alpar@9: /* glpipm.c */
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  This code is part of GLPK (GNU Linear Programming Kit).
alpar@9: *
alpar@9: *  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
alpar@9: *  2009, 2010, 2011 Andrew Makhorin, Department for Applied Informatics,
alpar@9: *  Moscow Aviation Institute, Moscow, Russia. All rights reserved.
alpar@9: *  E-mail: <mao@gnu.org>.
alpar@9: *
alpar@9: *  GLPK is free software: you can redistribute it and/or modify it
alpar@9: *  under the terms of the GNU General Public License as published by
alpar@9: *  the Free Software Foundation, either version 3 of the License, or
alpar@9: *  (at your option) any later version.
alpar@9: *
alpar@9: *  GLPK is distributed in the hope that it will be useful, but WITHOUT
alpar@9: *  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
alpar@9: *  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
alpar@9: *  License for more details.
alpar@9: *
alpar@9: *  You should have received a copy of the GNU General Public License
alpar@9: *  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
alpar@9: ***********************************************************************/
alpar@9: 
alpar@9: #include "glpipm.h"
alpar@9: #include "glpmat.h"
alpar@9: 
alpar@9: #define ITER_MAX 100
alpar@9: /* maximal number of iterations */
alpar@9: 
alpar@9: struct csa
alpar@9: {     /* common storage area */
alpar@9:       /*--------------------------------------------------------------*/
alpar@9:       /* LP data */
alpar@9:       int m;
alpar@9:       /* number of rows (equality constraints) */
alpar@9:       int n;
alpar@9:       /* number of columns (structural variables) */
alpar@9:       int *A_ptr; /* int A_ptr[1+m+1]; */
alpar@9:       int *A_ind; /* int A_ind[A_ptr[m+1]]; */
alpar@9:       double *A_val; /* double A_val[A_ptr[m+1]]; */
alpar@9:       /* mxn-matrix A in storage-by-rows format */
alpar@9:       double *b; /* double b[1+m]; */
alpar@9:       /* m-vector b of right-hand sides */
alpar@9:       double *c; /* double c[1+n]; */
alpar@9:       /* n-vector c of objective coefficients; c[0] is constant term of
alpar@9:          the objective function */
alpar@9:       /*--------------------------------------------------------------*/
alpar@9:       /* LP solution */
alpar@9:       double *x; /* double x[1+n]; */
alpar@9:       double *y; /* double y[1+m]; */
alpar@9:       double *z; /* double z[1+n]; */
alpar@9:       /* current point in primal-dual space; the best point on exit */
alpar@9:       /*--------------------------------------------------------------*/
alpar@9:       /* control parameters */
alpar@9:       const glp_iptcp *parm;
alpar@9:       /*--------------------------------------------------------------*/
alpar@9:       /* working arrays and variables */
alpar@9:       double *D; /* double D[1+n]; */
alpar@9:       /* diagonal nxn-matrix D = X*inv(Z), where X = diag(x[j]) and
alpar@9:          Z = diag(z[j]) */
alpar@9:       int *P; /* int P[1+m+m]; */
alpar@9:       /* permutation mxm-matrix P used to minimize fill-in in Cholesky
alpar@9:          factorization */
alpar@9:       int *S_ptr; /* int S_ptr[1+m+1]; */
alpar@9:       int *S_ind; /* int S_ind[S_ptr[m+1]]; */
alpar@9:       double *S_val; /* double S_val[S_ptr[m+1]]; */
alpar@9:       double *S_diag; /* double S_diag[1+m]; */
alpar@9:       /* symmetric mxm-matrix S = P*A*D*A'*P' whose upper triangular
alpar@9:          part without diagonal elements is stored in S_ptr, S_ind, and
alpar@9:          S_val in storage-by-rows format, diagonal elements are stored
alpar@9:          in S_diag */
alpar@9:       int *U_ptr; /* int U_ptr[1+m+1]; */
alpar@9:       int *U_ind; /* int U_ind[U_ptr[m+1]]; */
alpar@9:       double *U_val; /* double U_val[U_ptr[m+1]]; */
alpar@9:       double *U_diag; /* double U_diag[1+m]; */
alpar@9:       /* upper triangular mxm-matrix U defining Cholesky factorization
alpar@9:          S = U'*U; its non-diagonal elements are stored in U_ptr, U_ind,
alpar@9:          U_val in storage-by-rows format, diagonal elements are stored
alpar@9:          in U_diag */
alpar@9:       int iter;
alpar@9:       /* iteration number (0, 1, 2, ...); iter = 0 corresponds to the
alpar@9:          initial point */
alpar@9:       double obj;
alpar@9:       /* current value of the objective function */
alpar@9:       double rpi;
alpar@9:       /* relative primal infeasibility rpi = ||A*x-b||/(1+||b||) */
alpar@9:       double rdi;
alpar@9:       /* relative dual infeasibility rdi = ||A'*y+z-c||/(1+||c||) */
alpar@9:       double gap;
alpar@9:       /* primal-dual gap = |c'*x-b'*y|/(1+|c'*x|) which is a relative
alpar@9:          difference between primal and dual objective functions */
alpar@9:       double phi;
alpar@9:       /* merit function phi = ||A*x-b||/max(1,||b||) +
alpar@9:                             + ||A'*y+z-c||/max(1,||c||) +
alpar@9:                             + |c'*x-b'*y|/max(1,||b||,||c||) */
alpar@9:       double mu;
alpar@9:       /* duality measure mu = x'*z/n (used as barrier parameter) */
alpar@9:       double rmu;
alpar@9:       /* rmu = max(||A*x-b||,||A'*y+z-c||)/mu */
alpar@9:       double rmu0;
alpar@9:       /* the initial value of rmu on iteration 0 */
alpar@9:       double *phi_min; /* double phi_min[1+ITER_MAX]; */
alpar@9:       /* phi_min[k] = min(phi[k]), where phi[k] is the value of phi on
alpar@9:          k-th iteration, 0 <= k <= iter */
alpar@9:       int best_iter;
alpar@9:       /* iteration number, on which the value of phi reached its best
alpar@9:          (minimal) value */
alpar@9:       double *best_x; /* double best_x[1+n]; */
alpar@9:       double *best_y; /* double best_y[1+m]; */
alpar@9:       double *best_z; /* double best_z[1+n]; */
alpar@9:       /* best point (in the sense of the merit function phi) which has
alpar@9:          been reached on iteration iter_best */
alpar@9:       double best_obj;
alpar@9:       /* objective value at the best point */
alpar@9:       double *dx_aff; /* double dx_aff[1+n]; */
alpar@9:       double *dy_aff; /* double dy_aff[1+m]; */
alpar@9:       double *dz_aff; /* double dz_aff[1+n]; */
alpar@9:       /* affine scaling direction */
alpar@9:       double alfa_aff_p, alfa_aff_d;
alpar@9:       /* maximal primal and dual stepsizes in affine scaling direction,
alpar@9:          on which x and z are still non-negative */
alpar@9:       double mu_aff;
alpar@9:       /* duality measure mu_aff = x_aff'*z_aff/n in the boundary point
alpar@9:          x_aff' = x+alfa_aff_p*dx_aff, z_aff' = z+alfa_aff_d*dz_aff */
alpar@9:       double sigma;
alpar@9:       /* Mehrotra's heuristic parameter (0 <= sigma <= 1) */
alpar@9:       double *dx_cc; /* double dx_cc[1+n]; */
alpar@9:       double *dy_cc; /* double dy_cc[1+m]; */
alpar@9:       double *dz_cc; /* double dz_cc[1+n]; */
alpar@9:       /* centering corrector direction */
alpar@9:       double *dx; /* double dx[1+n]; */
alpar@9:       double *dy; /* double dy[1+m]; */
alpar@9:       double *dz; /* double dz[1+n]; */
alpar@9:       /* final combined direction dx = dx_aff+dx_cc, dy = dy_aff+dy_cc,
alpar@9:          dz = dz_aff+dz_cc */
alpar@9:       double alfa_max_p;
alpar@9:       double alfa_max_d;
alpar@9:       /* maximal primal and dual stepsizes in combined direction, on
alpar@9:          which x and z are still non-negative */
alpar@9: };
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  initialize - allocate and initialize common storage area
alpar@9: *
alpar@9: *  This routine allocates and initializes the common storage area (CSA)
alpar@9: *  used by interior-point method routines. */
alpar@9: 
alpar@9: static void initialize(struct csa *csa)
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       int i;
alpar@9:       if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:          xprintf("Matrix A has %d non-zeros\n", csa->A_ptr[m+1]-1);
alpar@9:       csa->D = xcalloc(1+n, sizeof(double));
alpar@9:       /* P := I */
alpar@9:       csa->P = xcalloc(1+m+m, sizeof(int));
alpar@9:       for (i = 1; i <= m; i++) csa->P[i] = csa->P[m+i] = i;
alpar@9:       /* S := A*A', symbolically */
alpar@9:       csa->S_ptr = xcalloc(1+m+1, sizeof(int));
alpar@9:       csa->S_ind = adat_symbolic(m, n, csa->P, csa->A_ptr, csa->A_ind,
alpar@9:          csa->S_ptr);
alpar@9:       if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:          xprintf("Matrix S = A*A' has %d non-zeros (upper triangle)\n",
alpar@9:             csa->S_ptr[m+1]-1 + m);
alpar@9:       /* determine P using specified ordering algorithm */
alpar@9:       if (csa->parm->ord_alg == GLP_ORD_NONE)
alpar@9:       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:             xprintf("Original ordering is being used\n");
alpar@9:          for (i = 1; i <= m; i++)
alpar@9:             csa->P[i] = csa->P[m+i] = i;
alpar@9:       }
alpar@9:       else if (csa->parm->ord_alg == GLP_ORD_QMD)
alpar@9:       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:             xprintf("Minimum degree ordering (QMD)...\n");
alpar@9:          min_degree(m, csa->S_ptr, csa->S_ind, csa->P);
alpar@9:       }
alpar@9:       else if (csa->parm->ord_alg == GLP_ORD_AMD)
alpar@9:       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:             xprintf("Approximate minimum degree ordering (AMD)...\n");
alpar@9:          amd_order1(m, csa->S_ptr, csa->S_ind, csa->P);
alpar@9:       }
alpar@9:       else if (csa->parm->ord_alg == GLP_ORD_SYMAMD)
alpar@9:       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:             xprintf("Approximate minimum degree ordering (SYMAMD)...\n")
alpar@9:                ;
alpar@9:          symamd_ord(m, csa->S_ptr, csa->S_ind, csa->P);
alpar@9:       }
alpar@9:       else
alpar@9:          xassert(csa != csa);
alpar@9:       /* S := P*A*A'*P', symbolically */
alpar@9:       xfree(csa->S_ind);
alpar@9:       csa->S_ind = adat_symbolic(m, n, csa->P, csa->A_ptr, csa->A_ind,
alpar@9:          csa->S_ptr);
alpar@9:       csa->S_val = xcalloc(csa->S_ptr[m+1], sizeof(double));
alpar@9:       csa->S_diag = xcalloc(1+m, sizeof(double));
alpar@9:       /* compute Cholesky factorization S = U'*U, symbolically */
alpar@9:       if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:          xprintf("Computing Cholesky factorization S = L*L'...\n");
alpar@9:       csa->U_ptr = xcalloc(1+m+1, sizeof(int));
alpar@9:       csa->U_ind = chol_symbolic(m, csa->S_ptr, csa->S_ind, csa->U_ptr);
alpar@9:       if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:          xprintf("Matrix L has %d non-zeros\n", csa->U_ptr[m+1]-1 + m);
alpar@9:       csa->U_val = xcalloc(csa->U_ptr[m+1], sizeof(double));
alpar@9:       csa->U_diag = xcalloc(1+m, sizeof(double));
alpar@9:       csa->iter = 0;
alpar@9:       csa->obj = 0.0;
alpar@9:       csa->rpi = 0.0;
alpar@9:       csa->rdi = 0.0;
alpar@9:       csa->gap = 0.0;
alpar@9:       csa->phi = 0.0;
alpar@9:       csa->mu = 0.0;
alpar@9:       csa->rmu = 0.0;
alpar@9:       csa->rmu0 = 0.0;
alpar@9:       csa->phi_min = xcalloc(1+ITER_MAX, sizeof(double));
alpar@9:       csa->best_iter = 0;
alpar@9:       csa->best_x = xcalloc(1+n, sizeof(double));
alpar@9:       csa->best_y = xcalloc(1+m, sizeof(double));
alpar@9:       csa->best_z = xcalloc(1+n, sizeof(double));
alpar@9:       csa->best_obj = 0.0;
alpar@9:       csa->dx_aff = xcalloc(1+n, sizeof(double));
alpar@9:       csa->dy_aff = xcalloc(1+m, sizeof(double));
alpar@9:       csa->dz_aff = xcalloc(1+n, sizeof(double));
alpar@9:       csa->alfa_aff_p = 0.0;
alpar@9:       csa->alfa_aff_d = 0.0;
alpar@9:       csa->mu_aff = 0.0;
alpar@9:       csa->sigma = 0.0;
alpar@9:       csa->dx_cc = xcalloc(1+n, sizeof(double));
alpar@9:       csa->dy_cc = xcalloc(1+m, sizeof(double));
alpar@9:       csa->dz_cc = xcalloc(1+n, sizeof(double));
alpar@9:       csa->dx = csa->dx_aff;
alpar@9:       csa->dy = csa->dy_aff;
alpar@9:       csa->dz = csa->dz_aff;
alpar@9:       csa->alfa_max_p = 0.0;
alpar@9:       csa->alfa_max_d = 0.0;
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  A_by_vec - compute y = A*x
alpar@9: *
alpar@9: *  This routine computes matrix-vector product y = A*x, where A is the
alpar@9: *  constraint matrix. */
alpar@9: 
alpar@9: static void A_by_vec(struct csa *csa, double x[], double y[])
alpar@9: {     /* compute y = A*x */
alpar@9:       int m = csa->m;
alpar@9:       int *A_ptr = csa->A_ptr;
alpar@9:       int *A_ind = csa->A_ind;
alpar@9:       double *A_val = csa->A_val;
alpar@9:       int i, t, beg, end;
alpar@9:       double temp;
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:       {  temp = 0.0;
alpar@9:          beg = A_ptr[i], end = A_ptr[i+1];
alpar@9:          for (t = beg; t < end; t++) temp += A_val[t] * x[A_ind[t]];
alpar@9:          y[i] = temp;
alpar@9:       }
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  AT_by_vec - compute y = A'*x
alpar@9: *
alpar@9: *  This routine computes matrix-vector product y = A'*x, where A' is a
alpar@9: *  matrix transposed to the constraint matrix A. */
alpar@9: 
alpar@9: static void AT_by_vec(struct csa *csa, double x[], double y[])
alpar@9: {     /* compute y = A'*x, where A' is transposed to A */
alpar@9:       int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       int *A_ptr = csa->A_ptr;
alpar@9:       int *A_ind = csa->A_ind;
alpar@9:       double *A_val = csa->A_val;
alpar@9:       int i, j, t, beg, end;
alpar@9:       double temp;
alpar@9:       for (j = 1; j <= n; j++) y[j] = 0.0;
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:       {  temp = x[i];
alpar@9:          if (temp == 0.0) continue;
alpar@9:          beg = A_ptr[i], end = A_ptr[i+1];
alpar@9:          for (t = beg; t < end; t++) y[A_ind[t]] += A_val[t] * temp;
alpar@9:       }
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  decomp_NE - numeric factorization of matrix S = P*A*D*A'*P'
alpar@9: *
alpar@9: *  This routine implements numeric phase of Cholesky factorization of
alpar@9: *  the matrix S = P*A*D*A'*P', which is a permuted matrix of the normal
alpar@9: *  equation system. Matrix D is assumed to be already computed. */
alpar@9: 
alpar@9: static void decomp_NE(struct csa *csa)
alpar@9: {     adat_numeric(csa->m, csa->n, csa->P, csa->A_ptr, csa->A_ind,
alpar@9:          csa->A_val, csa->D, csa->S_ptr, csa->S_ind, csa->S_val,
alpar@9:          csa->S_diag);
alpar@9:       chol_numeric(csa->m, csa->S_ptr, csa->S_ind, csa->S_val,
alpar@9:          csa->S_diag, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag);
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  solve_NE - solve normal equation system
alpar@9: *
alpar@9: *  This routine solves the normal equation system:
alpar@9: *
alpar@9: *     A*D*A'*y = h.
alpar@9: *
alpar@9: *  It is assumed that the matrix A*D*A' has been previously factorized
alpar@9: *  by the routine decomp_NE.
alpar@9: *
alpar@9: *  On entry the array y contains the vector of right-hand sides h. On
alpar@9: *  exit this array contains the computed vector of unknowns y.
alpar@9: *
alpar@9: *  Once the vector y has been computed the routine checks for numeric
alpar@9: *  stability. If the residual vector:
alpar@9: *
alpar@9: *     r = A*D*A'*y - h
alpar@9: *
alpar@9: *  is relatively small, the routine returns zero, otherwise non-zero is
alpar@9: *  returned. */
alpar@9: 
alpar@9: static int solve_NE(struct csa *csa, double y[])
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       int *P = csa->P;
alpar@9:       int i, j, ret = 0;
alpar@9:       double *h, *r, *w;
alpar@9:       /* save vector of right-hand sides h */
alpar@9:       h = xcalloc(1+m, sizeof(double));
alpar@9:       for (i = 1; i <= m; i++) h[i] = y[i];
alpar@9:       /* solve normal equation system (A*D*A')*y = h */
alpar@9:       /* since S = P*A*D*A'*P' = U'*U, then A*D*A' = P'*U'*U*P, so we
alpar@9:          have inv(A*D*A') = P'*inv(U)*inv(U')*P */
alpar@9:       /* w := P*h */
alpar@9:       w = xcalloc(1+m, sizeof(double));
alpar@9:       for (i = 1; i <= m; i++) w[i] = y[P[i]];
alpar@9:       /* w := inv(U')*w */
alpar@9:       ut_solve(m, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag, w);
alpar@9:       /* w := inv(U)*w */
alpar@9:       u_solve(m, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag, w);
alpar@9:       /* y := P'*w */
alpar@9:       for (i = 1; i <= m; i++) y[i] = w[P[m+i]];
alpar@9:       xfree(w);
alpar@9:       /* compute residual vector r = A*D*A'*y - h */
alpar@9:       r = xcalloc(1+m, sizeof(double));
alpar@9:       /* w := A'*y */
alpar@9:       w = xcalloc(1+n, sizeof(double));
alpar@9:       AT_by_vec(csa, y, w);
alpar@9:       /* w := D*w */
alpar@9:       for (j = 1; j <= n; j++) w[j] *= csa->D[j];
alpar@9:       /* r := A*w */
alpar@9:       A_by_vec(csa, w, r);
alpar@9:       xfree(w);
alpar@9:       /* r := r - h */
alpar@9:       for (i = 1; i <= m; i++) r[i] -= h[i];
alpar@9:       /* check for numeric stability */
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:       {  if (fabs(r[i]) / (1.0 + fabs(h[i])) > 1e-4)
alpar@9:          {  ret = 1;
alpar@9:             break;
alpar@9:          }
alpar@9:       }
alpar@9:       xfree(h);
alpar@9:       xfree(r);
alpar@9:       return ret;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  solve_NS - solve Newtonian system
alpar@9: *
alpar@9: *  This routine solves the Newtonian system:
alpar@9: *
alpar@9: *     A*dx               = p
alpar@9: *
alpar@9: *           A'*dy +   dz = q
alpar@9: *
alpar@9: *     Z*dx        + X*dz = r
alpar@9: *
alpar@9: *  where X = diag(x[j]), Z = diag(z[j]), by reducing it to the normal
alpar@9: *  equation system:
alpar@9: *
alpar@9: *     (A*inv(Z)*X*A')*dy = A*inv(Z)*(X*q-r)+p
alpar@9: *
alpar@9: *  (it is assumed that the matrix A*inv(Z)*X*A' has been factorized by
alpar@9: *  the routine decomp_NE).
alpar@9: *
alpar@9: *  Once vector dy has been computed the routine computes vectors dx and
alpar@9: *  dz as follows:
alpar@9: *
alpar@9: *     dx = inv(Z)*(X*(A'*dy-q)+r)
alpar@9: *
alpar@9: *     dz = inv(X)*(r-Z*dx)
alpar@9: *
alpar@9: *  The routine solve_NS returns the same code which was reported by the
alpar@9: *  routine solve_NE (see above). */
alpar@9: 
alpar@9: static int solve_NS(struct csa *csa, double p[], double q[], double r[],
alpar@9:       double dx[], double dy[], double dz[])
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       double *x = csa->x;
alpar@9:       double *z = csa->z;
alpar@9:       int i, j, ret;
alpar@9:       double *w = dx;
alpar@9:       /* compute the vector of right-hand sides A*inv(Z)*(X*q-r)+p for
alpar@9:          the normal equation system */
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:          w[j] = (x[j] * q[j] - r[j]) / z[j];
alpar@9:       A_by_vec(csa, w, dy);
alpar@9:       for (i = 1; i <= m; i++) dy[i] += p[i];
alpar@9:       /* solve the normal equation system to compute vector dy */
alpar@9:       ret = solve_NE(csa, dy);
alpar@9:       /* compute vectors dx and dz */
alpar@9:       AT_by_vec(csa, dy, dx);
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  dx[j] = (x[j] * (dx[j] - q[j]) + r[j]) / z[j];
alpar@9:          dz[j] = (r[j] - z[j] * dx[j]) / x[j];
alpar@9:       }
alpar@9:       return ret;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  initial_point - choose initial point using Mehrotra's heuristic
alpar@9: *
alpar@9: *  This routine chooses a starting point using a heuristic proposed in
alpar@9: *  the paper:
alpar@9: *
alpar@9: *  S. Mehrotra. On the implementation of a primal-dual interior point
alpar@9: *  method. SIAM J. on Optim., 2(4), pp. 575-601, 1992.
alpar@9: *
alpar@9: *  The starting point x in the primal space is chosen as a solution of
alpar@9: *  the following least squares problem:
alpar@9: *
alpar@9: *     minimize    ||x||
alpar@9: *
alpar@9: *     subject to  A*x = b
alpar@9: *
alpar@9: *  which can be computed explicitly as follows:
alpar@9: *
alpar@9: *     x = A'*inv(A*A')*b
alpar@9: *
alpar@9: *  Similarly, the starting point (y, z) in the dual space is chosen as
alpar@9: *  a solution of the following least squares problem:
alpar@9: *
alpar@9: *     minimize    ||z||
alpar@9: *
alpar@9: *     subject to  A'*y + z = c
alpar@9: *
alpar@9: *  which can be computed explicitly as follows:
alpar@9: *
alpar@9: *     y = inv(A*A')*A*c
alpar@9: *
alpar@9: *     z = c - A'*y
alpar@9: *
alpar@9: *  However, some components of the vectors x and z may be non-positive
alpar@9: *  or close to zero, so the routine uses a Mehrotra's heuristic to find
alpar@9: *  a more appropriate starting point. */
alpar@9: 
alpar@9: static void initial_point(struct csa *csa)
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       double *b = csa->b;
alpar@9:       double *c = csa->c;
alpar@9:       double *x = csa->x;
alpar@9:       double *y = csa->y;
alpar@9:       double *z = csa->z;
alpar@9:       double *D = csa->D;
alpar@9:       int i, j;
alpar@9:       double dp, dd, ex, ez, xz;
alpar@9:       /* factorize A*A' */
alpar@9:       for (j = 1; j <= n; j++) D[j] = 1.0;
alpar@9:       decomp_NE(csa);
alpar@9:       /* x~ = A'*inv(A*A')*b */
alpar@9:       for (i = 1; i <= m; i++) y[i] = b[i];
alpar@9:       solve_NE(csa, y);
alpar@9:       AT_by_vec(csa, y, x);
alpar@9:       /* y~ = inv(A*A')*A*c */
alpar@9:       A_by_vec(csa, c, y);
alpar@9:       solve_NE(csa, y);
alpar@9:       /* z~ = c - A'*y~ */
alpar@9:       AT_by_vec(csa, y,z);
alpar@9:       for (j = 1; j <= n; j++) z[j] = c[j] - z[j];
alpar@9:       /* use Mehrotra's heuristic in order to choose more appropriate
alpar@9:          starting point with positive components of vectors x and z */
alpar@9:       dp = dd = 0.0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  if (dp < -1.5 * x[j]) dp = -1.5 * x[j];
alpar@9:          if (dd < -1.5 * z[j]) dd = -1.5 * z[j];
alpar@9:       }
alpar@9:       /* note that b = 0 involves x = 0, and c = 0 involves y = 0 and
alpar@9:          z = 0, so we need to be careful */
alpar@9:       if (dp == 0.0) dp = 1.5;
alpar@9:       if (dd == 0.0) dd = 1.5;
alpar@9:       ex = ez = xz = 0.0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  ex += (x[j] + dp);
alpar@9:          ez += (z[j] + dd);
alpar@9:          xz += (x[j] + dp) * (z[j] + dd);
alpar@9:       }
alpar@9:       dp += 0.5 * (xz / ez);
alpar@9:       dd += 0.5 * (xz / ex);
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  x[j] += dp;
alpar@9:          z[j] += dd;
alpar@9:          xassert(x[j] > 0.0 && z[j] > 0.0);
alpar@9:       }
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  basic_info - perform basic computations at the current point
alpar@9: *
alpar@9: *  This routine computes the following quantities at the current point:
alpar@9: *
alpar@9: *  1) value of the objective function:
alpar@9: *
alpar@9: *     F = c'*x + c[0]
alpar@9: *
alpar@9: *  2) relative primal infeasibility:
alpar@9: *
alpar@9: *     rpi = ||A*x-b|| / (1+||b||)
alpar@9: *
alpar@9: *  3) relative dual infeasibility:
alpar@9: *
alpar@9: *     rdi = ||A'*y+z-c|| / (1+||c||)
alpar@9: *
alpar@9: *  4) primal-dual gap (relative difference between the primal and the
alpar@9: *     dual objective function values):
alpar@9: *
alpar@9: *     gap = |c'*x-b'*y| / (1+|c'*x|)
alpar@9: *
alpar@9: *  5) merit function:
alpar@9: *
alpar@9: *     phi = ||A*x-b|| / max(1,||b||) + ||A'*y+z-c|| / max(1,||c||) +
alpar@9: *
alpar@9: *         + |c'*x-b'*y| / max(1,||b||,||c||)
alpar@9: *
alpar@9: *  6) duality measure:
alpar@9: *
alpar@9: *     mu = x'*z / n
alpar@9: *
alpar@9: *  7) the ratio of infeasibility to mu:
alpar@9: *
alpar@9: *     rmu = max(||A*x-b||,||A'*y+z-c||) / mu
alpar@9: *
alpar@9: *  where ||*|| denotes euclidian norm, *' denotes transposition. */
alpar@9: 
alpar@9: static void basic_info(struct csa *csa)
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       double *b = csa->b;
alpar@9:       double *c = csa->c;
alpar@9:       double *x = csa->x;
alpar@9:       double *y = csa->y;
alpar@9:       double *z = csa->z;
alpar@9:       int i, j;
alpar@9:       double norm1, bnorm, norm2, cnorm, cx, by, *work, temp;
alpar@9:       /* compute value of the objective function */
alpar@9:       temp = c[0];
alpar@9:       for (j = 1; j <= n; j++) temp += c[j] * x[j];
alpar@9:       csa->obj = temp;
alpar@9:       /* norm1 = ||A*x-b|| */
alpar@9:       work = xcalloc(1+m, sizeof(double));
alpar@9:       A_by_vec(csa, x, work);
alpar@9:       norm1 = 0.0;
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:          norm1 += (work[i] - b[i]) * (work[i] - b[i]);
alpar@9:       norm1 = sqrt(norm1);
alpar@9:       xfree(work);
alpar@9:       /* bnorm = ||b|| */
alpar@9:       bnorm = 0.0;
alpar@9:       for (i = 1; i <= m; i++) bnorm += b[i] * b[i];
alpar@9:       bnorm = sqrt(bnorm);
alpar@9:       /* compute relative primal infeasibility */
alpar@9:       csa->rpi = norm1 / (1.0 + bnorm);
alpar@9:       /* norm2 = ||A'*y+z-c|| */
alpar@9:       work = xcalloc(1+n, sizeof(double));
alpar@9:       AT_by_vec(csa, y, work);
alpar@9:       norm2 = 0.0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:          norm2 += (work[j] + z[j] - c[j]) * (work[j] + z[j] - c[j]);
alpar@9:       norm2 = sqrt(norm2);
alpar@9:       xfree(work);
alpar@9:       /* cnorm = ||c|| */
alpar@9:       cnorm = 0.0;
alpar@9:       for (j = 1; j <= n; j++) cnorm += c[j] * c[j];
alpar@9:       cnorm = sqrt(cnorm);
alpar@9:       /* compute relative dual infeasibility */
alpar@9:       csa->rdi = norm2 / (1.0 + cnorm);
alpar@9:       /* by = b'*y */
alpar@9:       by = 0.0;
alpar@9:       for (i = 1; i <= m; i++) by += b[i] * y[i];
alpar@9:       /* cx = c'*x */
alpar@9:       cx = 0.0;
alpar@9:       for (j = 1; j <= n; j++) cx += c[j] * x[j];
alpar@9:       /* compute primal-dual gap */
alpar@9:       csa->gap = fabs(cx - by) / (1.0 + fabs(cx));
alpar@9:       /* compute merit function */
alpar@9:       csa->phi = 0.0;
alpar@9:       csa->phi += norm1 / (bnorm > 1.0 ? bnorm : 1.0);
alpar@9:       csa->phi += norm2 / (cnorm > 1.0 ? cnorm : 1.0);
alpar@9:       temp = 1.0;
alpar@9:       if (temp < bnorm) temp = bnorm;
alpar@9:       if (temp < cnorm) temp = cnorm;
alpar@9:       csa->phi += fabs(cx - by) / temp;
alpar@9:       /* compute duality measure */
alpar@9:       temp = 0.0;
alpar@9:       for (j = 1; j <= n; j++) temp += x[j] * z[j];
alpar@9:       csa->mu = temp / (double)n;
alpar@9:       /* compute the ratio of infeasibility to mu */
alpar@9:       csa->rmu = (norm1 > norm2 ? norm1 : norm2) / csa->mu;
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  make_step - compute next point using Mehrotra's technique
alpar@9: *
alpar@9: *  This routine computes the next point using the predictor-corrector
alpar@9: *  technique proposed in the paper:
alpar@9: *
alpar@9: *  S. Mehrotra. On the implementation of a primal-dual interior point
alpar@9: *  method. SIAM J. on Optim., 2(4), pp. 575-601, 1992.
alpar@9: *
alpar@9: *  At first, the routine computes so called affine scaling (predictor)
alpar@9: *  direction (dx_aff,dy_aff,dz_aff) which is a solution of the system:
alpar@9: *
alpar@9: *     A*dx_aff                       = b - A*x
alpar@9: *
alpar@9: *               A'*dy_aff +   dz_aff = c - A'*y - z
alpar@9: *
alpar@9: *     Z*dx_aff            + X*dz_aff = - X*Z*e
alpar@9: *
alpar@9: *  where (x,y,z) is the current point, X = diag(x[j]), Z = diag(z[j]),
alpar@9: *  e = (1,...,1)'.
alpar@9: *
alpar@9: *  Then, the routine computes the centering parameter sigma, using the
alpar@9: *  following Mehrotra's heuristic:
alpar@9: *
alpar@9: *     alfa_aff_p = inf{0 <= alfa <= 1 | x+alfa*dx_aff >= 0}
alpar@9: *
alpar@9: *     alfa_aff_d = inf{0 <= alfa <= 1 | z+alfa*dz_aff >= 0}
alpar@9: *
alpar@9: *     mu_aff = (x+alfa_aff_p*dx_aff)'*(z+alfa_aff_d*dz_aff)/n
alpar@9: *
alpar@9: *     sigma = (mu_aff/mu)^3
alpar@9: *
alpar@9: *  where alfa_aff_p is the maximal stepsize along the affine scaling
alpar@9: *  direction in the primal space, alfa_aff_d is the maximal stepsize
alpar@9: *  along the same direction in the dual space.
alpar@9: *
alpar@9: *  After determining sigma the routine computes so called centering
alpar@9: *  (corrector) direction (dx_cc,dy_cc,dz_cc) which is the solution of
alpar@9: *  the system:
alpar@9: *
alpar@9: *     A*dx_cc                     = 0
alpar@9: *
alpar@9: *              A'*dy_cc +   dz_cc = 0
alpar@9: *
alpar@9: *     Z*dx_cc           + X*dz_cc = sigma*mu*e - X*Z*e
alpar@9: *
alpar@9: *  Finally, the routine computes the combined direction
alpar@9: *
alpar@9: *     (dx,dy,dz) = (dx_aff,dy_aff,dz_aff) + (dx_cc,dy_cc,dz_cc)
alpar@9: *
alpar@9: *  and determines maximal primal and dual stepsizes along the combined
alpar@9: *  direction:
alpar@9: *
alpar@9: *     alfa_max_p = inf{0 <= alfa <= 1 | x+alfa*dx >= 0}
alpar@9: *
alpar@9: *     alfa_max_d = inf{0 <= alfa <= 1 | z+alfa*dz >= 0}
alpar@9: *
alpar@9: *  In order to prevent the next point to be too close to the boundary
alpar@9: *  of the positive ortant, the routine decreases maximal stepsizes:
alpar@9: *
alpar@9: *     alfa_p = gamma_p * alfa_max_p
alpar@9: *
alpar@9: *     alfa_d = gamma_d * alfa_max_d
alpar@9: *
alpar@9: *  where gamma_p and gamma_d are scaling factors, and computes the next
alpar@9: *  point:
alpar@9: *
alpar@9: *     x_new = x + alfa_p * dx
alpar@9: *
alpar@9: *     y_new = y + alfa_d * dy
alpar@9: *
alpar@9: *     z_new = z + alfa_d * dz
alpar@9: *
alpar@9: *  which becomes the current point on the next iteration. */
alpar@9: 
alpar@9: static int make_step(struct csa *csa)
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       double *b = csa->b;
alpar@9:       double *c = csa->c;
alpar@9:       double *x = csa->x;
alpar@9:       double *y = csa->y;
alpar@9:       double *z = csa->z;
alpar@9:       double *dx_aff = csa->dx_aff;
alpar@9:       double *dy_aff = csa->dy_aff;
alpar@9:       double *dz_aff = csa->dz_aff;
alpar@9:       double *dx_cc = csa->dx_cc;
alpar@9:       double *dy_cc = csa->dy_cc;
alpar@9:       double *dz_cc = csa->dz_cc;
alpar@9:       double *dx = csa->dx;
alpar@9:       double *dy = csa->dy;
alpar@9:       double *dz = csa->dz;
alpar@9:       int i, j, ret = 0;
alpar@9:       double temp, gamma_p, gamma_d, *p, *q, *r;
alpar@9:       /* allocate working arrays */
alpar@9:       p = xcalloc(1+m, sizeof(double));
alpar@9:       q = xcalloc(1+n, sizeof(double));
alpar@9:       r = xcalloc(1+n, sizeof(double));
alpar@9:       /* p = b - A*x */
alpar@9:       A_by_vec(csa, x, p);
alpar@9:       for (i = 1; i <= m; i++) p[i] = b[i] - p[i];
alpar@9:       /* q = c - A'*y - z */
alpar@9:       AT_by_vec(csa, y,q);
alpar@9:       for (j = 1; j <= n; j++) q[j] = c[j] - q[j] - z[j];
alpar@9:       /* r = - X * Z * e */
alpar@9:       for (j = 1; j <= n; j++) r[j] = - x[j] * z[j];
alpar@9:       /* solve the first Newtonian system */
alpar@9:       if (solve_NS(csa, p, q, r, dx_aff, dy_aff, dz_aff))
alpar@9:       {  ret = 1;
alpar@9:          goto done;
alpar@9:       }
alpar@9:       /* alfa_aff_p = inf{0 <= alfa <= 1 | x + alfa*dx_aff >= 0} */
alpar@9:       /* alfa_aff_d = inf{0 <= alfa <= 1 | z + alfa*dz_aff >= 0} */
alpar@9:       csa->alfa_aff_p = csa->alfa_aff_d = 1.0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  if (dx_aff[j] < 0.0)
alpar@9:          {  temp = - x[j] / dx_aff[j];
alpar@9:             if (csa->alfa_aff_p > temp) csa->alfa_aff_p = temp;
alpar@9:          }
alpar@9:          if (dz_aff[j] < 0.0)
alpar@9:          {  temp = - z[j] / dz_aff[j];
alpar@9:             if (csa->alfa_aff_d > temp) csa->alfa_aff_d = temp;
alpar@9:          }
alpar@9:       }
alpar@9:       /* mu_aff = (x+alfa_aff_p*dx_aff)' * (z+alfa_aff_d*dz_aff) / n */
alpar@9:       temp = 0.0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:          temp += (x[j] + csa->alfa_aff_p * dx_aff[j]) *
alpar@9:                  (z[j] + csa->alfa_aff_d * dz_aff[j]);
alpar@9:       csa->mu_aff = temp / (double)n;
alpar@9:       /* sigma = (mu_aff/mu)^3 */
alpar@9:       temp = csa->mu_aff / csa->mu;
alpar@9:       csa->sigma = temp * temp * temp;
alpar@9:       /* p = 0 */
alpar@9:       for (i = 1; i <= m; i++) p[i] = 0.0;
alpar@9:       /* q = 0 */
alpar@9:       for (j = 1; j <= n; j++) q[j] = 0.0;
alpar@9:       /* r = sigma * mu * e - X * Z * e */
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:          r[j] = csa->sigma * csa->mu - dx_aff[j] * dz_aff[j];
alpar@9:       /* solve the second Newtonian system with the same coefficients
alpar@9:          but with altered right-hand sides */
alpar@9:       if (solve_NS(csa, p, q, r, dx_cc, dy_cc, dz_cc))
alpar@9:       {  ret = 1;
alpar@9:          goto done;
alpar@9:       }
alpar@9:       /* (dx,dy,dz) = (dx_aff,dy_aff,dz_aff) + (dx_cc,dy_cc,dz_cc) */
alpar@9:       for (j = 1; j <= n; j++) dx[j] = dx_aff[j] + dx_cc[j];
alpar@9:       for (i = 1; i <= m; i++) dy[i] = dy_aff[i] + dy_cc[i];
alpar@9:       for (j = 1; j <= n; j++) dz[j] = dz_aff[j] + dz_cc[j];
alpar@9:       /* alfa_max_p = inf{0 <= alfa <= 1 | x + alfa*dx >= 0} */
alpar@9:       /* alfa_max_d = inf{0 <= alfa <= 1 | z + alfa*dz >= 0} */
alpar@9:       csa->alfa_max_p = csa->alfa_max_d = 1.0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  if (dx[j] < 0.0)
alpar@9:          {  temp = - x[j] / dx[j];
alpar@9:             if (csa->alfa_max_p > temp) csa->alfa_max_p = temp;
alpar@9:          }
alpar@9:          if (dz[j] < 0.0)
alpar@9:          {  temp = - z[j] / dz[j];
alpar@9:             if (csa->alfa_max_d > temp) csa->alfa_max_d = temp;
alpar@9:          }
alpar@9:       }
alpar@9:       /* determine scale factors (not implemented yet) */
alpar@9:       gamma_p = 0.90;
alpar@9:       gamma_d = 0.90;
alpar@9:       /* compute the next point */
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  x[j] += gamma_p * csa->alfa_max_p * dx[j];
alpar@9:          xassert(x[j] > 0.0);
alpar@9:       }
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:          y[i] += gamma_d * csa->alfa_max_d * dy[i];
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  z[j] += gamma_d * csa->alfa_max_d * dz[j];
alpar@9:          xassert(z[j] > 0.0);
alpar@9:       }
alpar@9: done: /* free working arrays */
alpar@9:       xfree(p);
alpar@9:       xfree(q);
alpar@9:       xfree(r);
alpar@9:       return ret;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  terminate - deallocate common storage area
alpar@9: *
alpar@9: *  This routine frees all memory allocated to the common storage area
alpar@9: *  used by interior-point method routines. */
alpar@9: 
alpar@9: static void terminate(struct csa *csa)
alpar@9: {     xfree(csa->D);
alpar@9:       xfree(csa->P);
alpar@9:       xfree(csa->S_ptr);
alpar@9:       xfree(csa->S_ind);
alpar@9:       xfree(csa->S_val);
alpar@9:       xfree(csa->S_diag);
alpar@9:       xfree(csa->U_ptr);
alpar@9:       xfree(csa->U_ind);
alpar@9:       xfree(csa->U_val);
alpar@9:       xfree(csa->U_diag);
alpar@9:       xfree(csa->phi_min);
alpar@9:       xfree(csa->best_x);
alpar@9:       xfree(csa->best_y);
alpar@9:       xfree(csa->best_z);
alpar@9:       xfree(csa->dx_aff);
alpar@9:       xfree(csa->dy_aff);
alpar@9:       xfree(csa->dz_aff);
alpar@9:       xfree(csa->dx_cc);
alpar@9:       xfree(csa->dy_cc);
alpar@9:       xfree(csa->dz_cc);
alpar@9:       return;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  ipm_main - main interior-point method routine
alpar@9: *
alpar@9: *  This is a main routine of the primal-dual interior-point method.
alpar@9: *
alpar@9: *  The routine ipm_main returns one of the following codes:
alpar@9: *
alpar@9: *  0 - optimal solution found;
alpar@9: *  1 - problem has no feasible (primal or dual) solution;
alpar@9: *  2 - no convergence;
alpar@9: *  3 - iteration limit exceeded;
alpar@9: *  4 - numeric instability on solving Newtonian system.
alpar@9: *
alpar@9: *  In case of non-zero return code the routine returns the best point,
alpar@9: *  which has been reached during optimization. */
alpar@9: 
alpar@9: static int ipm_main(struct csa *csa)
alpar@9: {     int m = csa->m;
alpar@9:       int n = csa->n;
alpar@9:       int i, j, status;
alpar@9:       double temp;
alpar@9:       /* choose initial point using Mehrotra's heuristic */
alpar@9:       if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:          xprintf("Guessing initial point...\n");
alpar@9:       initial_point(csa);
alpar@9:       /* main loop starts here */
alpar@9:       if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:          xprintf("Optimization begins...\n");
alpar@9:       for (;;)
alpar@9:       {  /* perform basic computations at the current point */
alpar@9:          basic_info(csa);
alpar@9:          /* save initial value of rmu */
alpar@9:          if (csa->iter == 0) csa->rmu0 = csa->rmu;
alpar@9:          /* accumulate values of min(phi[k]) and save the best point */
alpar@9:          xassert(csa->iter <= ITER_MAX);
alpar@9:          if (csa->iter == 0 || csa->phi_min[csa->iter-1] > csa->phi)
alpar@9:          {  csa->phi_min[csa->iter] = csa->phi;
alpar@9:             csa->best_iter = csa->iter;
alpar@9:             for (j = 1; j <= n; j++) csa->best_x[j] = csa->x[j];
alpar@9:             for (i = 1; i <= m; i++) csa->best_y[i] = csa->y[i];
alpar@9:             for (j = 1; j <= n; j++) csa->best_z[j] = csa->z[j];
alpar@9:             csa->best_obj = csa->obj;
alpar@9:          }
alpar@9:          else
alpar@9:             csa->phi_min[csa->iter] = csa->phi_min[csa->iter-1];
alpar@9:          /* display information at the current point */
alpar@9:          if (csa->parm->msg_lev >= GLP_MSG_ON)
alpar@9:             xprintf("%3d: obj = %17.9e; rpi = %8.1e; rdi = %8.1e; gap ="
alpar@9:                " %8.1e\n", csa->iter, csa->obj, csa->rpi, csa->rdi,
alpar@9:                csa->gap);
alpar@9:          /* check if the current point is optimal */
alpar@9:          if (csa->rpi < 1e-8 && csa->rdi < 1e-8 && csa->gap < 1e-8)
alpar@9:          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:                xprintf("OPTIMAL SOLUTION FOUND\n");
alpar@9:             status = 0;
alpar@9:             break;
alpar@9:          }
alpar@9:          /* check if the problem has no feasible solution */
alpar@9:          temp = 1e5 * csa->phi_min[csa->iter];
alpar@9:          if (temp < 1e-8) temp = 1e-8;
alpar@9:          if (csa->phi >= temp)
alpar@9:          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:                xprintf("PROBLEM HAS NO FEASIBLE PRIMAL/DUAL SOLUTION\n")
alpar@9:                   ;
alpar@9:             status = 1;
alpar@9:             break;
alpar@9:          }
alpar@9:          /* check for very slow convergence or divergence */
alpar@9:          if (((csa->rpi >= 1e-8 || csa->rdi >= 1e-8) && csa->rmu /
alpar@9:                csa->rmu0 >= 1e6) ||
alpar@9:                (csa->iter >= 30 && csa->phi_min[csa->iter] >= 0.5 *
alpar@9:                csa->phi_min[csa->iter - 30]))
alpar@9:          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:                xprintf("NO CONVERGENCE; SEARCH TERMINATED\n");
alpar@9:             status = 2;
alpar@9:             break;
alpar@9:          }
alpar@9:          /* check for maximal number of iterations */
alpar@9:          if (csa->iter == ITER_MAX)
alpar@9:          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:                xprintf("ITERATION LIMIT EXCEEDED; SEARCH TERMINATED\n");
alpar@9:             status = 3;
alpar@9:             break;
alpar@9:          }
alpar@9:          /* start the next iteration */
alpar@9:          csa->iter++;
alpar@9:          /* factorize normal equation system */
alpar@9:          for (j = 1; j <= n; j++) csa->D[j] = csa->x[j] / csa->z[j];
alpar@9:          decomp_NE(csa);
alpar@9:          /* compute the next point using Mehrotra's predictor-corrector
alpar@9:             technique */
alpar@9:          if (make_step(csa))
alpar@9:          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:                xprintf("NUMERIC INSTABILITY; SEARCH TERMINATED\n");
alpar@9:             status = 4;
alpar@9:             break;
alpar@9:          }
alpar@9:       }
alpar@9:       /* restore the best point */
alpar@9:       if (status != 0)
alpar@9:       {  for (j = 1; j <= n; j++) csa->x[j] = csa->best_x[j];
alpar@9:          for (i = 1; i <= m; i++) csa->y[i] = csa->best_y[i];
alpar@9:          for (j = 1; j <= n; j++) csa->z[j] = csa->best_z[j];
alpar@9:          if (csa->parm->msg_lev >= GLP_MSG_ALL)
alpar@9:             xprintf("Best point %17.9e was reached on iteration %d\n",
alpar@9:                csa->best_obj, csa->best_iter);
alpar@9:       }
alpar@9:       /* return to the calling program */
alpar@9:       return status;
alpar@9: }
alpar@9: 
alpar@9: /***********************************************************************
alpar@9: *  NAME
alpar@9: *
alpar@9: *  ipm_solve - core LP solver based on the interior-point method
alpar@9: *
alpar@9: *  SYNOPSIS
alpar@9: *
alpar@9: *  #include "glpipm.h"
alpar@9: *  int ipm_solve(glp_prob *P, const glp_iptcp *parm);
alpar@9: *
alpar@9: *  DESCRIPTION
alpar@9: *
alpar@9: *  The routine ipm_solve is a core LP solver based on the primal-dual
alpar@9: *  interior-point method.
alpar@9: *
alpar@9: *  The routine assumes the following standard formulation of LP problem
alpar@9: *  to be solved:
alpar@9: *
alpar@9: *     minimize
alpar@9: *
alpar@9: *        F = c[0] + c[1]*x[1] + c[2]*x[2] + ... + c[n]*x[n]
alpar@9: *
alpar@9: *     subject to linear constraints
alpar@9: *
alpar@9: *        a[1,1]*x[1] + a[1,2]*x[2] + ... + a[1,n]*x[n] = b[1]
alpar@9: *
alpar@9: *        a[2,1]*x[1] + a[2,2]*x[2] + ... + a[2,n]*x[n] = b[2]
alpar@9: *
alpar@9: *              . . . . . .
alpar@9: *
alpar@9: *        a[m,1]*x[1] + a[m,2]*x[2] + ... + a[m,n]*x[n] = b[m]
alpar@9: *
alpar@9: *     and non-negative variables
alpar@9: *
alpar@9: *        x[1] >= 0, x[2] >= 0, ..., x[n] >= 0
alpar@9: *
alpar@9: *  where:
alpar@9: *  F                    is the objective function;
alpar@9: *  x[1], ..., x[n]      are (structural) variables;
alpar@9: *  c[0]                 is a constant term of the objective function;
alpar@9: *  c[1], ..., c[n]      are objective coefficients;
alpar@9: *  a[1,1], ..., a[m,n]  are constraint coefficients;
alpar@9: *  b[1], ..., b[n]      are right-hand sides.
alpar@9: *
alpar@9: *  The solution is three vectors x, y, and z, which are stored by the
alpar@9: *  routine in the arrays x, y, and z, respectively. These vectors
alpar@9: *  correspond to the best primal-dual point found during optimization.
alpar@9: *  They are approximate solution of the following system (which is the
alpar@9: *  Karush-Kuhn-Tucker optimality conditions):
alpar@9: *
alpar@9: *     A*x      = b      (primal feasibility condition)
alpar@9: *
alpar@9: *     A'*y + z = c      (dual feasibility condition)
alpar@9: *
alpar@9: *     x'*z     = 0      (primal-dual complementarity condition)
alpar@9: *
alpar@9: *     x >= 0, z >= 0    (non-negativity condition)
alpar@9: *
alpar@9: *  where:
alpar@9: *  x[1], ..., x[n]      are primal (structural) variables;
alpar@9: *  y[1], ..., y[m]      are dual variables (Lagrange multipliers) for
alpar@9: *                       equality constraints;
alpar@9: *  z[1], ..., z[n]      are dual variables (Lagrange multipliers) for
alpar@9: *                       non-negativity constraints.
alpar@9: *
alpar@9: *  RETURNS
alpar@9: *
alpar@9: *  0  LP has been successfully solved.
alpar@9: *
alpar@9: *  GLP_ENOCVG
alpar@9: *     No convergence.
alpar@9: *
alpar@9: *  GLP_EITLIM
alpar@9: *     Iteration limit exceeded.
alpar@9: *
alpar@9: *  GLP_EINSTAB
alpar@9: *     Numeric instability on solving Newtonian system.
alpar@9: *
alpar@9: *  In case of non-zero return code the routine returns the best point,
alpar@9: *  which has been reached during optimization. */
alpar@9: 
alpar@9: int ipm_solve(glp_prob *P, const glp_iptcp *parm)
alpar@9: {     struct csa _dsa, *csa = &_dsa;
alpar@9:       int m = P->m;
alpar@9:       int n = P->n;
alpar@9:       int nnz = P->nnz;
alpar@9:       GLPROW *row;
alpar@9:       GLPCOL *col;
alpar@9:       GLPAIJ *aij;
alpar@9:       int i, j, loc, ret, *A_ind, *A_ptr;
alpar@9:       double dir, *A_val, *b, *c, *x, *y, *z;
alpar@9:       xassert(m > 0);
alpar@9:       xassert(n > 0);
alpar@9:       /* allocate working arrays */
alpar@9:       A_ptr = xcalloc(1+m+1, sizeof(int));
alpar@9:       A_ind = xcalloc(1+nnz, sizeof(int));
alpar@9:       A_val = xcalloc(1+nnz, sizeof(double));
alpar@9:       b = xcalloc(1+m, sizeof(double));
alpar@9:       c = xcalloc(1+n, sizeof(double));
alpar@9:       x = xcalloc(1+n, sizeof(double));
alpar@9:       y = xcalloc(1+m, sizeof(double));
alpar@9:       z = xcalloc(1+n, sizeof(double));
alpar@9:       /* prepare rows and constraint coefficients */
alpar@9:       loc = 1;
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:       {  row = P->row[i];
alpar@9:          xassert(row->type == GLP_FX);
alpar@9:          b[i] = row->lb * row->rii;
alpar@9:          A_ptr[i] = loc;
alpar@9:          for (aij = row->ptr; aij != NULL; aij = aij->r_next)
alpar@9:          {  A_ind[loc] = aij->col->j;
alpar@9:             A_val[loc] = row->rii * aij->val * aij->col->sjj;
alpar@9:             loc++;
alpar@9:          }
alpar@9:       }
alpar@9:       A_ptr[m+1] = loc;
alpar@9:       xassert(loc-1 == nnz);
alpar@9:       /* prepare columns and objective coefficients */
alpar@9:       if (P->dir == GLP_MIN)
alpar@9:          dir = +1.0;
alpar@9:       else if (P->dir == GLP_MAX)
alpar@9:          dir = -1.0;
alpar@9:       else
alpar@9:          xassert(P != P);
alpar@9:       c[0] = dir * P->c0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  col = P->col[j];
alpar@9:          xassert(col->type == GLP_LO && col->lb == 0.0);
alpar@9:          c[j] = dir * col->coef * col->sjj;
alpar@9:       }
alpar@9:       /* allocate and initialize the common storage area */
alpar@9:       csa->m = m;
alpar@9:       csa->n = n;
alpar@9:       csa->A_ptr = A_ptr;
alpar@9:       csa->A_ind = A_ind;
alpar@9:       csa->A_val = A_val;
alpar@9:       csa->b = b;
alpar@9:       csa->c = c;
alpar@9:       csa->x = x;
alpar@9:       csa->y = y;
alpar@9:       csa->z = z;
alpar@9:       csa->parm = parm;
alpar@9:       initialize(csa);
alpar@9:       /* solve LP with the interior-point method */
alpar@9:       ret = ipm_main(csa);
alpar@9:       /* deallocate the common storage area */
alpar@9:       terminate(csa);
alpar@9:       /* determine solution status */
alpar@9:       if (ret == 0)
alpar@9:       {  /* optimal solution found */
alpar@9:          P->ipt_stat = GLP_OPT;
alpar@9:          ret = 0;
alpar@9:       }
alpar@9:       else if (ret == 1)
alpar@9:       {  /* problem has no feasible (primal or dual) solution */
alpar@9:          P->ipt_stat = GLP_NOFEAS;
alpar@9:          ret = 0;
alpar@9:       }
alpar@9:       else if (ret == 2)
alpar@9:       {  /* no convergence */
alpar@9:          P->ipt_stat = GLP_INFEAS;
alpar@9:          ret = GLP_ENOCVG;
alpar@9:       }
alpar@9:       else if (ret == 3)
alpar@9:       {  /* iteration limit exceeded */
alpar@9:          P->ipt_stat = GLP_INFEAS;
alpar@9:          ret = GLP_EITLIM;
alpar@9:       }
alpar@9:       else if (ret == 4)
alpar@9:       {  /* numeric instability on solving Newtonian system */
alpar@9:          P->ipt_stat = GLP_INFEAS;
alpar@9:          ret = GLP_EINSTAB;
alpar@9:       }
alpar@9:       else
alpar@9:          xassert(ret != ret);
alpar@9:       /* store row solution components */
alpar@9:       for (i = 1; i <= m; i++)
alpar@9:       {  row = P->row[i];
alpar@9:          row->pval = row->lb;
alpar@9:          row->dval = dir * y[i] * row->rii;
alpar@9:       }
alpar@9:       /* store column solution components */
alpar@9:       P->ipt_obj = P->c0;
alpar@9:       for (j = 1; j <= n; j++)
alpar@9:       {  col = P->col[j];
alpar@9:          col->pval = x[j] * col->sjj;
alpar@9:          col->dval = dir * z[j] / col->sjj;
alpar@9:          P->ipt_obj += col->coef * col->pval;
alpar@9:       }
alpar@9:       /* free working arrays */
alpar@9:       xfree(A_ptr);
alpar@9:       xfree(A_ind);
alpar@9:       xfree(A_val);
alpar@9:       xfree(b);
alpar@9:       xfree(c);
alpar@9:       xfree(x);
alpar@9:       xfree(y);
alpar@9:       xfree(z);
alpar@9:       return ret;
alpar@9: }
alpar@9: 
alpar@9: /* eof */