/*
* 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 Xin Xu
*
*/
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<\lambda<=\alpha<br/>
* 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 .<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 & 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: 1.7 $
* @see #getTechnicalInformation()
*/
public abstract class Optimization
implements TechnicalInformationHandler {
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"+
"newS