/* glpipm.c */

 /***********************************************************************

 *  This code is part of GLPK (GNU Linear Programming Kit).

 *

 *  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,

 *  2009, 2010 Andrew Makhorin, Department for Applied Informatics,

 *  Moscow Aviation Institute, Moscow, Russia. All rights reserved.

 *  E-mail: <mao@gnu.org>.

 *

 *  GLPK 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.

 *

 *  GLPK 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 GLPK. If not, see <http://www.gnu.org/licenses/>.

 ***********************************************************************/

 #include "glpipm.h"

 #include "glpmat.h"

 #define ITER_MAX 100

 /* maximal number of iterations */

 struct csa

 {     /* common storage area */

       /*--------------------------------------------------------------*/

       /* LP data */

       int m;

       /* number of rows (equality constraints) */

       int n;

       /* number of columns (structural variables) */

       int *A_ptr; /* int A_ptr[1+m+1]; */

       int *A_ind; /* int A_ind[A_ptr[m+1]]; */

       double *A_val; /* double A_val[A_ptr[m+1]]; */

       /* mxn-matrix A in storage-by-rows format */

       double *b; /* double b[1+m]; */

       /* m-vector b of right-hand sides */

       double *c; /* double c[1+n]; */

       /* n-vector c of objective coefficients; c[0] is constant term of

          the objective function */

       /*--------------------------------------------------------------*/

       /* LP solution */

       double *x; /* double x[1+n]; */

       double *y; /* double y[1+m]; */

       double *z; /* double z[1+n]; */

       /* current point in primal-dual space; the best point on exit */

       /*--------------------------------------------------------------*/

       /* control parameters */

       const glp_iptcp *parm;

       /*--------------------------------------------------------------*/

       /* working arrays and variables */

       double *D; /* double D[1+n]; */

       /* diagonal nxn-matrix D = X*inv(Z), where X = diag(x[j]) and

          Z = diag(z[j]) */

       int *P; /* int P[1+m+m]; */

       /* permutation mxm-matrix P used to minimize fill-in in Cholesky

          factorization */

       int *S_ptr; /* int S_ptr[1+m+1]; */

       int *S_ind; /* int S_ind[S_ptr[m+1]]; */

       double *S_val; /* double S_val[S_ptr[m+1]]; */

       double *S_diag; /* double S_diag[1+m]; */

       /* symmetric mxm-matrix S = P*A*D*A'*P' whose upper triangular

          part without diagonal elements is stored in S_ptr, S_ind, and

          S_val in storage-by-rows format, diagonal elements are stored

          in S_diag */

       int *U_ptr; /* int U_ptr[1+m+1]; */

       int *U_ind; /* int U_ind[U_ptr[m+1]]; */

       double *U_val; /* double U_val[U_ptr[m+1]]; */

       double *U_diag; /* double U_diag[1+m]; */

       /* upper triangular mxm-matrix U defining Cholesky factorization

          S = U'*U; its non-diagonal elements are stored in U_ptr, U_ind,

          U_val in storage-by-rows format, diagonal elements are stored

          in U_diag */

       int iter;

       /* iteration number (0, 1, 2, ...); iter = 0 corresponds to the

          initial point */

       double obj;

       /* current value of the objective function */

       double rpi;

       /* relative primal infeasibility rpi = ||A*x-b||/(1+||b||) */

       double rdi;

       /* relative dual infeasibility rdi = ||A'*y+z-c||/(1+||c||) */

       double gap;

       /* primal-dual gap = |c'*x-b'*y|/(1+|c'*x|) which is a relative

          difference between primal and dual objective functions */

       double phi;

       /* merit function phi = ||A*x-b||/max(1,||b||) +

                             + ||A'*y+z-c||/max(1,||c||) +

                             + |c'*x-b'*y|/max(1,||b||,||c||) */

       double mu;

       /* duality measure mu = x'*z/n (used as barrier parameter) */

       double rmu;

       /* rmu = max(||A*x-b||,||A'*y+z-c||)/mu */

       double rmu0;

       /* the initial value of rmu on iteration 0 */

       double *phi_min; /* double phi_min[1+ITER_MAX]; */

       /* phi_min[k] = min(phi[k]), where phi[k] is the value of phi on

          k-th iteration, 0 <= k <= iter */

       int best_iter;

       /* iteration number, on which the value of phi reached its best

          (minimal) value */

       double *best_x; /* double best_x[1+n]; */

       double *best_y; /* double best_y[1+m]; */

       double *best_z; /* double best_z[1+n]; */

       /* best point (in the sense of the merit function phi) which has

          been reached on iteration iter_best */

       double best_obj;

       /* objective value at the best point */

       double *dx_aff; /* double dx_aff[1+n]; */

       double *dy_aff; /* double dy_aff[1+m]; */

       double *dz_aff; /* double dz_aff[1+n]; */

       /* affine scaling direction */

       double alfa_aff_p, alfa_aff_d;

       /* maximal primal and dual stepsizes in affine scaling direction,

          on which x and z are still non-negative */

       double mu_aff;

       /* duality measure mu_aff = x_aff'*z_aff/n in the boundary point

          x_aff' = x+alfa_aff_p*dx_aff, z_aff' = z+alfa_aff_d*dz_aff */

       double sigma;

       /* Mehrotra's heuristic parameter (0 <= sigma <= 1) */

       double *dx_cc; /* double dx_cc[1+n]; */

       double *dy_cc; /* double dy_cc[1+m]; */

       double *dz_cc; /* double dz_cc[1+n]; */

       /* centering corrector direction */

       double *dx; /* double dx[1+n]; */

       double *dy; /* double dy[1+m]; */

       double *dz; /* double dz[1+n]; */

       /* final combined direction dx = dx_aff+dx_cc, dy = dy_aff+dy_cc,

          dz = dz_aff+dz_cc */

       double alfa_max_p;

       double alfa_max_d;

       /* maximal primal and dual stepsizes in combined direction, on

          which x and z are still non-negative */

 };

 /***********************************************************************

 *  initialize - allocate and initialize common storage area

 *

 *  This routine allocates and initializes the common storage area (CSA)

 *  used by interior-point method routines. */

 static void initialize(struct csa *csa)

 {     int m = csa->m;

       int n = csa->n;

       int i;

       if (csa->parm->msg_lev >= GLP_MSG_ALL)

          xprintf("Matrix A has %d non-zeros\n", csa->A_ptr[m+1]-1);

       csa->D = xcalloc(1+n, sizeof(double));

       /* P := I */

       csa->P = xcalloc(1+m+m, sizeof(int));

       for (i = 1; i <= m; i++) csa->P[i] = csa->P[m+i] = i;

       /* S := A*A', symbolically */

       csa->S_ptr = xcalloc(1+m+1, sizeof(int));

       csa->S_ind = adat_symbolic(m, n, csa->P, csa->A_ptr, csa->A_ind,

          csa->S_ptr);

       if (csa->parm->msg_lev >= GLP_MSG_ALL)

          xprintf("Matrix S = A*A' has %d non-zeros (upper triangle)\n",

             csa->S_ptr[m+1]-1 + m);

       /* determine P using specified ordering algorithm */

       if (csa->parm->ord_alg == GLP_ORD_NONE)

       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

             xprintf("Original ordering is being used\n");

          for (i = 1; i <= m; i++)

             csa->P[i] = csa->P[m+i] = i;

       }

       else if (csa->parm->ord_alg == GLP_ORD_QMD)

       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

             xprintf("Minimum degree ordering (QMD)...\n");

          min_degree(m, csa->S_ptr, csa->S_ind, csa->P);

       }

       else if (csa->parm->ord_alg == GLP_ORD_AMD)

       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

             xprintf("Approximate minimum degree ordering (AMD)...\n");

          amd_order1(m, csa->S_ptr, csa->S_ind, csa->P);

       }

       else if (csa->parm->ord_alg == GLP_ORD_SYMAMD)

       {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

             xprintf("Approximate minimum degree ordering (SYMAMD)...\n")

                ;

          symamd_ord(m, csa->S_ptr, csa->S_ind, csa->P);

       }

       else

          xassert(csa != csa);

       /* S := P*A*A'*P', symbolically */

       xfree(csa->S_ind);

       csa->S_ind = adat_symbolic(m, n, csa->P, csa->A_ptr, csa->A_ind,

          csa->S_ptr);

       csa->S_val = xcalloc(csa->S_ptr[m+1], sizeof(double));

       csa->S_diag = xcalloc(1+m, sizeof(double));

       /* compute Cholesky factorization S = U'*U, symbolically */

       if (csa->parm->msg_lev >= GLP_MSG_ALL)

          xprintf("Computing Cholesky factorization S = L*L'...\n");

       csa->U_ptr = xcalloc(1+m+1, sizeof(int));

       csa->U_ind = chol_symbolic(m, csa->S_ptr, csa->S_ind, csa->U_ptr);

       if (csa->parm->msg_lev >= GLP_MSG_ALL)

          xprintf("Matrix L has %d non-zeros\n", csa->U_ptr[m+1]-1 + m);

       csa->U_val = xcalloc(csa->U_ptr[m+1], sizeof(double));

       csa->U_diag = xcalloc(1+m, sizeof(double));

       csa->iter = 0;

       csa->obj = 0.0;

       csa->rpi = 0.0;

       csa->rdi = 0.0;

       csa->gap = 0.0;

       csa->phi = 0.0;

       csa->mu = 0.0;

       csa->rmu = 0.0;

       csa->rmu0 = 0.0;

       csa->phi_min = xcalloc(1+ITER_MAX, sizeof(double));

       csa->best_iter = 0;

       csa->best_x = xcalloc(1+n, sizeof(double));

       csa->best_y = xcalloc(1+m, sizeof(double));

       csa->best_z = xcalloc(1+n, sizeof(double));

       csa->best_obj = 0.0;

       csa->dx_aff = xcalloc(1+n, sizeof(double));

       csa->dy_aff = xcalloc(1+m, sizeof(double));

       csa->dz_aff = xcalloc(1+n, sizeof(double));

       csa->alfa_aff_p = 0.0;

       csa->alfa_aff_d = 0.0;

       csa->mu_aff = 0.0;

       csa->sigma = 0.0;

       csa->dx_cc = xcalloc(1+n, sizeof(double));

       csa->dy_cc = xcalloc(1+m, sizeof(double));

       csa->dz_cc = xcalloc(1+n, sizeof(double));

       csa->dx = csa->dx_aff;

       csa->dy = csa->dy_aff;

       csa->dz = csa->dz_aff;

       csa->alfa_max_p = 0.0;

       csa->alfa_max_d = 0.0;

       return;

 }

 /***********************************************************************

 *  A_by_vec - compute y = A*x

 *

 *  This routine computes matrix-vector product y = A*x, where A is the

 *  constraint matrix. */

 static void A_by_vec(struct csa *csa, double x[], double y[])

 {     /* compute y = A*x */

       int m = csa->m;

       int *A_ptr = csa->A_ptr;

       int *A_ind = csa->A_ind;

       double *A_val = csa->A_val;

       int i, t, beg, end;

       double temp;

       for (i = 1; i <= m; i++)

       {  temp = 0.0;

          beg = A_ptr[i], end = A_ptr[i+1];

          for (t = beg; t < end; t++) temp += A_val[t] * x[A_ind[t]];

          y[i] = temp;

       }

       return;

 }

 /***********************************************************************

 *  AT_by_vec - compute y = A'*x

 *

 *  This routine computes matrix-vector product y = A'*x, where A' is a

 *  matrix transposed to the constraint matrix A. */

 static void AT_by_vec(struct csa *csa, double x[], double y[])

 {     /* compute y = A'*x, where A' is transposed to A */

       int m = csa->m;

       int n = csa->n;

       int *A_ptr = csa->A_ptr;

       int *A_ind = csa->A_ind;

       double *A_val = csa->A_val;

       int i, j, t, beg, end;

       double temp;

       for (j = 1; j <= n; j++) y[j] = 0.0;

       for (i = 1; i <= m; i++)

       {  temp = x[i];

          if (temp == 0.0) continue;

          beg = A_ptr[i], end = A_ptr[i+1];

          for (t = beg; t < end; t++) y[A_ind[t]] += A_val[t] * temp;

       }

       return;

 }

 /***********************************************************************

 *  decomp_NE - numeric factorization of matrix S = P*A*D*A'*P'

 *

 *  This routine implements numeric phase of Cholesky factorization of

 *  the matrix S = P*A*D*A'*P', which is a permuted matrix of the normal

 *  equation system. Matrix D is assumed to be already computed. */

 static void decomp_NE(struct csa *csa)

 {     adat_numeric(csa->m, csa->n, csa->P, csa->A_ptr, csa->A_ind,

          csa->A_val, csa->D, csa->S_ptr, csa->S_ind, csa->S_val,

          csa->S_diag);

       chol_numeric(csa->m, csa->S_ptr, csa->S_ind, csa->S_val,

          csa->S_diag, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag);

       return;

 }

 /***********************************************************************

 *  solve_NE - solve normal equation system

 *

 *  This routine solves the normal equation system:

 *

 *     A*D*A'*y = h.

 *

 *  It is assumed that the matrix A*D*A' has been previously factorized

 *  by the routine decomp_NE.

 *

 *  On entry the array y contains the vector of right-hand sides h. On

 *  exit this array contains the computed vector of unknowns y.

 *

 *  Once the vector y has been computed the routine checks for numeric

 *  stability. If the residual vector:

 *

 *     r = A*D*A'*y - h

 *

 *  is relatively small, the routine returns zero, otherwise non-zero is

 *  returned. */

 static int solve_NE(struct csa *csa, double y[])

 {     int m = csa->m;

       int n = csa->n;

       int *P = csa->P;

       int i, j, ret = 0;

       double *h, *r, *w;

       /* save vector of right-hand sides h */

       h = xcalloc(1+m, sizeof(double));

       for (i = 1; i <= m; i++) h[i] = y[i];

       /* solve normal equation system (A*D*A')*y = h */

       /* since S = P*A*D*A'*P' = U'*U, then A*D*A' = P'*U'*U*P, so we

          have inv(A*D*A') = P'*inv(U)*inv(U')*P */

       /* w := P*h */

       w = xcalloc(1+m, sizeof(double));

       for (i = 1; i <= m; i++) w[i] = y[P[i]];

       /* w := inv(U')*w */

       ut_solve(m, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag, w);

       /* w := inv(U)*w */

       u_solve(m, csa->U_ptr, csa->U_ind, csa->U_val, csa->U_diag, w);

       /* y := P'*w */

       for (i = 1; i <= m; i++) y[i] = w[P[m+i]];

       xfree(w);

       /* compute residual vector r = A*D*A'*y - h */

       r = xcalloc(1+m, sizeof(double));

       /* w := A'*y */

       w = xcalloc(1+n, sizeof(double));

       AT_by_vec(csa, y, w);

       /* w := D*w */

       for (j = 1; j <= n; j++) w[j] *= csa->D[j];

       /* r := A*w */

       A_by_vec(csa, w, r);

       xfree(w);

       /* r := r - h */

       for (i = 1; i <= m; i++) r[i] -= h[i];

       /* check for numeric stability */

       for (i = 1; i <= m; i++)

       {  if (fabs(r[i]) / (1.0 + fabs(h[i])) > 1e-4)

          {  ret = 1;

             break;

          }

       }

       xfree(h);

       xfree(r);

       return ret;

 }

 /***********************************************************************

 *  solve_NS - solve Newtonian system

 *

 *  This routine solves the Newtonian system:

 *

 *     A*dx               = p

 *

 *           A'*dy +   dz = q

 *

 *     Z*dx        + X*dz = r

 *

 *  where X = diag(x[j]), Z = diag(z[j]), by reducing it to the normal

 *  equation system:

 *

 *     (A*inv(Z)*X*A')*dy = A*inv(Z)*(X*q-r)+p

 *

 *  (it is assumed that the matrix A*inv(Z)*X*A' has been factorized by

 *  the routine decomp_NE).

 *

 *  Once vector dy has been computed the routine computes vectors dx and

 *  dz as follows:

 *

 *     dx = inv(Z)*(X*(A'*dy-q)+r)

 *

 *     dz = inv(X)*(r-Z*dx)

 *

 *  The routine solve_NS returns the same code which was reported by the

 *  routine solve_NE (see above). */

 static int solve_NS(struct csa *csa, double p[], double q[], double r[],

       double dx[], double dy[], double dz[])

 {     int m = csa->m;

       int n = csa->n;

       double *x = csa->x;

       double *z = csa->z;

       int i, j, ret;

       double *w = dx;

       /* compute the vector of right-hand sides A*inv(Z)*(X*q-r)+p for

          the normal equation system */

       for (j = 1; j <= n; j++)

          w[j] = (x[j] * q[j] - r[j]) / z[j];

       A_by_vec(csa, w, dy);

       for (i = 1; i <= m; i++) dy[i] += p[i];

       /* solve the normal equation system to compute vector dy */

       ret = solve_NE(csa, dy);

       /* compute vectors dx and dz */

       AT_by_vec(csa, dy, dx);

       for (j = 1; j <= n; j++)

       {  dx[j] = (x[j] * (dx[j] - q[j]) + r[j]) / z[j];

          dz[j] = (r[j] - z[j] * dx[j]) / x[j];

       }

       return ret;

 }

 /***********************************************************************

 *  initial_point - choose initial point using Mehrotra's heuristic

 *

 *  This routine chooses a starting point using a heuristic proposed in

 *  the paper:

 *

 *  S. Mehrotra. On the implementation of a primal-dual interior point

 *  method. SIAM J. on Optim., 2(4), pp. 575-601, 1992.

 *

 *  The starting point x in the primal space is chosen as a solution of

 *  the following least squares problem:

 *

 *     minimize    ||x||

 *

 *     subject to  A*x = b

 *

 *  which can be computed explicitly as follows:

 *

 *     x = A'*inv(A*A')*b

 *

 *  Similarly, the starting point (y, z) in the dual space is chosen as

 *  a solution of the following least squares problem:

 *

 *     minimize    ||z||

 *

 *     subject to  A'*y + z = c

 *

 *  which can be computed explicitly as follows:

 *

 *     y = inv(A*A')*A*c

 *

 *     z = c - A'*y

 *

 *  However, some components of the vectors x and z may be non-positive

 *  or close to zero, so the routine uses a Mehrotra's heuristic to find

 *  a more appropriate starting point. */

 static void initial_point(struct csa *csa)

 {     int m = csa->m;

       int n = csa->n;

       double *b = csa->b;

       double *c = csa->c;

       double *x = csa->x;

       double *y = csa->y;

       double *z = csa->z;

       double *D = csa->D;

       int i, j;

       double dp, dd, ex, ez, xz;

       /* factorize A*A' */

       for (j = 1; j <= n; j++) D[j] = 1.0;

       decomp_NE(csa);

       /* x~ = A'*inv(A*A')*b */

       for (i = 1; i <= m; i++) y[i] = b[i];

       solve_NE(csa, y);

       AT_by_vec(csa, y, x);

       /* y~ = inv(A*A')*A*c */

       A_by_vec(csa, c, y);

       solve_NE(csa, y);

       /* z~ = c - A'*y~ */

       AT_by_vec(csa, y,z);

       for (j = 1; j <= n; j++) z[j] = c[j] - z[j];

       /* use Mehrotra's heuristic in order to choose more appropriate

          starting point with positive components of vectors x and z */

       dp = dd = 0.0;

       for (j = 1; j <= n; j++)

       {  if (dp < -1.5 * x[j]) dp = -1.5 * x[j];

          if (dd < -1.5 * z[j]) dd = -1.5 * z[j];

       }

       /* note that b = 0 involves x = 0, and c = 0 involves y = 0 and

          z = 0, so we need to be careful */

       if (dp == 0.0) dp = 1.5;

       if (dd == 0.0) dd = 1.5;

       ex = ez = xz = 0.0;

       for (j = 1; j <= n; j++)

       {  ex += (x[j] + dp);

          ez += (z[j] + dd);

          xz += (x[j] + dp) * (z[j] + dd);

       }

       dp += 0.5 * (xz / ez);

       dd += 0.5 * (xz / ex);

       for (j = 1; j <= n; j++)

       {  x[j] += dp;

          z[j] += dd;

          xassert(x[j] > 0.0 && z[j] > 0.0);

       }

       return;

 }

 /***********************************************************************

 *  basic_info - perform basic computations at the current point

 *

 *  This routine computes the following quantities at the current point:

 *

 *  1) value of the objective function:

 *

 *     F = c'*x + c[0]

 *

 *  2) relative primal infeasibility:

 *

 *     rpi = ||A*x-b|| / (1+||b||)

 *

 *  3) relative dual infeasibility:

 *

 *     rdi = ||A'*y+z-c|| / (1+||c||)

 *

 *  4) primal-dual gap (relative difference between the primal and the

 *     dual objective function values):

 *

 *     gap = |c'*x-b'*y| / (1+|c'*x|)

 *

 *  5) merit function:

 *

 *     phi = ||A*x-b|| / max(1,||b||) + ||A'*y+z-c|| / max(1,||c||) +

 *

 *         + |c'*x-b'*y| / max(1,||b||,||c||)

 *

 *  6) duality measure:

 *

 *     mu = x'*z / n

 *

 *  7) the ratio of infeasibility to mu:

 *

 *     rmu = max(||A*x-b||,||A'*y+z-c||) / mu

 *

 *  where ||*|| denotes euclidian norm, *' denotes transposition. */

 static void basic_info(struct csa *csa)

 {     int m = csa->m;

       int n = csa->n;

       double *b = csa->b;

       double *c = csa->c;

       double *x = csa->x;

       double *y = csa->y;

       double *z = csa->z;

       int i, j;

       double norm1, bnorm, norm2, cnorm, cx, by, *work, temp;

       /* compute value of the objective function */

       temp = c[0];

       for (j = 1; j <= n; j++) temp += c[j] * x[j];

       csa->obj = temp;

       /* norm1 = ||A*x-b|| */

       work = xcalloc(1+m, sizeof(double));

       A_by_vec(csa, x, work);

       norm1 = 0.0;

       for (i = 1; i <= m; i++)

          norm1 += (work[i] - b[i]) * (work[i] - b[i]);

       norm1 = sqrt(norm1);

       xfree(work);

       /* bnorm = ||b|| */

       bnorm = 0.0;

       for (i = 1; i <= m; i++) bnorm += b[i] * b[i];

       bnorm = sqrt(bnorm);

       /* compute relative primal infeasibility */

       csa->rpi = norm1 / (1.0 + bnorm);

       /* norm2 = ||A'*y+z-c|| */

       work = xcalloc(1+n, sizeof(double));

       AT_by_vec(csa, y, work);

       norm2 = 0.0;

       for (j = 1; j <= n; j++)

          norm2 += (work[j] + z[j] - c[j]) * (work[j] + z[j] - c[j]);

       norm2 = sqrt(norm2);

       xfree(work);

       /* cnorm = ||c|| */

       cnorm = 0.0;

       for (j = 1; j <= n; j++) cnorm += c[j] * c[j];

       cnorm = sqrt(cnorm);

       /* compute relative dual infeasibility */

       csa->rdi = norm2 / (1.0 + cnorm);

       /* by = b'*y */

       by = 0.0;

       for (i = 1; i <= m; i++) by += b[i] * y[i];

       /* cx = c'*x */

       cx = 0.0;

       for (j = 1; j <= n; j++) cx += c[j] * x[j];

       /* compute primal-dual gap */

       csa->gap = fabs(cx - by) / (1.0 + fabs(cx));

       /* compute merit function */

       csa->phi = 0.0;

       csa->phi += norm1 / (bnorm > 1.0 ? bnorm : 1.0);

       csa->phi += norm2 / (cnorm > 1.0 ? cnorm : 1.0);

       temp = 1.0;

       if (temp < bnorm) temp = bnorm;

       if (temp < cnorm) temp = cnorm;

       csa->phi += fabs(cx - by) / temp;

       /* compute duality measure */

       temp = 0.0;

       for (j = 1; j <= n; j++) temp += x[j] * z[j];

       csa->mu = temp / (double)n;

       /* compute the ratio of infeasibility to mu */

       csa->rmu = (norm1 > norm2 ? norm1 : norm2) / csa->mu;

       return;

 }

 /***********************************************************************

 *  make_step - compute next point using Mehrotra's technique

 *

 *  This routine computes the next point using the predictor-corrector

 *  technique proposed in the paper:

 *

 *  S. Mehrotra. On the implementation of a primal-dual interior point

 *  method. SIAM J. on Optim., 2(4), pp. 575-601, 1992.

 *

 *  At first, the routine computes so called affine scaling (predictor)

 *  direction (dx_aff,dy_aff,dz_aff) which is a solution of the system:

 *

 *     A*dx_aff                       = b - A*x

 *

 *               A'*dy_aff +   dz_aff = c - A'*y - z

 *

 *     Z*dx_aff            + X*dz_aff = - X*Z*e

 *

 *  where (x,y,z) is the current point, X = diag(x[j]), Z = diag(z[j]),

 *  e = (1,...,1)'.

 *

 *  Then, the routine computes the centering parameter sigma, using the

 *  following Mehrotra's heuristic:

 *

 *     alfa_aff_p = inf{0 <= alfa <= 1 | x+alfa*dx_aff >= 0}

 *

 *     alfa_aff_d = inf{0 <= alfa <= 1 | z+alfa*dz_aff >= 0}

 *

 *     mu_aff = (x+alfa_aff_p*dx_aff)'*(z+alfa_aff_d*dz_aff)/n

 *

 *     sigma = (mu_aff/mu)^3

 *

 *  where alfa_aff_p is the maximal stepsize along the affine scaling

 *  direction in the primal space, alfa_aff_d is the maximal stepsize

 *  along the same direction in the dual space.

 *

 *  After determining sigma the routine computes so called centering

 *  (corrector) direction (dx_cc,dy_cc,dz_cc) which is the solution of

 *  the system:

 *

 *     A*dx_cc                     = 0

 *

 *              A'*dy_cc +   dz_cc = 0

 *

 *     Z*dx_cc           + X*dz_cc = sigma*mu*e - X*Z*e

 *

 *  Finally, the routine computes the combined direction

 *

 *     (dx,dy,dz) = (dx_aff,dy_aff,dz_aff) + (dx_cc,dy_cc,dz_cc)

 *

 *  and determines maximal primal and dual stepsizes along the combined

 *  direction:

 *

 *     alfa_max_p = inf{0 <= alfa <= 1 | x+alfa*dx >= 0}

 *

 *     alfa_max_d = inf{0 <= alfa <= 1 | z+alfa*dz >= 0}

 *

 *  In order to prevent the next point to be too close to the boundary

 *  of the positive ortant, the routine decreases maximal stepsizes:

 *

 *     alfa_p = gamma_p * alfa_max_p

 *

 *     alfa_d = gamma_d * alfa_max_d

 *

 *  where gamma_p and gamma_d are scaling factors, and computes the next

 *  point:

 *

 *     x_new = x + alfa_p * dx

 *

 *     y_new = y + alfa_d * dy

 *

 *     z_new = z + alfa_d * dz

 *

 *  which becomes the current point on the next iteration. */

 static int make_step(struct csa *csa)

 {     int m = csa->m;

       int n = csa->n;

       double *b = csa->b;

       double *c = csa->c;

       double *x = csa->x;

       double *y = csa->y;

       double *z = csa->z;

       double *dx_aff = csa->dx_aff;

       double *dy_aff = csa->dy_aff;

       double *dz_aff = csa->dz_aff;

       double *dx_cc = csa->dx_cc;

       double *dy_cc = csa->dy_cc;

       double *dz_cc = csa->dz_cc;

       double *dx = csa->dx;

       double *dy = csa->dy;

       double *dz = csa->dz;

       int i, j, ret = 0;

       double temp, gamma_p, gamma_d, *p, *q, *r;

       /* allocate working arrays */

       p = xcalloc(1+m, sizeof(double));

       q = xcalloc(1+n, sizeof(double));

       r = xcalloc(1+n, sizeof(double));

       /* p = b - A*x */

       A_by_vec(csa, x, p);

       for (i = 1; i <= m; i++) p[i] = b[i] - p[i];

       /* q = c - A'*y - z */

       AT_by_vec(csa, y,q);

       for (j = 1; j <= n; j++) q[j] = c[j] - q[j] - z[j];

       /* r = - X * Z * e */

       for (j = 1; j <= n; j++) r[j] = - x[j] * z[j];

       /* solve the first Newtonian system */

       if (solve_NS(csa, p, q, r, dx_aff, dy_aff, dz_aff))

       {  ret = 1;

          goto done;

       }

       /* alfa_aff_p = inf{0 <= alfa <= 1 | x + alfa*dx_aff >= 0} */

       /* alfa_aff_d = inf{0 <= alfa <= 1 | z + alfa*dz_aff >= 0} */

       csa->alfa_aff_p = csa->alfa_aff_d = 1.0;

       for (j = 1; j <= n; j++)

       {  if (dx_aff[j] < 0.0)

          {  temp = - x[j] / dx_aff[j];

             if (csa->alfa_aff_p > temp) csa->alfa_aff_p = temp;

          }

          if (dz_aff[j] < 0.0)

          {  temp = - z[j] / dz_aff[j];

             if (csa->alfa_aff_d > temp) csa->alfa_aff_d = temp;

          }

       }

       /* mu_aff = (x+alfa_aff_p*dx_aff)' * (z+alfa_aff_d*dz_aff) / n */

       temp = 0.0;

       for (j = 1; j <= n; j++)

          temp += (x[j] + csa->alfa_aff_p * dx_aff[j]) *

                  (z[j] + csa->alfa_aff_d * dz_aff[j]);

       csa->mu_aff = temp / (double)n;

       /* sigma = (mu_aff/mu)^3 */

       temp = csa->mu_aff / csa->mu;

       csa->sigma = temp * temp * temp;

       /* p = 0 */

       for (i = 1; i <= m; i++) p[i] = 0.0;

       /* q = 0 */

       for (j = 1; j <= n; j++) q[j] = 0.0;

       /* r = sigma * mu * e - X * Z * e */

       for (j = 1; j <= n; j++)

          r[j] = csa->sigma * csa->mu - dx_aff[j] * dz_aff[j];

       /* solve the second Newtonian system with the same coefficients

          but with altered right-hand sides */

       if (solve_NS(csa, p, q, r, dx_cc, dy_cc, dz_cc))

       {  ret = 1;

          goto done;

       }

       /* (dx,dy,dz) = (dx_aff,dy_aff,dz_aff) + (dx_cc,dy_cc,dz_cc) */

       for (j = 1; j <= n; j++) dx[j] = dx_aff[j] + dx_cc[j];

       for (i = 1; i <= m; i++) dy[i] = dy_aff[i] + dy_cc[i];

       for (j = 1; j <= n; j++) dz[j] = dz_aff[j] + dz_cc[j];

       /* alfa_max_p = inf{0 <= alfa <= 1 | x + alfa*dx >= 0} */

       /* alfa_max_d = inf{0 <= alfa <= 1 | z + alfa*dz >= 0} */

       csa->alfa_max_p = csa->alfa_max_d = 1.0;

       for (j = 1; j <= n; j++)

       {  if (dx[j] < 0.0)

          {  temp = - x[j] / dx[j];

             if (csa->alfa_max_p > temp) csa->alfa_max_p = temp;

          }

          if (dz[j] < 0.0)

          {  temp = - z[j] / dz[j];

             if (csa->alfa_max_d > temp) csa->alfa_max_d = temp;

          }

       }

       /* determine scale factors (not implemented yet) */

       gamma_p = 0.90;

       gamma_d = 0.90;

       /* compute the next point */

       for (j = 1; j <= n; j++)

       {  x[j] += gamma_p * csa->alfa_max_p * dx[j];

          xassert(x[j] > 0.0);

       }

       for (i = 1; i <= m; i++)

          y[i] += gamma_d * csa->alfa_max_d * dy[i];

       for (j = 1; j <= n; j++)

       {  z[j] += gamma_d * csa->alfa_max_d * dz[j];

          xassert(z[j] > 0.0);

       }

 done: /* free working arrays */

       xfree(p);

       xfree(q);

       xfree(r);

       return ret;

 }

 /***********************************************************************

 *  terminate - deallocate common storage area

 *

 *  This routine frees all memory allocated to the common storage area

 *  used by interior-point method routines. */

 static void terminate(struct csa *csa)

 {     xfree(csa->D);

       xfree(csa->P);

       xfree(csa->S_ptr);

       xfree(csa->S_ind);

       xfree(csa->S_val);

       xfree(csa->S_diag);

       xfree(csa->U_ptr);

       xfree(csa->U_ind);

       xfree(csa->U_val);

       xfree(csa->U_diag);

       xfree(csa->phi_min);

       xfree(csa->best_x);

       xfree(csa->best_y);

       xfree(csa->best_z);

       xfree(csa->dx_aff);

       xfree(csa->dy_aff);

       xfree(csa->dz_aff);

       xfree(csa->dx_cc);

       xfree(csa->dy_cc);

       xfree(csa->dz_cc);

       return;

 }

 /***********************************************************************

 *  ipm_main - main interior-point method routine

 *

 *  This is a main routine of the primal-dual interior-point method.

 *

 *  The routine ipm_main returns one of the following codes:

 *

 *  0 - optimal solution found;

 *  1 - problem has no feasible (primal or dual) solution;

 *  2 - no convergence;

 *  3 - iteration limit exceeded;

 *  4 - numeric instability on solving Newtonian system.

 *

 *  In case of non-zero return code the routine returns the best point,

 *  which has been reached during optimization. */

 static int ipm_main(struct csa *csa)

 {     int m = csa->m;

       int n = csa->n;

       int i, j, status;

       double temp;

       /* choose initial point using Mehrotra's heuristic */

       if (csa->parm->msg_lev >= GLP_MSG_ALL)

          xprintf("Guessing initial point...\n");

       initial_point(csa);

       /* main loop starts here */

       if (csa->parm->msg_lev >= GLP_MSG_ALL)

          xprintf("Optimization begins...\n");

       for (;;)

       {  /* perform basic computations at the current point */

          basic_info(csa);

          /* save initial value of rmu */

          if (csa->iter == 0) csa->rmu0 = csa->rmu;

          /* accumulate values of min(phi[k]) and save the best point */

          xassert(csa->iter <= ITER_MAX);

          if (csa->iter == 0 || csa->phi_min[csa->iter-1] > csa->phi)

          {  csa->phi_min[csa->iter] = csa->phi;

             csa->best_iter = csa->iter;

             for (j = 1; j <= n; j++) csa->best_x[j] = csa->x[j];

             for (i = 1; i <= m; i++) csa->best_y[i] = csa->y[i];

             for (j = 1; j <= n; j++) csa->best_z[j] = csa->z[j];

             csa->best_obj = csa->obj;

          }

          else

             csa->phi_min[csa->iter] = csa->phi_min[csa->iter-1];

          /* display information at the current point */

          if (csa->parm->msg_lev >= GLP_MSG_ON)

             xprintf("%3d: obj = %17.9e; rpi = %8.1e; rdi = %8.1e; gap ="

                " %8.1e\n", csa->iter, csa->obj, csa->rpi, csa->rdi,

                csa->gap);

          /* check if the current point is optimal */

          if (csa->rpi < 1e-8 && csa->rdi < 1e-8 && csa->gap < 1e-8)

          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

                xprintf("OPTIMAL SOLUTION FOUND\n");

             status = 0;

             break;

          }

          /* check if the problem has no feasible solution */

          temp = 1e5 * csa->phi_min[csa->iter];

          if (temp < 1e-8) temp = 1e-8;

          if (csa->phi >= temp)

          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

                xprintf("PROBLEM HAS NO FEASIBLE PRIMAL/DUAL SOLUTION\n")

                   ;

             status = 1;

             break;

          }

          /* check for very slow convergence or divergence */

          if (((csa->rpi >= 1e-8 || csa->rdi >= 1e-8) && csa->rmu /

                csa->rmu0 >= 1e6) ||

                (csa->iter >= 30 && csa->phi_min[csa->iter] >= 0.5 *

                csa->phi_min[csa->iter - 30]))

          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

                xprintf("NO CONVERGENCE; SEARCH TERMINATED\n");

             status = 2;

             break;

          }

          /* check for maximal number of iterations */

          if (csa->iter == ITER_MAX)

          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

                xprintf("ITERATION LIMIT EXCEEDED; SEARCH TERMINATED\n");

             status = 3;

             break;

          }

          /* start the next iteration */

          csa->iter++;

          /* factorize normal equation system */

          for (j = 1; j <= n; j++) csa->D[j] = csa->x[j] / csa->z[j];

          decomp_NE(csa);

          /* compute the next point using Mehrotra's predictor-corrector

             technique */

          if (make_step(csa))

          {  if (csa->parm->msg_lev >= GLP_MSG_ALL)

                xprintf("NUMERIC INSTABILITY; SEARCH TERMINATED\n");

             status = 4;

             break;

          }

       }

       /* restore the best point */

       if (status != 0)

       {  for (j = 1; j <= n; j++) csa->x[j] = csa->best_x[j];

          for (i = 1; i <= m; i++) csa->y[i] = csa->best_y[i];

          for (j = 1; j <= n; j++) csa->z[j] = csa->best_z[j];

          if (csa->parm->msg_lev >= GLP_MSG_ALL)

             xprintf("Best point %17.9e was reached on iteration %d\n",

                csa->best_obj, csa->best_iter);

       }

       /* return to the calling program */

       return status;

 }

 /***********************************************************************

 *  NAME

 *

 *  ipm_solve - core LP solver based on the interior-point method

 *

 *  SYNOPSIS

 *

 *  #include "glpipm.h"

 *  int ipm_solve(glp_prob *P, const glp_iptcp *parm);

 *

 *  DESCRIPTION

 *

 *  The routine ipm_solve is a core LP solver based on the primal-dual

 *  interior-point method.

 *

 *  The routine assumes the following standard formulation of LP problem

 *  to be solved:

 *

 *     minimize

 *

 *        F = c[0] + c[1]*x[1] + c[2]*x[2] + ... + c[n]*x[n]

 *

 *     subject to linear constraints

 *

 *        a[1,1]*x[1] + a[1,2]*x[2] + ... + a[1,n]*x[n] = b[1]

 *

 *        a[2,1]*x[1] + a[2,2]*x[2] + ... + a[2,n]*x[n] = b[2]

 *

 *              . . . . . .

 *

 *        a[m,1]*x[1] + a[m,2]*x[2] + ... + a[m,n]*x[n] = b[m]

 *

 *     and non-negative variables

 *

 *        x[1] >= 0, x[2] >= 0, ..., x[n] >= 0

 *

 *  where:

 *  F                    is the objective function;

 *  x[1], ..., x[n]      are (structural) variables;

 *  c[0]                 is a constant term of the objective function;

 *  c[1], ..., c[n]      are objective coefficients;

 *  a[1,1], ..., a[m,n]  are constraint coefficients;

 *  b[1], ..., b[n]      are right-hand sides.

 *

 *  The solution is three vectors x, y, and z, which are stored by the

 *  routine in the arrays x, y, and z, respectively. These vectors

 *  correspond to the best primal-dual point found during optimization.

 *  They are approximate solution of the following system (which is the

 *  Karush-Kuhn-Tucker optimality conditions):

 *

 *     A*x      = b      (primal feasibility condition)

 *

 *     A'*y + z = c      (dual feasibility condition)

 *

 *     x'*z     = 0      (primal-dual complementarity condition)

 *

 *     x >= 0, z >= 0    (non-negativity condition)

 *

 *  where:

 *  x[1], ..., x[n]      are primal (structural) variables;

 *  y[1], ..., y[m]      are dual variables (Lagrange multipliers) for

 *                       equality constraints;

 *  z[1], ..., z[n]      are dual variables (Lagrange multipliers) for

 *                       non-negativity constraints.

 *

 *  RETURNS

 *

 *  0  LP has been successfully solved.

 *

 *  GLP_ENOCVG

 *     No convergence.

 *

 *  GLP_EITLIM

 *     Iteration limit exceeded.

 *

 *  GLP_EINSTAB

 *     Numeric instability on solving Newtonian system.

 *

 *  In case of non-zero return code the routine returns the best point,

 *  which has been reached during optimization. */

 int ipm_solve(glp_prob *P, const glp_iptcp *parm)

 {     struct csa _dsa, *csa = &_dsa;

       int m = P->m;

       int n = P->n;

       int nnz = P->nnz;

       GLPROW *row;

       GLPCOL *col;

       GLPAIJ *aij;

       int i, j, loc, ret, *A_ind, *A_ptr;

       double dir, *A_val, *b, *c, *x, *y, *z;

       xassert(m > 0);

       xassert(n > 0);

       /* allocate working arrays */

       A_ptr = xcalloc(1+m+1, sizeof(int));

       A_ind = xcalloc(1+nnz, sizeof(int));

       A_val = xcalloc(1+nnz, sizeof(double));

       b = xcalloc(1+m, sizeof(double));

       c = xcalloc(1+n, sizeof(double));

       x = xcalloc(1+n, sizeof(double));

       y = xcalloc(1+m, sizeof(double));

       z = xcalloc(1+n, sizeof(double));

       /* prepare rows and constraint coefficients */

       loc = 1;

       for (i = 1; i <= m; i++)

       {  row = P->row[i];

          xassert(row->type == GLP_FX);

          b[i] = row->lb * row->rii;

          A_ptr[i] = loc;

          for (aij = row->ptr; aij != NULL; aij = aij->r_next)

          {  A_ind[loc] = aij->col->j;

             A_val[loc] = row->rii * aij->val * aij->col->sjj;

             loc++;

          }

       }

       A_ptr[m+1] = loc;

       xassert(loc-1 == nnz);

       /* prepare columns and objective coefficients */

       if (P->dir == GLP_MIN)

          dir = +1.0;

       else if (P->dir == GLP_MAX)

          dir = -1.0;

       else

          xassert(P != P);

       c[0] = dir * P->c0;

       for (j = 1; j <= n; j++)

       {  col = P->col[j];

          xassert(col->type == GLP_LO && col->lb == 0.0);

          c[j] = dir * col->coef * col->sjj;

       }

       /* allocate and initialize the common storage area */

       csa->m = m;

       csa->n = n;

       csa->A_ptr = A_ptr;

       csa->A_ind = A_ind;

       csa->A_val = A_val;

       csa->b = b;

       csa->c = c;

       csa->x = x;

       csa->y = y;

       csa->z = z;

       csa->parm = parm;

       initialize(csa);

       /* solve LP with the interior-point method */

       ret = ipm_main(csa);

       /* deallocate the common storage area */

       terminate(csa);

       /* determine solution status */

       if (ret == 0)

       {  /* optimal solution found */

          P->ipt_stat = GLP_OPT;

          ret = 0;

       }

       else if (ret == 1)

       {  /* problem has no feasible (primal or dual) solution */

          P->ipt_stat = GLP_NOFEAS;

          ret = 0;

       }

       else if (ret == 2)

       {  /* no convergence */

          P->ipt_stat = GLP_INFEAS;

          ret = GLP_ENOCVG;

       }

       else if (ret == 3)

       {  /* iteration limit exceeded */

          P->ipt_stat = GLP_INFEAS;

          ret = GLP_EITLIM;

       }

       else if (ret == 4)

       {  /* numeric instability on solving Newtonian system */

          P->ipt_stat = GLP_INFEAS;

          ret = GLP_EINSTAB;

       }

       else

          xassert(ret != ret);

       /* store row solution components */

       for (i = 1; i <= m; i++)

       {  row = P->row[i];

          row->pval = row->lb;

          row->dval = dir * y[i] * row->rii;

       }

       /* store column solution components */

       P->ipt_obj = P->c0;

       for (j = 1; j <= n; j++)

       {  col = P->col[j];

          col->pval = x[j] * col->sjj;

          col->dval = dir * z[j] / col->sjj;

          P->ipt_obj += col->coef * col->pval;

       }

       /* free working arrays */

       xfree(A_ptr);

       xfree(A_ind);

       xfree(A_val);

       xfree(b);

       xfree(c);