epsp.h

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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 <http://www.gnu.org/licenses/>.
 *
 *   Copyright (C) 2009 - 2012 Jun Liu and Jieping Ye 
 */

#ifndef  EPSP_SLEP
#define  EPSP_SLEP

#include <stdlib.h>
#include <stdio.h>
#include <time.h>
#include <math.h>

#define delta 1e-12

/*
 Euclidean Projection onto the simplex (epsp)
 
        min  1/2 ||x- y||_2^2
        s.t. ||x||_1 = z, x >=0
 
which is converted to the following zero finding problem
 
        f(lambda)= sum( max( x-lambda,0) )-z=0
 
 Usage:
 [x, lambda, iter_step]=epsp(y, n, z, lambda0);
 
 */

void epsp(double * x, double *root, int * stepsp, double * v,
int n, double z, double lambda0)
{
    
    int i, j, flag=0;
    int rho_1, rho_2, rho, rho_T, rho_S;
    int V_i_b, V_i_e, V_i;
    double lambda_1, lambda_2, lambda_T, lambda_S, lambda;
    double s_1, s_2, s, s_T, s_S, v_max, temp;
    double f_lambda_1, f_lambda_2, f_lambda, f_lambda_T, f_lambda_S;
    int iter_step=0;
        


    if (z< 0){
        printf("\n z should be nonnegative!");
        exit(-1);
    }
         
    
    /* 
     * find the maximal value in v
     */
    v_max=v[0];    
    for (i=1;i<n; i++){            
            if (v[i] > v_max)
                v_max=v[i];
    }
       
   
    lambda_1=v_max - z; lambda_2=v_max;
    /*
     * copy v to x
     * compute f_lambda_1, rho_1, s_1
     */
    V_i=0;s_1=0; rho_1=0;
    for (i=0;i<n;i++){
        if (v[i] > lambda_1){
            x[V_i]=v[i];

            s_1+=x[V_i]; rho_1++;

            V_i++;
        }
    }
    f_lambda_1=s_1-rho_1* lambda_1 -z;

    rho_2=0; s_2=0; f_lambda_2=-z; 
    V_i_b=0; V_i_e=V_i-1;
    
    lambda=lambda0; 
    if ( (lambda<lambda_2) && (lambda> lambda_1) ){ 
    /*-------------------------------------------------------------------
                  Initialization with the root
     *-------------------------------------------------------------------
     */
           
        i=V_i_b; j=V_i_e; rho=0; s=0;
        while (i <= j){            
            while( (i <= V_i_e) && (x[i] <= lambda) ){
                i++;
            }
            while( (j>=V_i_b) && (x[j] > lambda) ){
                s+=x[j];                
                j--;
            }
            if (i<j){
                s+=x[i];
                
                temp=x[i];  x[i]=x[j];  x[j]=temp;
                i++;  j--;
            }
        }
        
        rho=V_i_e-j;  rho+=rho_2;  s+=s_2;        
        f_lambda=s-rho*lambda-z;
        
        if ( fabs(f_lambda)< delta ){
            flag=1;
        }
        
        if (f_lambda <0){
            lambda_2=lambda; s_2=s; rho_2=rho; f_lambda_2=f_lambda;

            V_i_e=j;  V_i=V_i_e-V_i_b+1;
        }
        else{
            lambda_1=lambda; rho_1=rho; s_1=s; f_lambda_1=f_lambda;

            V_i_b=i; V_i=V_i_e-V_i_b+1;
        }

        if (V_i==0){
            /*printf("\n rho=%d, rho_1=%d, rho_2=%d",rho, rho_1, rho_2);*/

            /*printf("\n V_i=%d",V_i);*/
            
            lambda=(s - z)/ rho;
            flag=1;
        }       
     /*-------------------------------------------------------------------
                          End of initialization
      *--------------------------------------------------------------------
      */       
        
    }/* end of if(!flag) */
    
    while (!flag){
        iter_step++;
        
        /* compute lambda_T  */
        lambda_T=lambda_1 + f_lambda_1 /rho_1;
        if(rho_2 !=0){
            if (lambda_2 + f_lambda_2 /rho_2 >  lambda_T)
                lambda_T=lambda_2 + f_lambda_2 /rho_2;
        }
        
        /* compute lambda_S */
        lambda_S=lambda_2 - f_lambda_2 *(lambda_2-lambda_1)/(f_lambda_2-f_lambda_1);
        
        if (fabs(lambda_T-lambda_S) <= delta){
            lambda=lambda_T; flag=1;
            break;
        }
        
        /* set lambda as the middle point of lambda_T and lambda_S */
        lambda=(lambda_T+lambda_S)/2;
        
        s_T=s_S=s=0;
        rho_T=rho_S=rho=0;
        i=V_i_b; j=V_i_e;
        while (i <= j){            
            while( (i <= V_i_e) && (x[i] <= lambda) ){
                if (x[i]> lambda_T){
                    s_T+=x[i]; rho_T++;
                }
                i++;
            }
            while( (j>=V_i_b) && (x[j] > lambda) ){
                if (x[j] > lambda_S){
                    s_S+=x[j]; rho_S++;
                }
                else{
                    s+=x[j];  rho++;
                }
                j--;
            }
            if (i<j){
                if (x[i] > lambda_S){
                    s_S+=x[i]; rho_S++;
                }
                else{
                    s+=x[i]; rho++;
                }
                
                if (x[j]> lambda_T){
                    s_T+=x[j]; rho_T++;
                }
                
                temp=x[i]; x[i]=x[j];  x[j]=temp;
                i++; j--;
            }
        }
        
        s_S+=s_2; rho_S+=rho_2;
        s+=s_S; rho+=rho_S;
        s_T+=s; rho_T+=rho;
        f_lambda_S=s_S-rho_S*lambda_S-z;
        f_lambda=s-rho*lambda-z;
        f_lambda_T=s_T-rho_T*lambda_T-z;
        
        /*printf("\n %d & %d  & %5.6f & %5.6f & %5.6f & %5.6f & %5.6f \\\\ \n \\hline ", iter_step, V_i, lambda_1, lambda_T, lambda, lambda_S, lambda_2);*/
                
        if ( fabs(f_lambda)< delta ){
            /*printf("\n lambda");*/
            flag=1;
            break;
        }
        if ( fabs(f_lambda_S)< delta ){
           /* printf("\n lambda_S");*/
            lambda=lambda_S; flag=1;
            break;
        }
        if ( fabs(f_lambda_T)< delta ){
           /* printf("\n lambda_T");*/
            lambda=lambda_T; flag=1;
            break;
        }        
        
        /*
        printf("\n\n f_lambda_1=%5.6f, f_lambda_2=%5.6f, f_lambda=%5.6f",f_lambda_1,f_lambda_2, f_lambda);
        printf("\n lambda_1=%5.6f, lambda_2=%5.6f, lambda=%5.6f",lambda_1, lambda_2, lambda);
        printf("\n rho_1=%d, rho_2=%d, rho=%d ",rho_1, rho_2, rho);
         */
        
        if (f_lambda <0){
            lambda_2=lambda;  s_2=s;  rho_2=rho;
            f_lambda_2=f_lambda;            
            
            lambda_1=lambda_T; s_1=s_T; rho_1=rho_T;
            f_lambda_1=f_lambda_T;
            
            V_i_e=j;  i=V_i_b;
            while (i <= j){
                while( (i <= V_i_e) && (x[i] <= lambda_T) ){
                    i++;
                }
                while( (j>=V_i_b) && (x[j] > lambda_T) ){
                    j--;
                }
                if (i<j){                    
                    x[j]=x[i];
                    i++;   j--;
                }
            }            
            V_i_b=i; V_i=V_i_e-V_i_b+1;
        }
        else{
            lambda_1=lambda;  s_1=s; rho_1=rho;
            f_lambda_1=f_lambda;
            
            lambda_2=lambda_S; s_2=s_S; rho_2=rho_S;
            f_lambda_2=f_lambda_S;
            
            V_i_b=i;  j=V_i_e;
            while (i <= j){
                while( (i <= V_i_e) && (x[i] <= lambda_S) ){
                    i++;
                }
                while( (j>=V_i_b) && (x[j] > lambda_S) ){
                    j--;
                }
                if (i<j){
                    x[i]=x[j];
                    i++;   j--;
                }
            }
            V_i_e=j; V_i=V_i_e-V_i_b+1;
        }
        
        if (V_i==0){
            lambda=(s - z)/ rho; flag=1;
            /*printf("\n V_i=0, lambda=%5.6f",lambda);*/
            break;
        }
    }/* end of while */
    
    
    for(i=0;i<n;i++){        
        if (v[i] > lambda)
            x[i]=v[i]-lambda;
        else
            x[i]=0;
    }
    *root=lambda;
    *stepsp=iter_step;
}
#endif   /* ----- #ifndef EPSP_SLEP  ----- */