG, IPRINT and MAXFUN are the * same as the corresponding arguments in SUBROUTINE BOBYQA. * The arguments XBASE, XPT, FVAL, HQ, PQ, BMAT, ZMAT, NDIM, SL and SU * are the same as the corresponding arguments in BOBYQB, the elements * of SL and SU being set in BOBYQA. * GOPT is usually the gradient of the quadratic model at XOPT+XBASE, but * it is set by PRELIM to the gradient of the quadratic model at XBASE. * If XOPT is nonzero, BOBYQB will change it to its usual value later. * NF is maintaned as the number of calls of CALFUN so far. * KOPT will be such that the least calculated value of F so far is at * the point XPT(KOPT,.)+XBASE in the space of the variables. * * @param lowerBound Lower bounds. * @param upperBound Upper bounds. */ private void prelim(double[] lowerBound, double[] upperBound) { printMethod(); // XXX final int n = currentBest.getDimension(); final int npt = numberOfInterpolationPoints; final int ndim = bMatrix.getRowDimension(); final double rhosq = initialTrustRegionRadius * initialTrustRegionRadius; final double recip = 1d / rhosq; final int np = n + 1; // Set XBASE to the initial vector of variables, and set the initial // elements of XPT, BMAT, HQ, PQ and ZMAT to zero. for (int j = 0; j < n; j++) { originShift.setEntry(j, currentBest.getEntry(j)); for (int k = 0; k < npt; k++) { interpolationPoints.setEntry(k, j, ZERO); } for (int i = 0; i < ndim; i++) { bMatrix.setEntry(i, j, ZERO); } } for (int i = 0, max = n * np / 2; i < max; i++) { modelSecondDerivativesValues.setEntry(i, ZERO); } for (int k = 0; k < npt; k++) { modelSecondDerivativesParameters.setEntry(k, ZERO); for (int j = 0, max = npt - np; j < max; j++) { zMatrix.setEntry(k, j, ZERO); } } // Begin the initialization procedure. NF becomes one more than the number // of function values so far. The coordinates of the displacement of the // next initial interpolation point from XBASE are set in XPT(NF+1,.). int ipt = 0; int jpt = 0; double fbeg = Double.NaN; do { final int nfm = getEvaluations(); final int nfx = nfm - n; final int nfmm = nfm - 1; final int nfxm = nfx - 1; double stepa = 0; double stepb = 0; if (nfm <= 2 * n) { if (nfm >= 1 && nfm <= n) { stepa = initialTrustRegionRadius; if (upperDifference.getEntry(nfmm) == ZERO) { stepa = -stepa; // throw new PathIsExploredException(); // XXX } interpolationPoints.setEntry(nfm, nfmm, stepa); } else if (nfm > n) { stepa = interpolationPoints.getEntry(nfx, nfxm); stepb = -initialTrustRegionRadius; if (lowerDifference.getEntry(nfxm) == ZERO) { stepb = FastMath.min(TWO * initialTrustRegionRadius, upperDifference.getEntry(nfxm)); // throw new PathIsExploredException(); // XXX } if (upperDifference.getEntry(nfxm) == ZERO) { stepb = FastMath.max(-TWO * initialTrustRegionRadius, lowerDifference.getEntry(nfxm)); // throw new PathIsExploredException(); // XXX } interpolationPoints.setEntry(nfm, nfxm, stepb); } } else { final int tmp1 = (nfm - np) / n; jpt = nfm - tmp1 * n - n; ipt = jpt + tmp1; if (ipt > n) { final int tmp2 = jpt; jpt = ipt - n; ipt = tmp2; // throw new PathIsExploredException(); // XXX } final int iptMinus1 = ipt - 1; final int jptMinus1 = jpt - 1; interpolationPoints.setEntry(nfm, iptMinus1, interpolationPoints.getEntry(ipt, iptMinus1)); interpolationPoints.setEntry(nfm, jptMinus1, interpolationPoints.getEntry(jpt, jptMinus1)); } // Calculate the next value of F. The least function value so far and // its index are required. for (int j = 0; j < n; j++) { currentBest.setEntry(j, FastMath.min(FastMath.max(lowerBound[j], originShift.getEntry(j) + interpolationPoints.getEntry(nfm, j)), upperBound[j])); if (interpolationPoints.getEntry(nfm, j) == lowerDifference.getEntry(j)) { currentBest.setEntry(j, lowerBound[j]); } if (interpolationPoints.getEntry(nfm, j) == upperDifference.getEntry(j)) { currentBest.setEntry(j, upperBound[j]); } } final double objectiveValue = computeObjectiveValue(currentBest.toArray()); final double f = isMinimize ? objectiveValue : -objectiveValue; final int numEval = getEvaluations(); // nfm + 1 fAtInterpolationPoints.setEntry(nfm, f); if (numEval == 1) { fbeg = f; trustRegionCenterInterpolationPointIndex = 0; } else if (f < fAtInterpolationPoints.getEntry(trustRegionCenterInterpolationPointIndex)) { trustRegionCenterInterpolationPointIndex = nfm; } // Set the nonzero initial elements of BMAT and the quadratic model in the // cases when NF is at most 2*N+1. If NF exceeds N+1, then the positions // of the NF-th and (NF-N)-th interpolation points may be switched, in // order that the function value at the first of them contributes to the // off-diagonal second derivative terms of the initial quadratic model. if (numEval <= 2 * n + 1) { if (numEval >= 2 && numEval <= n + 1) { gradientAtTrustRegionCenter.setEntry(nfmm, (f - fbeg) / stepa); if (npt < numEval + n) { final double oneOverStepA = ONE / stepa; bMatrix.setEntry(0, nfmm, -oneOverStepA); bMatrix.setEntry(nfm, nfmm, oneOverStepA); bMatrix.setEntry(npt + nfmm, nfmm, -HALF * rhosq); // throw new PathIsExploredException(); // XXX } } else if (numEval >= n + 2) { final int ih = nfx * (nfx + 1) / 2 - 1; final double tmp = (f - fbeg) / stepb; final double diff = stepb - stepa; modelSecondDerivativesValues.setEntry(ih, TWO * (tmp - gradientAtTrustRegionCenter.getEntry(nfxm)) / diff); gradientAtTrustRegionCenter.setEntry(nfxm, (gradientAtTrustRegionCenter.getEntry(nfxm) * stepb - tmp * stepa) / diff); if (stepa * stepb < ZERO && f < fAtInterpolationPoints.getEntry(nfm - n)) { fAtInterpolationPoints.setEntry(nfm, fAtInterpolationPoints.getEntry(nfm - n)); fAtInterpolationPoints.setEntry(nfm - n, f); if (trustRegionCenterInterpolationPointIndex == nfm) { trustRegionCenterInterpolationPointIndex = nfm - n; } interpolationPoints.setEntry(nfm - n, nfxm, stepb); interpolationPoints.setEntry(nfm, nfxm, stepa); } bMatrix.setEntry(0, nfxm, -(stepa + stepb) / (stepa * stepb)); bMatrix.setEntry(nfm, nfxm, -HALF / interpolationPoints.getEntry(nfm - n, nfxm)); bMatrix.setEntry(nfm - n, nfxm, -bMatrix.getEntry(0, nfxm) - bMatrix.getEntry(nfm, nfxm)); zMatrix.setEntry(0, nfxm, FastMath.sqrt(TWO) / (stepa * stepb)); zMatrix.setEntry(nfm, nfxm, FastMath.sqrt(HALF) / rhosq); // zMatrix.setEntry(nfm, nfxm, FastMath.sqrt(HALF) * recip); // XXX "testAckley" and "testDiffPow" fail. zMatrix.setEntry(nfm - n, nfxm, -zMatrix.getEntry(0, nfxm) - zMatrix.getEntry(nfm, nfxm)); } // Set the off-diagonal second derivatives of the Lagrange functions and // the initial quadratic model. } else { zMatrix.setEntry(0, nfxm, recip); zMatrix.setEntry(nfm, nfxm, recip); zMatrix.setEntry(ipt, nfxm, -recip); zMatrix.setEntry(jpt, nfxm, -recip); final int ih = ipt * (ipt - 1) / 2 + jpt - 1; final double tmp = interpolationPoints.getEntry(nfm, ipt - 1) * interpolationPoints.getEntry(nfm, jpt - 1); modelSecondDerivativesValues.setEntry(ih, (fbeg - fAtInterpolationPoints.getEntry(ipt) - fAtInterpolationPoints.getEntry(jpt) + f) / tmp); // throw new PathIsExploredException(); // XXX } } while (getEvaluations() < npt); } // prelim // ---------------------------------------------------------------------------------------- /** * A version of the truncated conjugate gradient is applied. If a line * search is restricted by a constraint, then the procedure is restarted, * the values of the variables that are at their bounds being fixed. If * the trust region boundary is reached, then further changes may be made * to D, each one being in the two dimensional space that is spanned * by the current D and the gradient of Q at XOPT+D, staying on the trust * region boundary. Termination occurs when the reduction in Q seems to * be close to the greatest reduction that can be achieved. * The arguments N, NPT, XPT, XOPT, GOPT, HQ, PQ, SL and SU have the same * meanings as the corresponding arguments of BOBYQB. * DELTA is the trust region radius for the present calculation, which * seeks a small value of the quadratic model within distance DELTA of * XOPT subject to the bounds on the variables. * XNEW will be set to a new vector of variables that is approximately * the one that minimizes the quadratic model within the trust region * subject to the SL and SU constraints on the variables. It satisfies * as equations the bounds that become active during the calculation. * D is the calculated trial step from XOPT, generated iteratively from an * initial value of zero. Thus XNEW is XOPT+D after the final iteration. * GNEW holds the gradient of the quadratic model at XOPT+D. It is updated * when D is updated. * xbdi.get( is a working space vector. For I=1,2,...,N, the element xbdi.get((I) is * set to -1.0, 0.0, or 1.0, the value being nonzero if and only if the * I-th variable has become fixed at a bound, the bound being SL(I) or * SU(I) in the case xbdi.get((I)=-1.0 or xbdi.get((I)=1.0, respectively. This * information is accumulated during the construction of XNEW. * The arrays S, HS and HRED are also used for working space. They hold the * current search direction, and the changes in the gradient of Q along S * and the reduced D, respectively, where the reduced D is the same as D, * except that the components of the fixed variables are zero. * DSQ will be set to the square of the length of XNEW-XOPT. * CRVMIN is set to zero if D reaches the trust region boundary. Otherwise * it is set to the least curvature of H that occurs in the conjugate * gradient searches that are not restricted by any constraints. The * value CRVMIN=-1.0D0 is set, however, if all of these searches are * constrained. * @param delta * @param gnew * @param xbdi * @param s * @param hs * @param hred */ private double[] trsbox( double delta, ArrayRealVector gnew, ArrayRealVector xbdi, ArrayRealVector s, ArrayRealVector hs, ArrayRealVector hred ) { printMethod(); // XXX final int n = currentBest.getDimension(); final int npt = numberOfInterpolationPoints; double dsq = Double.NaN; double crvmin = Double.NaN; // Local variables double ds; int iu; double dhd, dhs, cth, shs, sth, ssq, beta=0, sdec, blen; int iact = -1; int nact = 0; double angt = 0, qred; int isav; double temp = 0, xsav = 0, xsum = 0, angbd = 0, dredg = 0, sredg = 0; int iterc; double resid = 0, delsq = 0, ggsav = 0, tempa = 0, tempb = 0, redmax = 0, dredsq = 0, redsav = 0, gredsq = 0, rednew = 0; int itcsav = 0; double rdprev = 0, rdnext = 0, stplen = 0, stepsq = 0; int itermax = 0; // Set some constants. // Function Body // The sign of GOPT(I) gives the sign of the change to the I-th variable // that will reduce Q from its value at XOPT. Thus xbdi.get((I) shows whether // or not to fix the I-th variable at one of its bounds initially, with // NACT being set to the number of fixed variables. D and GNEW are also // set for the first iteration. DELSQ is the upper bound on the sum of // squares of the free variables. QRED is the reduction in Q so far. iterc = 0; nact = 0; for (int i = 0; i < n; i++) { xbdi.setEntry(i, ZERO); if (trustRegionCenterOffset.getEntry(i) <= lowerDifference.getEntry(i)) { if (gradientAtTrustRegionCenter.getEntry(i) >= ZERO) { xbdi.setEntry(i, MINUS_ONE); } } else if (trustRegionCenterOffset.getEntry(i) >= upperDifference.getEntry(i) && gradientAtTrustRegionCenter.getEntry(i) <= ZERO) { xbdi.setEntry(i, ONE); } if (xbdi.getEntry(i) != ZERO) { ++nact; } trialStepPoint.setEntry(i, ZERO); gnew.setEntry(i, gradientAtTrustRegionCenter.getEntry(i)); } delsq = delta * delta; qred = ZERO; crvmin = MINUS_ONE; // Set the next search direction of the conjugate gradient method. It is // the steepest descent direction initially and when the iterations are // restarted because a variable has just been fixed by a bound, and of // course the components of the fixed variables are zero. ITERMAX is an // upper bound on the indices of the conjugate gradient iterations. int state = 20; for(;;) { switch (state) { case 20: { printState(20); // XXX beta = ZERO; } case 30: { printState(30); // XXX stepsq = ZERO; for (int i = 0; i < n; i++) { if (xbdi.getEntry(i) != ZERO) { s.setEntry(i, ZERO); } else if (beta == ZERO) { s.setEntry(i, -gnew.getEntry(i)); } else { s.setEntry(i, beta * s.getEntry(i) - gnew.getEntry(i)); } // Computing 2nd power final double d1 = s.getEntry(i); stepsq += d1 * d1; } if (stepsq == ZERO) { state = 190; break; } if (beta == ZERO) { gredsq = stepsq; itermax = iterc + n - nact; } if (gredsq * delsq <= qred * 1e-4 * qred) { state = 190; break; } // Multiply the search direction by the second derivative matrix of Q and // calculate some scalars for the choice of steplength. Then set BLEN to // the length of the the step to the trust region boundary and STPLEN to // the steplength, ignoring the simple bounds. state = 210; break; } case 50: { printState(50); // XXX resid = delsq; ds = ZERO; shs = ZERO; for (int i = 0; i < n; i++) { if (xbdi.getEntry(i) == ZERO) { // Computing 2nd power final double d1 = trialStepPoint.getEntry(i); resid -= d1 * d1; ds += s.getEntry(i) * trialStepPoint.getEntry(i); shs += s.getEntry(i) * hs.getEntry(i); } } if (resid <= ZERO) { state = 90; break; } temp = FastMath.sqrt(stepsq * resid + ds * ds); if (ds < ZERO) { blen = (temp - ds) / stepsq; } else { blen = resid / (temp + ds); } stplen = blen; if (shs > ZERO) { // Computing MIN stplen = FastMath.min(blen, gredsq / shs); } // Reduce STPLEN if necessary in order to preserve the simple bounds, // letting IACT be the index of the new constrained variable. iact = -1; for (int i = 0; i < n; i++) { if (s.getEntry(i) != ZERO) { xsum = trustRegionCenterOffset.getEntry(i) + trialStepPoint.getEntry(i); if (s.getEntry(i) > ZERO) { temp = (upperDifference.getEntry(i) - xsum) / s.getEntry(i); } else { temp = (lowerDifference.getEntry(i) - xsum) / s.getEntry(i); } if (temp < stplen) { stplen = temp; iact = i; } } } // Update CRVMIN, GNEW and D. Set SDEC to the decrease that occurs in Q. sdec = ZERO; if (stplen > ZERO) { ++iterc; temp = shs / stepsq; if (iact == -1 && temp > ZERO) { crvmin = FastMath.min(crvmin,temp); if (crvmin == MINUS_ONE) { crvmin = temp; } } ggsav = gredsq; gredsq = ZERO; for (int i = 0; i < n; i++) { gnew.setEntry(i, gnew.getEntry(i) + stplen * hs.getEntry(i)); if (xbdi.getEntry(i) == ZERO) { // Computing 2nd power final double d1 = gnew.getEntry(i); gredsq += d1 * d1; } trialStepPoint.setEntry(i, trialStepPoint.getEntry(i) + stplen * s.getEntry(i)); } // Computing MAX final double d1 = stplen * (ggsav - HALF * stplen * shs); sdec = FastMath.max(d1, ZERO); qred += sdec; } // Restart the conjugate gradient method if it has hit a new bound. if (iact >= 0) { ++nact; xbdi.setEntry(iact, ONE); if (s.getEntry(iact) < ZERO) { xbdi.setEntry(iact, MINUS_ONE); } // Computing 2nd power final double d1 = trialStepPoint.getEntry(iact); delsq -= d1 * d1; if (delsq <= ZERO) { state = 190; break; } state = 20; break; } // If STPLEN is less than BLEN, then either apply another conjugate // gradient iteration or RETURN. if (stplen < blen) { if (iterc == itermax) { state = 190; break; } if (sdec <= qred * .01) { state = 190; break; } beta = gredsq / ggsav; state = 30; break; } } case 90: { printState(90); // XXX crvmin = ZERO; // Prepare for the alternative iteration by calculating some scalars // and by multiplying the reduced D by the second derivative matrix of // Q, where S holds the reduced D in the call of GGMULT. } case 100: { printState(100); // XXX if (nact >= n - 1) { state = 190; break; } dredsq = ZERO; dredg = ZERO; gredsq = ZERO; for (int i = 0; i < n; i++) { if (xbdi.getEntry(i) == ZERO) { // Computing 2nd power double d1 = trialStepPoint.getEntry(i); dredsq += d1 * d1; dredg += trialStepPoint.getEntry(i) * gnew.getEntry(i); // Computing 2nd power d1 = gnew.getEntry(i); gredsq += d1 * d1; s.setEntry(i, trialStepPoint.getEntry(i)); } else { s.setEntry(i, ZERO); } } itcsav = iterc; state = 210; break; // Let the search direction S be a linear combination of the reduced D // and the reduced G that is orthogonal to the reduced D. } case 120: { printState(120); // XXX ++iterc; temp = gredsq * dredsq - dredg * dredg; if (temp <= qred * 1e-4 * qred) { state = 190; break; } temp = FastMath.sqrt(temp); for (int i = 0; i < n; i++) { if (xbdi.getEntry(i) == ZERO) { s.setEntry(i, (dredg * trialStepPoint.getEntry(i) - dredsq * gnew.getEntry(i)) / temp); } else { s.setEntry(i, ZERO); } } sredg = -temp; // By considering the simple bounds on the variables, calculate an upper // bound on the tangent of half the angle of the alternative iteration, // namely ANGBD, except that, if already a free variable has reached a // bound, there is a branch back to label 100 after fixing that variable. angbd = ONE; iact = -1; for (int i = 0; i < n; i++) { if (xbdi.getEntry(i) == ZERO) { tempa = trustRegionCenterOffset.getEntry(i) + trialStepPoint.getEntry(i) - lowerDifference.getEntry(i); tempb = upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i) - trialStepPoint.getEntry(i); if (tempa <= ZERO) { ++nact; xbdi.setEntry(i, MINUS_ONE); state = 100; break; } else if (tempb <= ZERO) { ++nact; xbdi.setEntry(i, ONE); state = 100; break; } // Computing 2nd power double d1 = trialStepPoint.getEntry(i); // Computing 2nd power double d2 = s.getEntry(i); ssq = d1 * d1 + d2 * d2; // Computing 2nd power d1 = trustRegionCenterOffset.getEntry(i) - lowerDifference.getEntry(i); temp = ssq - d1 * d1; if (temp > ZERO) { temp = FastMath.sqrt(temp) - s.getEntry(i); if (angbd * temp > tempa) { angbd = tempa / temp; iact = i; xsav = MINUS_ONE; } } // Computing 2nd power d1 = upperDifference.getEntry(i) - trustRegionCenterOffset.getEntry(i); temp = ssq - d1 * d1; if (temp > ZERO) { temp = FastMath.sqrt(temp) + s.getEntry(i); if (angbd * temp > tempb) { angbd = tempb / temp; iact = i; xsav = ONE; } } } } // Calculate HHD and some curvatures for the alternative iteration. state = 210; break; } case 150: { printState(150); // XXX shs = ZERO; dhs = ZERO; dhd = ZERO; for (int i = 0; i < n; i++) { if (xbdi.getEntry(i) == ZERO) { shs += s.getEntry(i) * hs.getEntry(i); dhs += trialStepPoint.getEntry(i) * hs.getEntry(i); dhd += trialStepPoint.getEntry(i) * hred.getEntry(i); } } // Seek the greatest reduction in Q for a range of equally spaced values // of ANGT in [0,ANGBD], where ANGT is the tangent of half the angle of // the alternative iteration. redmax = ZERO; isav = -1; redsav = ZERO; iu = (int) (angbd * 17. + 3.1); for (int i = 0; i < iu; i++) { angt = angbd * i / iu; sth = (angt + angt) / (ONE + angt * angt); temp = shs + angt * (angt * dhd - dhs - dhs); rednew = sth * (angt * dredg - sredg - HALF * sth * temp); if (rednew > redmax) { redmax = rednew; isav = i; rdprev = redsav; } else if (i == isav + 1) { rdnext = rednew; } redsav = rednew; } // Return if the reduction is zero. Otherwise, set the sine and cosine // of the angle of the alternative iteration, and calculate SDEC. if (isav < 0) { state = 190; break; } if (isav < iu) { temp = (rdnext - rdprev) / (redmax + redmax - rdprev - rdnext); angt = angbd * (isav + HALF * temp) / iu; } cth = (ONE - angt * angt) / (ONE + angt * angt); sth = (angt + angt) / (ONE + angt * angt); temp = shs + angt * (angt * dhd - dhs - dhs); sdec = sth * (angt * dredg - sredg - HALF * sth * temp); if (sdec <= ZERO) { state = 190; break; } // Update GNEW, D and HRED. If the angle of the alternative iteration // is restricted by a bound on a free variable, that variable is fixed // at the bound. dredg = ZERO; gredsq = ZERO; for (int i = 0; i < n; i++) { gnew.setEntry(i, gnew.getEntry(i) + (cth - ONE) * hred.getEntry(i) + sth * hs.getEntry(i)); if (xbdi.getEntry(i) == ZERO) { trialStepPoint.setEntry(i, cth * trialStepPoint.getEntry(i) + sth * s.getEntry(i)); dredg += trialStepPoint.getEntry(i) * gnew.getEntry(i); // Computing 2nd power final double d1 = gnew.getEntry(i); gredsq += d1 * d1; } hred.setEntry(i, cth * hred.getEntry(i) + sth * hs.getEntry(i)); } qred += sdec; if (iact >= 0 && isav == iu) { ++nact; xbdi.setEntry(iact, xsav); state = 100; break; } // If SDEC is sufficiently small, then RETURN after setting XNEW to // XOPT+D, giving careful attention to the bounds. if (sdec > qred * .01) { state = 120; break; } } case 190: { printState(190); // XXX dsq = ZERO; for (int i = 0; i < n; i++) { // Computing MAX // Computing MIN final double min = FastMath.min(trustRegionCenterOffset.getEntry(i) + trialStepPoint.getEntry(i), upperDifference.getEntry(i)); newPoint.setEntry(i, FastMath.max(min, lowerDifference.getEntry(i))); if (xbdi.getEntry(i) == MINUS_ONE) { newPoint.setEntry(i, lowerDifference.getEntry(i)); } if (xbdi.getEntry(i) == ONE) { newPoint.setEntry(i, upperDifference.getEntry(i)); } trialStepPoint.setEntry(i, newPoint.getEntry(i) - trustRegionCenterOffset.getEntry(i)); // Computing 2nd power final double d1 = trialStepPoint.getEntry(i); dsq += d1 * d1; } return new double[] { dsq, crvmin }; // The following instructions multiply the current S-vector by the second // derivative matrix of the quadratic model, putting the product in HS. // They are reached from three different parts of the software above and // they can be regarded as an external subroutine. } case 210: { printState(210); // XXX int ih = 0; for (int j = 0; j < n; j++) { hs.setEntry(j, ZERO); for (int i = 0; i <= j; i++) { if (i < j) { hs.setEntry(j, hs.getEntry(j) + modelSecondDerivativesValues.getEntry(ih) * s.getEntry(i)); } hs.setEntry(i, hs.getEntry(i) + modelSecondDerivativesValues.getEntry(ih) * s.getEntry(j)); ih++; } } final RealVector tmp = interpolationPoints.operate(s).ebeMultiply(modelSecondDerivativesParameters); for (int k = 0; k < npt; k++) { if (modelSecondDerivativesParameters.getEntry(k) != ZERO) { for (int i = 0; i < n; i++) { hs.setEntry(i, hs.getEntry(i) + tmp.getEntry(k) * interpolationPoints.getEntry(k, i)); } } } if (crvmin != ZERO) { state = 50; break; } if (iterc > itcsav) { state = 150; break; } for (int i = 0; i < n; i++) { hred.setEntry(i, hs.getEntry(i)); } state = 120; break; } default: { throw new MathIllegalStateException(LocalizedFormats.SIMPLE_MESSAGE, "trsbox"); }} } } // trsbox // ---------------------------------------------------------------------------------------- /** * The arrays BMAT and ZMAT are updated, as required by the new position * of the interpolation point that has the index KNEW. The vector VLAG has * N+NPT components, set on entry to the first NPT and last N components * of the product Hw in equation (4.11) of the Powell (2006) paper on * NEWUOA. Further, BETA is set on entry to the value of the parameter * with that name, and DENOM is set to the denominator of the updating * formula. Elements of ZMAT may be treated as zero if their moduli are * at most ZTEST. The first NDIM elements of W are used for working space. * @param beta * @param denom * @param knew */ private void update( double beta, double denom, int knew ) { printMethod(); // XXX final int n = currentBest.getDimension(); final int npt = numberOfInterpolationPoints; final int nptm = npt - n - 1; // XXX Should probably be split into two arrays. final ArrayRealVector work = new ArrayRealVector(npt + n); double ztest = ZERO; for (int k = 0; k < npt; k++) { for (int j = 0; j < nptm; j++) { // Computing MAX ztest = FastMath.max(ztest, FastMath.abs(zMatrix.getEntry(k, j))); } } ztest *= 1e-20; // Apply the rotations that put zeros in the KNEW-th row of ZMAT. for (int j = 1; j < nptm; j++) { final double d1 = zMatrix.getEntry(knew, j); if (FastMath.abs(d1) > ztest) { // Computing 2nd power final double d2 = zMatrix.getEntry(knew, 0); // Computing 2nd power final double d3 = zMatrix.getEntry(knew, j); final double d4 = FastMath.sqrt(d2 * d2 + d3 * d3); final double d5 = zMatrix.getEntry(knew, 0) / d4; final double d6 = zMatrix.getEntry(knew, j) / d4; for (int i = 0; i < npt; i++) { final double d7 = d5 * zMatrix.getEntry(i, 0) + d6 * zMatrix.getEntry(i, j); zMatrix.setEntry(i, j, d5 * zMatrix.getEntry(i, j) - d6 * zMatrix.getEntry(i, 0)); zMatrix.setEntry(i, 0, d7); } } zMatrix.setEntry(knew, j, ZERO); } // Put the first NPT components of the KNEW-th column of HLAG into W, // and calculate the parameters of the updating formula. for (int i = 0; i < npt; i++) { work.setEntry(i, zMatrix.getEntry(knew, 0) * zMatrix.getEntry(i, 0)); } final double alpha = work.getEntry(knew); final double tau = lagrangeValuesAtNewPoint.getEntry(knew); lagrangeValuesAtNewPoint.setEntry(knew, lagrangeValuesAtNewPoint.getEntry(knew) - ONE); // Complete the updating of ZMAT. final double sqrtDenom = FastMath.sqrt(denom); final double d1 = tau / sqrtDenom; final double d2 = zMatrix.getEntry(knew, 0) / sqrtDenom; for (int i = 0; i < npt; i++) { zMatrix.setEntry(i, 0, d1 * zMatrix.getEntry(i, 0) - d2 * lagrangeValuesAtNewPoint.getEntry(i)); } // Finally, update the matrix BMAT. for (int j = 0; j < n; j++) { final int jp = npt + j; work.setEntry(jp, bMatrix.getEntry(knew, j)); final double d3 = (alpha * lagrangeValuesAtNewPoint.getEntry(jp) - tau * work.getEntry(jp)) / denom; final double d4 = (-beta * work.getEntry(jp) - tau * lagrangeValuesAtNewPoint.getEntry(jp)) / denom; for (int i = 0; i <= jp; i++) { bMatrix.setEntry(i, j, bMatrix.getEntry(i, j) + d3 * lagrangeValuesAtNewPoint.getEntry(i) + d4 * work.getEntry(i)); if (i >= npt) { bMatrix.setEntry(jp, (i - npt), bMatrix.getEntry(i, j)); } } } } // update /** * Performs validity checks. * * @param lowerBound Lower bounds (constraints) of the objective variables. * @param upperBound Upperer bounds (constraints) of the objective variables. */ private void setup(double[] lowerBound, double[] upperBound) { printMethod(); // XXX double[] init = getStartPoint(); final int dimension = init.length; // Check problem dimension. if (dimension < MINIMUM_PROBLEM_DIMENSION) { throw new NumberIsTooSmallException(dimension, MINIMUM_PROBLEM_DIMENSION, true); } // Check number of interpolation points. final int[] nPointsInterval = { dimension + 2, (dimension + 2) * (dimension + 1) / 2 }; if (numberOfInterpolationPoints < nPointsInterval[0] || numberOfInterpolationPoints > nPointsInterval[1]) { throw new OutOfRangeException(LocalizedFormats.NUMBER_OF_INTERPOLATION_POINTS, numberOfInterpolationPoints, nPointsInterval[0], nPointsInterval[1]); } // Initialize bound differences. boundDifference = new double[dimension]; double requiredMinDiff = 2 * initialTrustRegionRadius; double minDiff = Double.POSITIVE_INFINITY; for (int i = 0; i < dimension; i++) { boundDifference[i] = upperBound[i] - lowerBound[i]; minDiff = FastMath.min(minDiff, boundDifference[i]); } if (minDiff < requiredMinDiff) { initialTrustRegionRadius = minDiff / 3.0; } // Initialize the data structures used by the "bobyqa" method. bMatrix = new Array2DRowRealMatrix(dimension + numberOfInterpolationPoints, dimension); zMatrix = new Array2DRowRealMatrix(numberOfInterpolationPoints, numberOfInterpolationPoints - dimension - 1); interpolationPoints = new Array2DRowRealMatrix(numberOfInterpolationPoints, dimension); originShift = new ArrayRealVector(dimension); fAtInterpolationPoints = new ArrayRealVector(numberOfInterpolationPoints); trustRegionCenterOffset = new ArrayRealVector(dimension); gradientAtTrustRegionCenter = new ArrayRealVector(dimension); lowerDifference = new ArrayRealVector(dimension); upperDifference = new ArrayRealVector(dimension); modelSecondDerivativesParameters = new ArrayRealVector(numberOfInterpolationPoints); newPoint = new ArrayRealVector(dimension); alternativeNewPoint = new ArrayRealVector(dimension); trialStepPoint = new ArrayRealVector(dimension); lagrangeValuesAtNewPoint = new ArrayRealVector(dimension + numberOfInterpolationPoints); modelSecondDerivativesValues = new ArrayRealVector(dimension * (dimension + 1) / 2); } // XXX utility for figuring out call sequence. private static String caller(int n) { final Throwable t = new Throwable(); final StackTraceElement[] elements = t.getStackTrace(); final StackTraceElement e = elements[n]; return e.getMethodName() + " (at line " + e.getLineNumber() + ")"; } // XXX utility for figuring out call sequence. private static void printState(int s) { // System.out.println(caller(2) + ": state " + s); } // XXX utility for figuring out call sequence. private static void printMethod() { // System.out.println(caller(2)); } /** * Marker for code paths that are not explored with the current unit tests. * If the path becomes explored, it should just be removed from the code. */ private static class PathIsExploredException extends RuntimeException { /** Serializable UID. */ private static final long serialVersionUID = 745350979634801853L; /** Message string. */ private static final String PATH_IS_EXPLORED = "If this exception is thrown, just remove it from the code"; PathIsExploredException() { super(PATH_IS_EXPLORED + " " + BOBYQAOptimizer.caller(3)); } } } //CHECKSTYLE: resume all

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