source: src/main/java/weka/core/Optimization.java @ 17

/*
 *    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 2 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, write to the Free Software
 *    Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 */

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

package weka.core;

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

/**
 * 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. <p/>
 *
 * 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:<p/>
 * Initialize everything, incl. initial value, direction, etc.<p/>
 * LOOP (main algorithm):<br/>
 * 
 * 1. Perform the line search using the directions for free variables<br/>
 * 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<br/>
 * 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&lt;\lambda&lt;=\alpha<br/>
 * 1.3  Search for any possible step length&lt;=\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 .<br/>
 * 1.4  Fix the variable(s) that are newly bound to its constraint(s).<p/>     
 *
 * 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.<p/>
 *
 * 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.<p/>  
 *
 * 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.<p/>
 *
 * ENDLOOP<p/>
 *
 * A typical usage of this class is to create your own subclass of this class
 * and provide the objective function and gradients as follows:<p/>
 * <pre>
 * 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 index^{th} row of the Hessian matrix
 *   protected double[] evaluateHessian(double[] x, int index) {
 *      // How to calculate the index^th variable's second derivative
 *      // ... 
 *   }
 * }
 * </pre>
 *
 * When it's the time to use it, in some routine(s) of other class...
 * <pre>
 * 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();
 * ...
 * </pre>
 * 
 * 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 <code>evaluateHessian()</code>
 * function.<p/>
 *
 * REFERENCES (see also the <code>getTechnicalInformation()</code> method):<br/>
 * 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 &amp; Sons, Inc. provides us a brief but helpful
 * introduction to the method. <p/>
 *
 * 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.  <p/>
 *
 * The Hessian modification in BFGS update uses Cholesky factorization and two
 * rank-one modifications:<br/>
 * Bk+1 = Bk + (Gk*Gk')/(Gk'Dk) + (dGk*(dGk)'))/[alpha*(dGk)'*Dk]. <br/>
 * where Gk is the gradient vector, Dk is the direction vector and alpha is the
 * step rate. <br/>
 * 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. <p/>
 *
 * @author Xin Xu (xx5@cs.waikato.ac.nz)
 * @version $Revision: 5953 $ 
 * @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 static boolean m_Debug = false;
    
    /** function value */
    protected double m_f;    
 
    /** G'*p */
    private double m_Slope;
    
    /** Test if zero step in lnsrch */
    private boolean m_IsZeroStep = false;
    
    /** Used when iteration overflow occurs */
    private 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);
        if (m_Debug)
            System.err.print("Machine precision is "+m_Epsilon+
                             " and zero set to "+m_Zero);
    }
    
    /**
     * 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
     */
    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 {
        
        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]) // For fixed variables, direction = 0
                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) // Not converge
            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(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)  // Avoid rounding errors
                            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]);
            L.setElement(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);
        
        iterates:
        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<lamda<=alpha
            oldX = x;
            oldGrad = grad;
            
            // Also update grad
            if (m_Debug)
                System.err.println("Line search ... ");
            m_IsZeroStep = false;
            x=lnsrch(x, grad, direct, stpmax, 
                     isFixed, nwsBounds, wsBdsIndx);
            if (m_Debug)
                System.err.println("Line search finished.");
            
            if(m_IsZeroStep){ // Zero step, simply delete rows/cols of D and L
                for(int f=0; f<wsBdsIndx.size(); f++){
                    int idx=wsBdsIndx.elementAt(f);
                    L.setRow(idx, new double[l]);
                    L.setColumn(idx, new double[l]);
                    D[idx] = 0.0;
                }               
                grad = evaluateGradient(x);
                step--;
            }
            else{
                // Check converge on x
                boolean finish = false;
                double test=0.0;
                for(int h=0; h<l; h++){
                    deltaX[h] = x[h]-oldX[h];
                    double tmp=Math.abs(deltaX[h])/
                        Math.max(Math.abs(x[h]), 1.0);
                    if(tmp > 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 // Only newly bounded variables will be non-zero
                        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]) // Free variable
                                    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]) // Upper bound
                            L1 = -grad[index];
                        else if(x[index] <= constraints[0][index])// Lower bound
                            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.setElement(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.setElement(j, k, L.getElement(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(int m=0; m<direct.length; m++){
                if(Double.isNaN(direct[m]))
                    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.getElement(j,j);
                
                for(; j<n; j++){
                    if(!isZero[j]){
                        double numerator=b[j];
                        for(int k=0; k<j; k++)
                            numerator -= t.getElement(j,k)*result[k];
                        result[j] = numerator/t.getElement(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.getElement(j,j);
                
                for(; j>=0; j--){
                    if(!isZero[j]){
                        double numerator=b[j];
                        for(int k=j+1; k<n; k++)
                            numerator -= t.getElement(k,j)*result[k];
                        result[j] = numerator/t.getElement(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.<br/>
     * 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.getElement(r, j);
                        vp[r] -= p*l;
                        L.setElement(r, j, l+b*vp[r]);
                    }
                    else
                        L.setElement(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.getElement(r, j);
                        vp[r] -= P[j]*l;
                        L.setElement(r, j, l+b*vp[r]);
                    }
                    else
                        L.setElement(r, j, 0.0);
                }
            }
        }       
    }

  /**
   * Implements a simple dynamic array for ints.
   */
  private 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
     */
    public String getRevision() {
      return RevisionUtils.extract("$Revision: 5953 $");
    }
  }
}