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The Waikato Environment for Knowledge Analysis (WEKA), a machine learning workbench. This is the stable version. Apart from bugfixes, this version does not receive any other updates.

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/*
 *   This program is free software: you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License as published by
 *   the Free Software Foundation, either version 3 of the License, or
 *   (at your option) any later version.
 *
 *   This program is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with this program.  If not, see .
 */

/*
 *    Optimization.java
 *    Copyright (C) 2003-2012 University of Waikato, Hamilton, New Zealand
 *
 */

package weka.core;

import weka.core.TechnicalInformation.Field;
import weka.core.TechnicalInformation.Type;
import weka.core.matrix.Matrix;

/**
 * Implementation of Active-sets method with BFGS update to solve optimization
 * problem with only bounds constraints in multi-dimensions. In this
 * implementation we consider both the lower and higher bound constraints.
 * 

* * Here is the sketch of our searching strategy, and the detailed description of * the algorithm can be found in the Appendix of Xin Xu's MSc thesis: *

* Initialize everything, incl. initial value, direction, etc. *

* LOOP (main algorithm):
* * 1. Perform the line search using the directions for free variables
* 1.1 Check all the bounds that are not "active" (i.e. binding variables) and * compute the feasible step length to the bound for each of them
* 1.2 Pick up the least feasible step length, say \alpha, and set it as the * upper bound of the current step length, i.e. 0<\lambda<=\alpha
* 1.3 Search for any possible step length<=\alpha that can result the * "sufficient function decrease" (\alpha condition) AND "positive definite * inverse Hessian" (\beta condition), if possible, using SAFEGUARDED polynomial * interpolation. This step length is "safe" and thus is used to compute the * next value of the free variables .
* 1.4 Fix the variable(s) that are newly bound to its constraint(s). *

* * 2. Check whether there is convergence of all variables or their gradients. If * there is, check the possibilities to release any current bindings of the * fixed variables to their bounds based on the "reliable" second-order * Lagarange multipliers if available. If it's available and negative for one * variable, then release it. If not available, use first-order Lagarange * multiplier to test release. If there is any released variables, STOP the * loop. Otherwise update the inverse of Hessian matrix and gradient for the * newly released variables and CONTINUE LOOP. *

* * 3. Use BFGS formula to update the inverse of Hessian matrix. Note the * already-fixed variables must have zeros in the corresponding entries in the * inverse Hessian. *

* * 4. Compute the new (newton) search direction d=H^{-1}*g, where H^{-1} is the * inverse Hessian and g is the Jacobian. Note that again, the already- fixed * variables will have zero direction. *

* * ENDLOOP *

* * A typical usage of this class is to create your own subclass of this class * and provide the objective function and gradients as follows: *

* *

 * class MyOpt extends Optimization {
 *   // Provide the objective function
 *   protected double objectiveFunction(double[] x) {
 *     // How to calculate your objective function...
 *     // ...
 *   }
 * 
 *   // Provide the first derivatives
 *   protected double[] evaluateGradient(double[] x) {
 *     // How to calculate the gradient of the objective function...
 *     // ...
 *   }
 * 
 *   // If possible, provide the indexˆ{th} row of the Hessian matrix
 *   protected double[] evaluateHessian(double[] x, int index) {
 *     // How to calculate the indexˆth variable's second derivative
 *     // ...
 *   }
 * }
 * 
* * When it's the time to use it, in some routine(s) of other class... * *
 * MyOpt opt = new MyOpt();
 * 
 * // Set up initial variable values and bound constraints
 * double[] x = new double[numVariables];
 * // Lower and upper bounds: 1st row is lower bounds, 2nd is upper
 * double[] constraints = new double[2][numVariables];
 * ...
 * 
 * // Find the minimum, 200 iterations as default
 * x = opt.findArgmin(x, constraints); 
 * while(x == null){  // 200 iterations are not enough
 *    x = opt.getVarbValues();  // Try another 200 iterations
 *    x = opt.findArgmin(x, constraints);
 * }
 * 
 * // The minimal function value
 * double minFunction = opt.getMinFunction();
 * ...
 * 
* * It is recommended that Hessian values be provided so that the second-order * Lagrangian multiplier estimate can be calcluated. However, if it is not * provided, there is no need to override the evaluateHessian() * function. *

* * REFERENCES (see also the getTechnicalInformation() method):
* The whole model algorithm is adapted from Chapter 5 and other related * chapters in Gill, Murray and Wright(1981) "Practical Optimization", Academic * Press. and Gill and Murray(1976) "Minimization Subject to Bounds on the * Variables", NPL Report NAC72, while Chong and Zak(1996) "An Introduction to * Optimization", John Wiley & Sons, Inc. provides us a brief but helpful * introduction to the method. *

* * Dennis and Schnabel(1983) "Numerical Methods for Unconstrained Optimization * and Nonlinear Equations", Prentice-Hall Inc. and Press et al.(1992) "Numeric * Recipe in C", Second Edition, Cambridge University Press. are consulted for * the polynomial interpolation used in the line search implementation. *

* * The Hessian modification in BFGS update uses Cholesky factorization and two * rank-one modifications:
* Bk+1 = Bk + (Gk*Gk')/(Gk'Dk) + (dGk*(dGk)'))/[alpha*(dGk)'*Dk].
* where Gk is the gradient vector, Dk is the direction vector and alpha is the * step rate.
* This method is due to Gill, Golub, Murray and Saunders(1974) ``Methods for * Modifying Matrix Factorizations'', Mathematics of Computation, Vol.28, * No.126, pp 505-535. *

* * @author Xin Xu ([email protected]) * @version $Revision: 11271 $ * @see #getTechnicalInformation() */ public abstract class Optimization implements TechnicalInformationHandler, RevisionHandler { protected double m_ALF = 1.0e-4; protected double m_BETA = 0.9; protected double m_TOLX = 1.0e-6; protected double m_STPMX = 100.0; protected int m_MAXITS = 200; protected boolean m_Debug = false; /** function value */ protected double m_f; /** G'*p */ private double m_Slope; /** Test if zero step in lnsrch */ protected boolean m_IsZeroStep = false; /** Used when iteration overflow occurs */ protected double[] m_X; /** Compute machine precision */ protected static double m_Epsilon, m_Zero; static { m_Epsilon = 1.0; while (1.0 + m_Epsilon > 1.0) { m_Epsilon /= 2.0; } m_Epsilon *= 2.0; m_Zero = Math.sqrt(m_Epsilon); } /** * Returns an instance of a TechnicalInformation object, containing detailed * information about the technical background of this class, e.g., paper * reference or book this class is based on. * * @return the technical information about this class */ @Override public TechnicalInformation getTechnicalInformation() { TechnicalInformation result; TechnicalInformation additional; result = new TechnicalInformation(Type.MASTERSTHESIS); result.setValue(Field.AUTHOR, "Xin Xu"); result.setValue(Field.YEAR, "2003"); result.setValue(Field.TITLE, "Statistical learning in multiple instance problem"); result.setValue(Field.SCHOOL, "University of Waikato"); result.setValue(Field.ADDRESS, "Hamilton, NZ"); result.setValue(Field.NOTE, "0657.594"); additional = result.add(Type.BOOK); additional.setValue(Field.AUTHOR, "P. E. Gill and W. Murray and M. H. Wright"); additional.setValue(Field.YEAR, "1981"); additional.setValue(Field.TITLE, "Practical Optimization"); additional.setValue(Field.PUBLISHER, "Academic Press"); additional.setValue(Field.ADDRESS, "London and New York"); additional = result.add(Type.TECHREPORT); additional.setValue(Field.AUTHOR, "P. E. Gill and W. Murray"); additional.setValue(Field.YEAR, "1976"); additional.setValue(Field.TITLE, "Minimization subject to bounds on the variables"); additional.setValue(Field.INSTITUTION, "National Physical Laboratory"); additional.setValue(Field.NUMBER, "NAC 72"); additional = result.add(Type.BOOK); additional.setValue(Field.AUTHOR, "E. K. P. Chong and S. H. Zak"); additional.setValue(Field.YEAR, "1996"); additional.setValue(Field.TITLE, "An Introduction to Optimization"); additional.setValue(Field.PUBLISHER, "John Wiley and Sons"); additional.setValue(Field.ADDRESS, "New York"); additional = result.add(Type.BOOK); additional.setValue(Field.AUTHOR, "J. E. Dennis and R. B. Schnabel"); additional.setValue(Field.YEAR, "1983"); additional .setValue(Field.TITLE, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations"); additional.setValue(Field.PUBLISHER, "Prentice-Hall"); additional = result.add(Type.BOOK); additional .setValue(Field.AUTHOR, "W. H. Press and B. P. Flannery and S. A. Teukolsky and W. T. Vetterling"); additional.setValue(Field.YEAR, "1992"); additional.setValue(Field.TITLE, "Numerical Recipes in C"); additional.setValue(Field.PUBLISHER, "Cambridge University Press"); additional.setValue(Field.EDITION, "Second"); additional = result.add(Type.ARTICLE); additional.setValue(Field.AUTHOR, "P. E. Gill and G. H. Golub and W. Murray and M. A. Saunders"); additional.setValue(Field.YEAR, "1974"); additional.setValue(Field.TITLE, "Methods for modifying matrix factorizations"); additional.setValue(Field.JOURNAL, "Mathematics of Computation"); additional.setValue(Field.VOLUME, "28"); additional.setValue(Field.NUMBER, "126"); additional.setValue(Field.PAGES, "505-535"); return result; } /** * Subclass should implement this procedure to evaluate objective function to * be minimized * * @param x the variable values * @return the objective function value * @throws Exception if something goes wrong */ protected abstract double objectiveFunction(double[] x) throws Exception; /** * Subclass should implement this procedure to evaluate gradient of the * objective function * * @param x the variable values * @return the gradient vector * @throws Exception if something goes wrong */ protected abstract double[] evaluateGradient(double[] x) throws Exception; /** * Subclass is recommended to override this procedure to evaluate second-order * gradient of the objective function. If it's not provided, it returns null. * * @param x the variables * @param index the row index in the Hessian matrix * @return one row (the row #index) of the Hessian matrix, null as default * @throws Exception if something goes wrong */ protected double[] evaluateHessian(double[] x, int index) throws Exception { return null; } /** * Get the minimal function value * * @return minimal function value found */ public double getMinFunction() { return m_f; } /** * Set the maximal number of iterations in searching (Default 200) * * @param it the maximal number of iterations */ public void setMaxIteration(int it) { m_MAXITS = it; } /** * Set whether in debug mode * * @param db use debug or not */ public void setDebug(boolean db) { m_Debug = db; } /** * Get the variable values. Only needed when iterations exceeds the max * threshold. * * @return the current variable values */ public double[] getVarbValues() { return m_X; } /** * Find a new point x in the direction p from a point xold at which the value * of the function has decreased sufficiently, the positive definiteness of B * matrix (approximation of the inverse of the Hessian) is preserved and no * bound constraints are violated. Details see "Numerical Methods for * Unconstrained Optimization and Nonlinear Equations". "Numeric Recipes in C" * was also consulted. * * @param xold old x value * @param gradient gradient at that point * @param direct direction vector * @param stpmax maximum step length * @param isFixed indicating whether a variable has been fixed * @param nwsBounds non-working set bounds. Means these variables are free and * subject to the bound constraints in this step * @param wsBdsIndx index of variables that has working-set bounds. Means * these variables are already fixed and no longer subject to the * constraints * @return new value along direction p from xold, null if no step was taken * @throws Exception if an error occurs */ public double[] lnsrch(double[] xold, double[] gradient, double[] direct, double stpmax, boolean[] isFixed, double[][] nwsBounds, DynamicIntArray wsBdsIndx) throws Exception { if (m_Debug) { System.err.print("Machine precision is " + m_Epsilon + " and zero set to " + m_Zero); } int i, k, len = xold.length, fixedOne = -1; // idx of variable to be fixed double alam, alamin; // lambda to be found, and its lower bound // For convergence and bound test double temp, test, alpha = Double.POSITIVE_INFINITY, fold = m_f, sum; // For cubic interpolation double a, alam2 = 0, b, disc = 0, maxalam = 1.0, rhs1, rhs2, tmplam; double[] x = new double[len]; // New variable values // Scale the step for (sum = 0.0, i = 0; i < len; i++) { if (!isFixed[i]) { sum += direct[i] * direct[i]; } } sum = Math.sqrt(sum); if (m_Debug) { System.err.println("fold: " + Utils.doubleToString(fold, 10, 7) + "\n" + "sum: " + Utils.doubleToString(sum, 10, 7) + "\n" + "stpmax: " + Utils.doubleToString(stpmax, 10, 7)); } if (sum > stpmax) { for (i = 0; i < len; i++) { if (!isFixed[i]) { direct[i] *= stpmax / sum; } } } else { maxalam = stpmax / sum; } // Compute initial rate of decrease, g'*d m_Slope = 0.0; for (i = 0; i < len; i++) { x[i] = xold[i]; if (!isFixed[i]) { m_Slope += gradient[i] * direct[i]; } } if (m_Debug) { System.err .print("slope: " + Utils.doubleToString(m_Slope, 10, 7) + "\n"); } // Slope too small if (Math.abs(m_Slope) <= m_Zero) { if (m_Debug) { System.err.println("Gradient and direction orthogonal -- " + "Min. found with current fixed variables" + " (or all variables fixed). Try to release" + " some variables now."); } return x; } // Err: slope > 0 if (m_Slope > m_Zero) { if (m_Debug) { for (int h = 0; h < x.length; h++) { System.err.println(h + ": isFixed=" + isFixed[h] + ", x=" + x[h] + ", grad=" + gradient[h] + ", direct=" + direct[h]); } } throw new Exception("g'*p positive! -- Try to debug from here: line 327."); } // Compute LAMBDAmin and upper bound of lambda--alpha test = 0.0; for (i = 0; i < len; i++) { if (!isFixed[i]) {// No need for fixed variables temp = Math.abs(direct[i]) / Math.max(Math.abs(x[i]), 1.0); if (temp > test) { test = temp; } } } if (test > m_Zero) { alamin = m_TOLX / test; } else { if (m_Debug) { System.err.println("Zero directions for all free variables -- " + "Min. found with current fixed variables" + " (or all variables fixed). Try to release" + " some variables now."); } return x; } // Check whether any non-working-set bounds are "binding" for (i = 0; i < len; i++) { if (!isFixed[i]) {// No need for fixed variables double alpi; if ((direct[i] < -m_Epsilon) && !Double.isNaN(nwsBounds[0][i])) {// Not // feasible alpi = (nwsBounds[0][i] - xold[i]) / direct[i]; if (alpi <= m_Zero) { // Zero if (m_Debug) { System.err.println("Fix variable " + i + " to lower bound " + nwsBounds[0][i] + " from value " + xold[i]); } x[i] = nwsBounds[0][i]; isFixed[i] = true; // Fix this variable alpha = 0.0; nwsBounds[0][i] = Double.NaN; // Add cons. to working set wsBdsIndx.addElement(i); } else if (alpha > alpi) { // Fix one variable in one iteration alpha = alpi; fixedOne = i; } } else if ((direct[i] > m_Epsilon) && !Double.isNaN(nwsBounds[1][i])) {// Not // feasible alpi = (nwsBounds[1][i] - xold[i]) / direct[i]; if (alpi <= m_Zero) { // Zero if (m_Debug) { System.err.println("Fix variable " + i + " to upper bound " + nwsBounds[1][i] + " from value " + xold[i]); } x[i] = nwsBounds[1][i]; isFixed[i] = true; // Fix this variable alpha = 0.0; nwsBounds[1][i] = Double.NaN; // Add cons. to working set wsBdsIndx.addElement(i); } else if (alpha > alpi) { alpha = alpi; fixedOne = i; } } } } if (m_Debug) { System.err.println("alamin: " + Utils.doubleToString(alamin, 10, 7)); System.err.println("alpha: " + Utils.doubleToString(alpha, 10, 7)); } if (alpha <= m_Zero) { // Zero m_IsZeroStep = true; if (m_Debug) { System.err.println("Alpha too small, try again"); } return x; } alam = alpha; // Always try full feasible newton step if (alam > 1.0) { alam = 1.0; } // Iteration of one newton step, if necessary, backtracking is done double initF = fold, // Initial function value hi = alam, lo = alam, newSlope = 0, fhi = m_f, flo = m_f;// Variables used // for beta // condition double[] newGrad; // Gradient on the new variable values kloop: for (k = 0;; k++) { if (m_Debug) { System.err.println("\nLine search iteration: " + k); } for (i = 0; i < len; i++) { if (!isFixed[i]) { x[i] = xold[i] + alam * direct[i]; // Compute xnew if (!Double.isNaN(nwsBounds[0][i]) && (x[i] < nwsBounds[0][i])) { x[i] = nwsBounds[0][i]; // Rounding error } else if (!Double.isNaN(nwsBounds[1][i]) && (x[i] > nwsBounds[1][i])) { x[i] = nwsBounds[1][i]; // Rounding error } } } m_f = objectiveFunction(x); // Compute fnew if (Double.isNaN(m_f)) { throw new Exception("Objective function value is NaN!"); } while (Double.isInfinite(m_f)) { // Avoid infinity if (m_Debug) { System.err.println("Too large m_f. Shrink step by half."); } alam *= 0.5; // Shrink by half if (alam <= m_Epsilon) { if (m_Debug) { System.err.println("Wrong starting points, change them!"); } return x; } for (i = 0; i < len; i++) { if (!isFixed[i]) { x[i] = xold[i] + alam * direct[i]; } } m_f = objectiveFunction(x); if (Double.isNaN(m_f)) { throw new Exception("Objective function value is NaN!"); } initF = Double.POSITIVE_INFINITY; } if (m_Debug) { System.err .println("obj. function: " + Utils.doubleToString(m_f, 10, 7)); System.err.println("threshold: " + Utils.doubleToString(fold + m_ALF * alam * m_Slope, 10, 7)); } if (m_f <= fold + m_ALF * alam * m_Slope) {// Alpha condition: sufficient // function decrease if (m_Debug) { System.err .println("Sufficient function decrease (alpha condition): "); } newGrad = evaluateGradient(x); for (newSlope = 0.0, i = 0; i < len; i++) { if (!isFixed[i]) { newSlope += newGrad[i] * direct[i]; } } if (m_Debug) { System.err.println("newSlope: " + newSlope); } if (newSlope >= m_BETA * m_Slope) { // Beta condition: ensure pos. // defnty. if (m_Debug) { System.err.println("Increasing derivatives (beta condition): "); } if ((fixedOne != -1) && (alam >= alpha)) { // Has bounds and over if (direct[fixedOne] > 0) { x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error nwsBounds[1][fixedOne] = Double.NaN; // Add cons. to working set } else { x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error nwsBounds[0][fixedOne] = Double.NaN; // Add cons. to working set } if (m_Debug) { System.err.println("Fix variable " + fixedOne + " to bound " + x[fixedOne] + " from value " + xold[fixedOne]); } isFixed[fixedOne] = true; // Fix the variable wsBdsIndx.addElement(fixedOne); } return x; } else if (k == 0) { // First time: increase alam // Search for the smallest value not complying with alpha condition double upper = Math.min(alpha, maxalam); if (m_Debug) { System.err.println("Alpha condition holds, increase alpha... "); } while (!((alam >= upper) || (m_f > fold + m_ALF * alam * m_Slope))) { lo = alam; flo = m_f; alam *= 2.0; if (alam >= upper) { alam = upper; } for (i = 0; i < len; i++) { if (!isFixed[i]) { x[i] = xold[i] + alam * direct[i]; } } m_f = objectiveFunction(x); if (Double.isNaN(m_f)) { throw new Exception("Objective function value is NaN!"); } newGrad = evaluateGradient(x); for (newSlope = 0.0, i = 0; i < len; i++) { if (!isFixed[i]) { newSlope += newGrad[i] * direct[i]; } } if (newSlope >= m_BETA * m_Slope) { if (m_Debug) { System.err .println("Increasing derivatives (beta condition): \n" + "newSlope = " + Utils.doubleToString(newSlope, 10, 7)); } if ((fixedOne != -1) && (alam >= alpha)) { // Has bounds and over if (direct[fixedOne] > 0) { x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error nwsBounds[1][fixedOne] = Double.NaN; // Add cons. to working // set } else { x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error nwsBounds[0][fixedOne] = Double.NaN; // Add cons. to working // set } if (m_Debug) { System.err.println("Fix variable " + fixedOne + " to bound " + x[fixedOne] + " from value " + xold[fixedOne]); } isFixed[fixedOne] = true; // Fix the variable wsBdsIndx.addElement(fixedOne); } return x; } } hi = alam; fhi = m_f; break kloop; } else { if (m_Debug) { System.err.println("Alpha condition holds."); } hi = alam2; lo = alam; flo = m_f; break kloop; } } else if (alam < alamin) { // No feasible lambda found if (initF < fold) { alam = Math.min(1.0, alpha); for (i = 0; i < len; i++) { if (!isFixed[i]) { x[i] = xold[i] + alam * direct[i]; // Still take Alpha } } if (m_Debug) { System.err.println("No feasible lambda: still take" + " alpha=" + alam); } if ((fixedOne != -1) && (alam >= alpha)) { // Has bounds and over if (direct[fixedOne] > 0) { x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error nwsBounds[1][fixedOne] = Double.NaN; // Add cons. to working set } else { x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error nwsBounds[0][fixedOne] = Double.NaN; // Add cons. to working set } if (m_Debug) { System.err.println("Fix variable " + fixedOne + " to bound " + x[fixedOne] + " from value " + xold[fixedOne]); } isFixed[fixedOne] = true; // Fix the variable wsBdsIndx.addElement(fixedOne); } } else { // Convergence on delta(x) for (i = 0; i < len; i++) { x[i] = xold[i]; } m_f = fold; if (m_Debug) { System.err.println("Cannot find feasible lambda"); } } return x; } else { // Backtracking by polynomial interpolation if (k == 0) { // First time backtrack: quadratic interpolation if (!Double.isInfinite(initF)) { initF = m_f; } // lambda = -g'(0)/(2*g''(0)) tmplam = -0.5 * alam * m_Slope / ((m_f - fold) / alam - m_Slope); } else { // Subsequent backtrack: cubic interpolation rhs1 = m_f - fold - alam * m_Slope; rhs2 = fhi - fold - alam2 * m_Slope; a = (rhs1 / (alam * alam) - rhs2 / (alam2 * alam2)) / (alam - alam2); b = (-alam2 * rhs1 / (alam * alam) + alam * rhs2 / (alam2 * alam2)) / (alam - alam2); if (a == 0.0) { tmplam = -m_Slope / (2.0 * b); } else { disc = b * b - 3.0 * a * m_Slope; if (disc < 0.0) { disc = 0.0; } double numerator = -b + Math.sqrt(disc); if (numerator >= Double.MAX_VALUE) { numerator = Double.MAX_VALUE; if (m_Debug) { System.err .print("-b+sqrt(disc) too large! Set it to MAX_VALUE."); } } tmplam = numerator / (3.0 * a); } if (m_Debug) { System.err.print("Cubic interpolation: \n" + "a: " + Utils.doubleToString(a, 10, 7) + "\n" + "b: " + Utils.doubleToString(b, 10, 7) + "\n" + "disc: " + Utils.doubleToString(disc, 10, 7) + "\n" + "tmplam: " + tmplam + "\n" + "alam: " + Utils.doubleToString(alam, 10, 7) + "\n"); } if (tmplam > 0.5 * alam) { tmplam = 0.5 * alam; // lambda <= 0.5*lambda_old } } } alam2 = alam; fhi = m_f; alam = Math.max(tmplam, 0.1 * alam); // lambda >= 0.1*lambda_old if (alam > alpha) { throw new Exception("Sth. wrong in lnsrch:" + "Lambda infeasible!(lambda=" + alam + ", alpha=" + alpha + ", upper=" + tmplam + "|" + (-alpha * m_Slope / (2.0 * ((m_f - fold) / alpha - m_Slope))) + ", m_f=" + m_f + ", fold=" + fold + ", slope=" + m_Slope); } } // Endfor(k=0;;k++) // Quadratic interpolation between lamda values between lo and hi. // If cannot find a value satisfying beta condition, use lo. double ldiff = hi - lo, lincr; if (m_Debug) { System.err .println("Last stage of searching for beta condition (alam between " + Utils.doubleToString(lo, 10, 7) + " and " + Utils.doubleToString(hi, 10, 7) + ")...\n" + "Quadratic Interpolation(QI):\n" + "Last newSlope = " + Utils.doubleToString(newSlope, 10, 7)); } while ((newSlope < m_BETA * m_Slope) && (ldiff >= alamin)) { lincr = -0.5 * newSlope * ldiff * ldiff / (fhi - flo - newSlope * ldiff); if (m_Debug) { System.err.println("fhi = " + fhi + "\n" + "flo = " + flo + "\n" + "ldiff = " + ldiff + "\n" + "lincr (using QI) = " + lincr + "\n"); } if (lincr < 0.2 * ldiff) { lincr = 0.2 * ldiff; } alam = lo + lincr; if (alam >= hi) { // We cannot go beyond the bounds, so the best we can // try is hi alam = hi; lincr = ldiff; } for (i = 0; i < len; i++) { if (!isFixed[i]) { x[i] = xold[i] + alam * direct[i]; } } m_f = objectiveFunction(x); if (Double.isNaN(m_f)) { throw new Exception("Objective function value is NaN!"); } if (m_f > fold + m_ALF * alam * m_Slope) { // Alpha condition fails, shrink lambda_upper ldiff = lincr; fhi = m_f; } else { // Alpha condition holds newGrad = evaluateGradient(x); for (newSlope = 0.0, i = 0; i < len; i++) { if (!isFixed[i]) { newSlope += newGrad[i] * direct[i]; } } if (newSlope < m_BETA * m_Slope) { // Beta condition fails, shrink lambda_lower lo = alam; ldiff -= lincr; flo = m_f; } } } if (newSlope < m_BETA * m_Slope) { // Cannot satisfy beta condition, take lo if (m_Debug) { System.err .println("Beta condition cannot be satisfied, take alpha condition"); } alam = lo; for (i = 0; i < len; i++) { if (!isFixed[i]) { x[i] = xold[i] + alam * direct[i]; } } m_f = flo; } else if (m_Debug) { System.err.println("Both alpha and beta conditions are satisfied. alam=" + Utils.doubleToString(alam, 10, 7)); } if ((fixedOne != -1) && (alam >= alpha)) { // Has bounds and over if (direct[fixedOne] > 0) { x[fixedOne] = nwsBounds[1][fixedOne]; // Avoid rounding error nwsBounds[1][fixedOne] = Double.NaN; // Add cons. to working set } else { x[fixedOne] = nwsBounds[0][fixedOne]; // Avoid rounding error nwsBounds[0][fixedOne] = Double.NaN; // Add cons. to working set } if (m_Debug) { System.err.println("Fix variable " + fixedOne + " to bound " + x[fixedOne] + " from value " + xold[fixedOne]); } isFixed[fixedOne] = true; // Fix the variable wsBdsIndx.addElement(fixedOne); } return x; } /** * Main algorithm. Descriptions see "Practical Optimization" * * @param initX initial point of x, assuming no value's on the bound! * @param constraints the bound constraints of each variable constraints[0] is * the lower bounds and constraints[1] is the upper bounds * @return the solution of x, null if number of iterations not enough * @throws Exception if an error occurs */ public double[] findArgmin(double[] initX, double[][] constraints) throws Exception { int l = initX.length; // Initially all variables are free, all bounds are constraints of // non-working-set constraints boolean[] isFixed = new boolean[l]; double[][] nwsBounds = new double[2][l]; // Record indice of fixed variables, simply for efficiency DynamicIntArray wsBdsIndx = new DynamicIntArray(constraints.length); // Vectors used to record the variable indices to be freed DynamicIntArray toFree = null, oldToFree = null; // Initial value of obj. function, gradient and inverse of the Hessian m_f = objectiveFunction(initX); if (Double.isNaN(m_f)) { throw new Exception("Objective function value is NaN!"); } double sum = 0; double[] grad = evaluateGradient(initX), oldGrad, oldX, deltaGrad = new double[l], deltaX = new double[l], direct = new double[l], x = new double[l]; Matrix L = new Matrix(l, l); // Lower triangle of Cholesky factor double[] D = new double[l]; // Diagonal of Cholesky factor for (int i = 0; i < l; i++) { // L.setRow(i, new double[l]); Not necessary L.set(i, i, 1.0); D[i] = 1.0; direct[i] = -grad[i]; sum += grad[i] * grad[i]; x[i] = initX[i]; nwsBounds[0][i] = constraints[0][i]; nwsBounds[1][i] = constraints[1][i]; isFixed[i] = false; } double stpmax = m_STPMX * Math.max(Math.sqrt(sum), l); for (int step = 0; step < m_MAXITS; step++) { if (m_Debug) { System.err.println("\nIteration # " + step + ":"); } // Try at most one feasible newton step, i.e. 0 test) { test = tmp; } } if (test < m_Zero) { if (m_Debug) { System.err.println("\nDeltaX converge: " + test); } finish = true; } // Check zero gradient grad = evaluateGradient(x); test = 0.0; double denom = 0.0, dxSq = 0.0, dgSq = 0.0, newlyBounded = 0.0; for (int g = 0; g < l; g++) { if (!isFixed[g]) { deltaGrad[g] = grad[g] - oldGrad[g]; // Calculate the denominators denom += deltaX[g] * deltaGrad[g]; dxSq += deltaX[g] * deltaX[g]; dgSq += deltaGrad[g] * deltaGrad[g]; } else { newlyBounded += deltaX[g] * (grad[g] - oldGrad[g]); } // Note: CANNOT use projected gradient for testing // convergence because of newly bounded variables double tmp = Math.abs(grad[g]) * Math.max(Math.abs(direct[g]), 1.0) / Math.max(Math.abs(m_f), 1.0); if (tmp > test) { test = tmp; } } if (test < m_Zero) { if (m_Debug) { System.err.println("Gradient converge: " + test); } finish = true; } // dg'*dx could be < 0 using inexact lnsrch if (m_Debug) { System.err.println("dg'*dx=" + (denom + newlyBounded)); } // dg'*dx = 0 if (Math.abs(denom + newlyBounded) < m_Zero) { finish = true; } int size = wsBdsIndx.size(); boolean isUpdate = true; // Whether to update BFGS formula // Converge: check whether release any current constraints if (finish) { if (m_Debug) { System.err.println("Test any release possible ..."); } if (toFree != null) { oldToFree = (DynamicIntArray) toFree.copy(); } toFree = new DynamicIntArray(wsBdsIndx.size()); for (int m = size - 1; m >= 0; m--) { int index = wsBdsIndx.elementAt(m); double[] hessian = evaluateHessian(x, index); double deltaL = 0.0; if (hessian != null) { for (int mm = 0; mm < hessian.length; mm++) { if (!isFixed[mm]) { deltaL += hessian[mm] * direct[mm]; } } } // First and second order Lagrangian multiplier estimate // If user didn't provide Hessian, use first-order only double L1, L2; if (x[index] >= constraints[1][index]) { L1 = -grad[index]; } else if (x[index] <= constraints[0][index]) { L1 = grad[index]; } else { throw new Exception("x[" + index + "] not fixed on the" + " bounds where it should have been!"); } // L2 = L1 + deltaL L2 = L1 + deltaL; if (m_Debug) { System.err.println("Variable " + index + ": Lagrangian=" + L1 + "|" + L2); } // Check validity of Lagrangian multiplier estimate boolean isConverge = (2.0 * Math.abs(deltaL)) < Math.min( Math.abs(L1), Math.abs(L2)); if ((L1 * L2 > 0.0) && isConverge) { // Same sign and converge: // valid if (L2 < 0.0) {// Negative Lagrangian: feasible toFree.addElement(index); wsBdsIndx.removeElementAt(m); finish = false; // Not optimal, cannot finish } } // Although hardly happen, better check it // If the first-order Lagrangian multiplier estimate is wrong, // avoid zigzagging if ((hessian == null) && (toFree != null) && toFree.equal(oldToFree)) { finish = true; } } if (finish) {// Min. found if (m_Debug) { System.err.println("Minimum found."); } m_f = objectiveFunction(x); if (Double.isNaN(m_f)) { throw new Exception("Objective function value is NaN!"); } return x; } // Free some variables for (int mmm = 0; mmm < toFree.size(); mmm++) { int freeIndx = toFree.elementAt(mmm); isFixed[freeIndx] = false; // Free this variable if (x[freeIndx] <= constraints[0][freeIndx]) {// Lower bound nwsBounds[0][freeIndx] = constraints[0][freeIndx]; if (m_Debug) { System.err.println("Free variable " + freeIndx + " from bound " + nwsBounds[0][freeIndx]); } } else { // Upper bound nwsBounds[1][freeIndx] = constraints[1][freeIndx]; if (m_Debug) { System.err.println("Free variable " + freeIndx + " from bound " + nwsBounds[1][freeIndx]); } } L.set(freeIndx, freeIndx, 1.0); D[freeIndx] = 1.0; isUpdate = false; } } if (denom < Math .max(m_Zero * Math.sqrt(dxSq) * Math.sqrt(dgSq), m_Zero)) { if (m_Debug) { System.err.println("dg'*dx negative!"); } isUpdate = false; // Do not update } // If Hessian will be positive definite, update it if (isUpdate) { // modify once: dg*dg'/(dg'*dx) double coeff = 1.0 / denom; // 1/(dg'*dx) updateCholeskyFactor(L, D, deltaGrad, coeff, isFixed); // modify twice: g*g'/(g'*p) coeff = 1.0 / m_Slope; // 1/(g'*p) updateCholeskyFactor(L, D, oldGrad, coeff, isFixed); } } // Find new direction Matrix LD = new Matrix(l, l); // L*D double[] b = new double[l]; for (int k = 0; k < l; k++) { if (!isFixed[k]) { b[k] = -grad[k]; } else { b[k] = 0.0; } for (int j = k; j < l; j++) { // Lower triangle if (!isFixed[j] && !isFixed[k]) { LD.set(j, k, L.get(j, k) * D[k]); } } } // Solve (LD)*y = -g, where y=L'*direct double[] LDIR = solveTriangle(LD, b, true, isFixed); LD = null; for (int m = 0; m < LDIR.length; m++) { if (Double.isNaN(LDIR[m])) { throw new Exception("L*direct[" + m + "] is NaN!" + "|-g=" + b[m] + "|" + isFixed[m] + "|diag=" + D[m]); } } // Solve L'*direct = y direct = solveTriangle(L, LDIR, false, isFixed); for (double element : direct) { if (Double.isNaN(element)) { throw new Exception("direct is NaN!"); } } // System.gc(); } if (m_Debug) { System.err.println("Cannot find minimum" + " -- too many interations!"); } m_X = x; return null; } /** * Solve the linear equation of TX=B where T is a triangle matrix It can be * solved using back/forward substitution, with O(N^2) complexity * * @param t the matrix T * @param b the vector B * @param isLower whether T is a lower or higher triangle matrix * @param isZero which row(s) of T are not used when solving the equation. If * it's null or all 'false', then every row is used. * @return the solution of X */ public static double[] solveTriangle(Matrix t, double[] b, boolean isLower, boolean[] isZero) { int n = b.length; double[] result = new double[n]; if (isZero == null) { isZero = new boolean[n]; } if (isLower) { // lower triangle, forward-substitution int j = 0; while ((j < n) && isZero[j]) { result[j] = 0.0; j++; } // go to the first row if (j < n) { result[j] = b[j] / t.get(j, j); for (; j < n; j++) { if (!isZero[j]) { double numerator = b[j]; for (int k = 0; k < j; k++) { numerator -= t.get(j, k) * result[k]; } result[j] = numerator / t.get(j, j); } else { result[j] = 0.0; } } } } else { // Upper triangle, back-substitution int j = n - 1; while ((j >= 0) && isZero[j]) { result[j] = 0.0; j--; } // go to the last row if (j >= 0) { result[j] = b[j] / t.get(j, j); for (; j >= 0; j--) { if (!isZero[j]) { double numerator = b[j]; for (int k = j + 1; k < n; k++) { numerator -= t.get(k, j) * result[k]; } result[j] = numerator / t.get(j, j); } else { result[j] = 0.0; } } } } return result; } /** * One rank update of the Cholesky factorization of B matrix in BFGS updates, * i.e. B = LDL', and B_{new} = LDL' + coeff*(vv') where L is a unit lower * triangle matrix and D is a diagonal matrix, and v is a vector.
* When coeff > 0, we use C1 algorithm, and otherwise we use C2 algorithm * described in ``Methods for Modifying Matrix Factorizations'' * * @param L the unit triangle matrix L * @param D the diagonal matrix D * @param v the update vector v * @param coeff the coeffcient of update * @param isFixed which variables are not to be updated */ protected void updateCholeskyFactor(Matrix L, double[] D, double[] v, double coeff, boolean[] isFixed) throws Exception { double t, p, b; int n = v.length; double[] vp = new double[n]; for (int i = 0; i < v.length; i++) { if (!isFixed[i]) { vp[i] = v[i]; } else { vp[i] = 0.0; } } if (coeff > 0.0) { t = coeff; for (int j = 0; j < n; j++) { if (isFixed[j]) { continue; } p = vp[j]; double d = D[j], dbarj = d + t * p * p; D[j] = dbarj; b = p * t / dbarj; t *= d / dbarj; for (int r = j + 1; r < n; r++) { if (!isFixed[r]) { double l = L.get(r, j); vp[r] -= p * l; L.set(r, j, l + b * vp[r]); } else { L.set(r, j, 0.0); } } } } else { double[] P = solveTriangle(L, v, true, isFixed); t = 0.0; for (int i = 0; i < n; i++) { if (!isFixed[i]) { t += P[i] * P[i] / D[i]; } } double sqrt = 1.0 + coeff * t; sqrt = (sqrt < 0.0) ? 0.0 : Math.sqrt(sqrt); double alpha = coeff, sigma = coeff / (1.0 + sqrt), rho, theta; for (int j = 0; j < n; j++) { if (isFixed[j]) { continue; } double d = D[j]; p = P[j] * P[j] / d; theta = 1.0 + sigma * p; t -= p; if (t < 0.0) { t = 0.0; // Rounding error } double plus = sigma * sigma * p * t; if ((j < n - 1) && (plus <= m_Zero)) { plus = m_Zero; // Avoid rounding error } rho = theta * theta + plus; D[j] = rho * d; if (Double.isNaN(D[j])) { throw new Exception("d[" + j + "] NaN! P=" + P[j] + ",d=" + d + ",t=" + t + ",p=" + p + ",sigma=" + sigma + ",sclar=" + coeff); } b = alpha * P[j] / (rho * d); alpha /= rho; rho = Math.sqrt(rho); double sigmaOld = sigma; sigma *= (1.0 + rho) / (rho * (theta + rho)); if ((j < n - 1) && (Double.isNaN(sigma) || Double.isInfinite(sigma))) { throw new Exception("sigma NaN/Inf! rho=" + rho + ",theta=" + theta + ",P[" + j + "]=" + P[j] + ",p=" + p + ",d=" + d + ",t=" + t + ",oldsigma=" + sigmaOld); } for (int r = j + 1; r < n; r++) { if (!isFixed[r]) { double l = L.get(r, j); vp[r] -= P[j] * l; L.set(r, j, l + b * vp[r]); } else { L.set(r, j, 0.0); } } } } } /** * Implements a simple dynamic array for ints. */ protected class DynamicIntArray implements RevisionHandler { /** The int array. */ private int[] m_Objects; /** The current size; */ private int m_Size = 0; /** The capacity increment */ private int m_CapacityIncrement = 1; /** The capacity multiplier. */ private int m_CapacityMultiplier = 2; /** * Constructs a vector with the given capacity. * * @param capacity the vector's initial capacity */ public DynamicIntArray(int capacity) { m_Objects = new int[capacity]; } /** * Adds an element to this vector. Increases its capacity if its not large * enough. * * @param element the element to add */ public final void addElement(int element) { if (m_Size == m_Objects.length) { int[] newObjects; newObjects = new int[m_CapacityMultiplier * (m_Objects.length + m_CapacityIncrement)]; System.arraycopy(m_Objects, 0, newObjects, 0, m_Size); m_Objects = newObjects; } m_Objects[m_Size] = element; m_Size++; } /** * Produces a copy of this vector. * * @return the new vector */ public final Object copy() { DynamicIntArray copy = new DynamicIntArray(m_Objects.length); copy.m_Size = m_Size; copy.m_CapacityIncrement = m_CapacityIncrement; copy.m_CapacityMultiplier = m_CapacityMultiplier; System.arraycopy(m_Objects, 0, copy.m_Objects, 0, m_Size); return copy; } /** * Returns the element at the given position. * * @param index the element's index * @return the element with the given index */ public final int elementAt(int index) { return m_Objects[index]; } /** * Check whether the two integer vectors equal to each other Two integer * vectors are equal if all the elements are the same, regardless of the * order of the elements * * @param b another integer vector * @return whether they are equal */ private boolean equal(DynamicIntArray b) { if ((b == null) || (size() != b.size())) { return false; } int size = size(); // Only values matter, order does not matter int[] sorta = Utils.sort(m_Objects), sortb = Utils.sort(b.m_Objects); for (int j = 0; j < size; j++) { if (m_Objects[sorta[j]] != b.m_Objects[sortb[j]]) { return false; } } return true; } /** * Deletes an element from this vector. * * @param index the index of the element to be deleted */ public final void removeElementAt(int index) { System.arraycopy(m_Objects, index + 1, m_Objects, index, m_Size - index - 1); m_Size--; } /** * Removes all components from this vector and sets its size to zero. */ public final void removeAllElements() { m_Objects = new int[m_Objects.length]; m_Size = 0; } /** * Returns the vector's current size. * * @return the vector's current size */ public final int size() { return m_Size; } /** * Returns the revision string. * * @return the revision */ @Override public String getRevision() { return RevisionUtils.extract("$Revision: 11271 $"); } } }





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