230 lines
8.2 KiB
Mathematica
230 lines
8.2 KiB
Mathematica
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%%***************************************************************
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%% linsysolve: solve linear system to get dy, and direction
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%% corresponding to unrestricted variables.
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%%
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%% [xx,coeff,L,resnrm] = linsysolve(schur,UU,Afree,EE,rhs);
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%%
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%% child functions: symqmr.m, mybicgstable.m, linsysolvefun.m
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%%
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%% SDPT3: version 3.1
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%% Copyright (c) 1997 by
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%% K.C. Toh, M.J. Todd, R.H. Tutuncu
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%% Last Modified: 16 Sep 2004
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%%***************************************************************
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function [xx,coeff,L,resnrm] = linsysolve(par,schur,UU,Afree,EE,rhs);
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global solve_ok exist_analytic_term
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global nnzmat nnzmatold matfct_options matfct_options_old use_LU
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global msg diagR diagRold numpertdiagschur
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spdensity = par.spdensity;
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printlevel = par.printlevel;
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iter = par.iter;
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if isfield(par,'relgap') & isfield(par,'pinfeas') & isfield(par,'dinfeas')
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err = max([par.relgap,par.pinfeas,par.dinfeas]);
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else
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err = inf;
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end
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%%
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m = length(schur);
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if (iter==1);
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use_LU = 0; matfct_options_old = '';
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diagR = ones(m,1);
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numpertdiagschur = 0;
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end
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if isempty(nnzmatold); nnzmatold = 0; end
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diagRold = diagR;
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%%
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%% schur = schur + rho*diagschur + lam*AAt
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%%
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diagschur = abs(full(diag(schur)));
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if (par.ublksize)
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minrho(1) = 1e-15;
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else
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minrho(1) = 1e-17;
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end
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minrho(1) = max(minrho(1), 1e-6/3.0^iter); %% old: 1e-6/3.0^iter
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minrho(2) = max(1e-04, 0.7^iter);
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minlam = max(1e-10, 1e-4/2.0^iter);
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rho = min(minrho(1), minrho(2)*(1+norm(rhs))/(1+norm(diagschur.*par.y)));
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lam = min(minlam, 0.1*rho*norm(diagschur)/par.normAAt);
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if (exist_analytic_term); rho = 0; end; %% important
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ratio = max(diagR)/min(diagR);
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if (par.depconstr) | (ratio > 1e10) | (iter < 5)
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%% important: do not perturb beyond certain threshold
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%% since it will adversely affect prim_infeas of fp43
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%%
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pertdiagschur = min(rho*diagschur,1e-4./max(1,abs(par.dy)));
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mexschurfun(schur,full(pertdiagschur));
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%%if (printlevel>2); fprintf(' %2.1e',rho); end
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end
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if (par.depconstr) | (par.ZpATynorm > 1e10) | (par.ublksize) | (iter < 10)
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%% Note: do not add this perturbation even if ratio is large.
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%% It adversely affects hinf15.
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%%
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lam = min(lam,1e-4/max(1,norm(par.AAt*par.dy)));
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if (exist_analytic_term); lam = 0; end
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mexschurfun(schur,lam*par.AAt);
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%%if (printlevel>2); fprintf('*'); end
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end
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if (max(diagschur)/min(diagschur) > 1e14) & (par.blkdim(2) == 0) ...
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& (iter > 10)
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tol = 1e-8;
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idx = find(diagschur < tol); len = length(idx);
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pertdiagschur = zeros(m,1);
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if (len > 0 & len < 5) & (norm(rhs(idx)) < tol)
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pertdiagschur(idx) = 1*ones(length(idx),1);
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mexschurfun(schur,pertdiagschur);
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numpertdiagschur = numpertdiagschur + 1;
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if (printlevel>2); fprintf('#'); end
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end
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end
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%%
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%% assemble coefficient matrix
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%%
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len = size(Afree,2);
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if ~isempty(EE)
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EE(:,[1 2]) = len + EE(:,[1 2]); %% adjust for ublk
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end
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EE = [(1:len)' (1:len)' zeros(len,1); EE];
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if isempty(EE)
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coeff.mat22 = [];
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else
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coeff.mat22 = spconvert(EE);
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end
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if (size(Afree,2) | size(UU,2))
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coeff.mat12 = [Afree, UU];
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else
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coeff.mat12 = [];
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end
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coeff.mat11 = schur; %% important to use perturbed schur matrix
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ncolU = size(coeff.mat12,2);
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%%
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%% pad rhs with zero vector
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%% decide which solution methods to use
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%%
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rhs = [rhs; zeros(m+ncolU-length(rhs),1)];
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if (ncolU > 300); use_LU = 1; end
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%%
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%% Cholesky factorization
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%%
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L = []; resnrm = norm(rhs); xx = inf*ones(m,1);
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if (~use_LU)
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solve_ok = 1; solvesys = 1;
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nnzmat = mexnnz(coeff.mat11);
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nnzmatdiff = (nnzmat ~= nnzmatold);
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if (nnzmat > spdensity*m^2) | (m < 500)
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matfct_options = 'chol';
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else
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matfct_options = 'spchol';
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end
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if (printlevel>2); fprintf(' %s ',matfct_options); end
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if strcmp(matfct_options,'chol')
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if issparse(schur); schur = full(schur); end;
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if (iter<=5); %%--- to fix strange anonmaly in Matlab
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mexschurfun(schur,1e-20,2);
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end
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L.matfct_options = 'chol';
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[L.R,indef] = chol(schur);
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L.perm = [1:m];
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diagR = diag(L.R).^2;
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elseif strcmp(matfct_options,'spchol')
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if ~issparse(schur); schur = sparse(schur); end;
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L.matfct_options = 'spchol';
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[L.R,indef,L.perm] = chol(schur,'vector');
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L.Rt = L.R';
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diagR = full(diag(L.R)).^2;
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end
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if (indef)
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diagR = diagRold;
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solve_ok = -2; solvesys = 0;
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msg = 'linsysolve: Schur complement matrix not positive definite';
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if (printlevel); fprintf('\n %s',msg); end
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end
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if (solvesys)
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if (ncolU)
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tmp = coeff.mat12'*linsysolvefun(L,coeff.mat12)-coeff.mat22;
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if issparse(tmp); tmp = full(tmp); end
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tmp = 0.5*(tmp + tmp');
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[L.Ml,L.Mu,L.Mp] = lu(tmp);
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tol = 1e-16;
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condest = max(abs(diag(L.Mu)))/min(abs(diag(L.Mu)));
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idx = find(abs(diag(L.Mu)) < tol);
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if ~isempty(idx) | (condest > 1e30); %%old: 1e18
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solvesys = 0; solve_ok = -4;
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use_LU = 1;
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msg = 'SMW too ill-conditioned, switch to LU factor';
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if (printlevel); fprintf('\n %s, %2.1e.',msg,condest); end
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end
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end
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[xx,resnrm,solve_ok] = symqmr(coeff,rhs,L,[],[],printlevel);
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if (solve_ok <= 0.3) & (printlevel)
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fprintf('\n warning: symqmr failed: %3.1f ',solve_ok);
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end
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end
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if (solve_ok <= 0.3)
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tol = 1e-10;
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if (m < 1e4 & strcmp(matfct_options,'chol') & (err > tol)) ...
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| (m < 2e5 & strcmp(matfct_options,'spchol') & (err > tol))
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use_LU = 1;
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if (printlevel); fprintf('\n switch to LU factor.'); end
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end
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end
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end
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%%
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%% LU factorization
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%%
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if (use_LU)
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nnzmat = mexnnz(coeff.mat11)+mexnnz(coeff.mat12);
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nnzmatdiff = (nnzmat ~= nnzmatold);
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solve_ok = 1; solvesys = 1;
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if ~isempty(coeff.mat22)
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raugmat = [coeff.mat11, coeff.mat12; coeff.mat12', coeff.mat22];
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else
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raugmat = coeff.mat11;
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end
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if (nnzmat > spdensity*m^2) | (m+ncolU < 500)
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matfct_options = 'lu'; %% lu is better than ldl
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else
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matfct_options = 'splu';
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end
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if (printlevel>2); fprintf(' %s ',matfct_options); end
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if strcmp(matfct_options,'lu')
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if issparse(raugmat); raugmat = full(raugmat); end
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L.matfct_options = 'lu';
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[L.l,L.u,L.p] = lu(raugmat);
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L.p = sparse(L.p);
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[ii,jj] = find(L.p); [dummy,idx] = sort(ii); L.perm = jj(idx);
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elseif strcmp(matfct_options,'splu')
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if ~issparse(raugmat); raugmat = sparse(raugmat); end
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L.matfct_options = 'splu';
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L.symmatrix = 0;
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[L.l,L.u,L.p,L.q] = lu(raugmat);
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L.perm = [1:length(raugmat)];
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elseif strcmp(matfct_options,'ldl')
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if issparse(raugmat); raugmat = full(raugmat); end
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L.matfct_options = 'ldl';
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[L.L,L.D,L.p] = ldl(raugmat);
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L.D = sparse(L.D);
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L.p = sparse(L.p);
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[ii,jj] = find(L.p); [dummy,idx] = sort(jj); L.perm = ii(idx);
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elseif strcmp(matfct_options,'spldl')
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if ~issparse(raugmat); raugmat = sparse(raugmat); end
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L.matfct_options = 'spldl';
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[L.L,L.D,L.perm] = ldl(raugmat,'vector');
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L.Lt = L.L';
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end
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if (solvesys)
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[xx,resnrm,solve_ok] = symqmr(coeff,rhs,L,[],[],printlevel);
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%%[xx,resnrm,solve_ok] = mybicgstab(coeff,rhs,L,[],[],printlevel);
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if (solve_ok<=0) & (printlevel)
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fprintf('\n warning: bicgstab fails: %3.1f,',solve_ok);
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end
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end
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end
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if (printlevel>2); fprintf('%2.0d ',length(resnrm)-1); end
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%%
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nnzmatold = nnzmat; matfct_options_old = matfct_options;
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%%***************************************************************
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