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sdpt3.m
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sdpt3.m
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%%*****************************************************************************
%% sdpt3: solve an semidefinite-quadratic-linear program
%% by infeasible path-following method on the original problem
%% or the 3-parameter homogeneous self-dual model.
%%
%% [obj,X,y,Z,info,runhist] = sdpt3(blk,At,C,b,OPTIONS,X0,y0,Z0);
%%
%% Input: blk: a cell array describing the block diagonal structure of SQL data.
%% At: a cell array with At{p} = [svec(Ap1) ... svec(Apm)]
%% b,C: data for the SQL instance.
%% (X0,y0,Z0): an initial iterate (if it is not given, the default is used).
%% OPTIONS: a structure that specifies parameters required in sdpt3.m,
%% (if it is not given, the default in sqlparameters.m is used).
%%
%% Output: obj = [<C,X>, <b,y>].
%% (X,y,Z): an approximately optimal solution or a primal or dual
%% infeasibility certificate.
%% info.termcode = termination-code
%% info.iter = number of iterations
%% info.obj = [primal-obj, dual-obj]
%% info.cputime = total-time
%% info.gap = gap
%% info.pinfeas = primal_infeas
%% info.dinfeas = dual_infeas
%% runhist.pobj = history of primal objective value.
%% runhist.dobj = history of dual objective value.
%% runhist.gap = history of <X,Z>.
%% runhist.pinfeas = history of primal infeasibility.
%% runhist.dinfeas = history of dual infeasibility.
%% runhist.cputime = history of cputime spent.
%%----------------------------------------------------------------------------
%% The OPTIONS structure specifies the required parameters:
%% vers gam predcorr expon gaptol inftol steptol
%% maxit printlevel scale_data ...
%% (all have default values set in sqlparameters.m).
%%*************************************************************************
%% SDPT3: version 4.0
%% Copyright (c) 1997 by
%% K.C. Toh, M.J. Todd, R.H. Tutuncu
%% Last Modified: 20 Dec 2007
%%*************************************************************************
function [obj,X,y,Z,info,runhist] = sdpt3(blk,At,C,b,OPTIONS,X0,y0,Z0)
%%
%%-----------------------------------------
%% get parameters from the OPTIONS structure
%%-----------------------------------------
%%
warning off;
if (nargin < 5); OPTIONS = []; end
matlabversion = sscanf(version,'%f');
if strcmp(computer,'PCWIN64') | strcmp(computer,'GLNXA64')
par.computer = 64;
else
par.computer = 32;
end
model = 0; %% automatically decide between sqlp.m and HSDsqlp.m
par.matlabversion = matlabversion(1);
par.vers = 0;
par.predcorr = 1;
par.gam = 0;
par.expon = 1;
par.gaptol = 1e-8;
par.inftol = 1e-8;
par.steptol = 1e-6;
par.maxit = 100;
par.printlevel = 3;
par.stoplevel = 1;
par.scale_data = 0;
par.spdensity = 0.4;
par.rmdepconstr = 0;
par.smallblkdim = 40;
par.schurfun = cell(size(blk,1),1);
par.schurfun_par = cell(size(blk,1),1);
%%
parbarrier = cell(size(blk,1),1);
for p = 1:size(blk,1)
pblk = blk(p,:);
if strcmp(pblk{1},'s') | strcmp(pblk{1},'q')
parbarrier{p} = zeros(1,length(pblk{2}));
elseif strcmp(pblk{1},'l') | strcmp(pblk{1},'u' )
parbarrier{p} = zeros(1,sum(pblk{2}));
end
end
parbarrier_0 = parbarrier;
%%
if exist('OPTIONS')
if isfield(OPTIONS,'model')
model = OPTIONS.model;
if all(model-[0,1,2]); error(' model must be 0, 1 or 2'); end
end
if isfield(OPTIONS,'vers'); par.vers = OPTIONS.vers; end
if isfield(OPTIONS,'predcorr'); par.predcorr = OPTIONS.predcorr; end
if isfield(OPTIONS,'gam'); par.gam = OPTIONS.gam; end
if isfield(OPTIONS,'expon'); par.expon = OPTIONS.expon; end
if isfield(OPTIONS,'gaptol'); par.gaptol = OPTIONS.gaptol; end
if isfield(OPTIONS,'inftol'); par.inftol = OPTIONS.inftol; end
if isfield(OPTIONS,'steptol'); par.steptol = OPTIONS.steptol; end
if isfield(OPTIONS,'maxit'); par.maxit = OPTIONS.maxit; end
if isfield(OPTIONS,'printlevel'); par.printlevel = OPTIONS.printlevel; end
if isfield(OPTIONS,'stoplevel'); par.stoplevel = OPTIONS.stoplevel; end
if isfield(OPTIONS,'scale_data'); par.scale_data = OPTIONS.scale_data; end
if isfield(OPTIONS,'spdensity'); par.spdensity = OPTIONS.spdensity; end
if isfield(OPTIONS,'rmdepconstr'); par.rmdepconstr = OPTIONS.rmdepconstr; end
if isfield(OPTIONS,'smallblkdim'); par.smallblkdim = OPTIONS.smallblkdim; end
if isfield(OPTIONS,'parbarrier');
parbarrier = OPTIONS.parbarrier;
if isempty(parbarrier); parbarrier = parbarrier_0; end
if ~iscell(parbarrier);
tmp = parbarrier; clear parbarrier; parbarrier{1} = tmp;
end
if (length(parbarrier) < size(blk,1))
len = length(parbarrier);
parbarrier(len+1:size(blk,1)) = parbarrier_0(len+1:size(blk,1));
end
end
if isfield(OPTIONS,'schurfun');
par.schurfun = OPTIONS.schurfun;
if ~isempty(par.schurfun); par.scale_data = 0; end
end
if isfield(OPTIONS,'schurfun_par'); par.schurfun_par = OPTIONS.schurfun_par; end
if isempty(par.schurfun); par.schurfun = cell(size(blk,1),1); end
if isempty(par.schurfun_par); par.schurfun_par = cell(size(blk,1),1); end
end
if (size(blk,2) > 2); par.smallblkdim = 0; end
%%
%%-----------------------------------------
%% Add a redundant constraint if there is
%% no linear constraint
%%-----------------------------------------
%%
isemptyAtb = 0;
if isempty(At) & isempty(b);
%% Add redundant constraint: <-I,X> <= 0
b = 0;
At = ops(ops(blk,'identity'),'*',-1);
numblk = size(blk,1);
blk{numblk+1,1} = 'l'; blk{numblk+1,2} = 1;
At{numblk+1,1} = 1; C{numblk+1,1} = 0;
parbarrier{numblk+1} = 0;
isemptyAtb = 1;
end
%%
%%-----------------------------------------
%% convert matrices to cell arrays
%%-----------------------------------------
%%
if ~iscell(At); At = {At}; end;
if ~iscell(C); C = {C}; end;
if all(size(At) == [size(blk,1), length(b)]);
convertyes = zeros(size(blk,1),1);
for p = 1:size(blk,1)
if strcmp(blk{p,1},'s') & all(size(At{p,1}) == sum(blk{p,2}))
convertyes(p) = 1;
end
end
if any(convertyes)
if (par.printlevel);
fprintf('\n sdpt3: converting At into required format');
end
At = svec(blk,At,ones(size(blk,1),1));
end
end
%%
%%-----------------------------------------
%% validate SQLP data
%%-----------------------------------------
%%
tstart = cputime;
[blk,At,C,b,blkdim,numblk,parbarrier] = validate(blk,At,C,b,par,parbarrier);
[blk,At,C,b,iscmp] = convertcmpsdp(blk,At,C,b);
if (iscmp) & (par.printlevel>=2);
fprintf('\n SQLP has complex data');
end
exist_analytic_term = 0;
for p = 1:size(blk,1);
idx = find(parbarrier{p} > 0);
if ~isempty(idx); exist_analytic_term = 1; end
end
if (par.printlevel>=2)
fprintf('\n num. of constraints = %2.0d',length(b));
if blkdim(1);
fprintf('\n dim. of sdp var = %2.0d,',blkdim(1));
fprintf(' num. of sdp blk = %2.0d',numblk(1));
end
if blkdim(2);
fprintf('\n dim. of socp var = %2.0d,',blkdim(2));
fprintf(' num. of socp blk = %2.0d',numblk(2));
end
if blkdim(3); fprintf('\n dim. of linear var = %2.0d',blkdim(3)); end
if blkdim(4); fprintf('\n dim. of free var = %2.0d',blkdim(4)); end
end
%%
%%-----------------------------------------
%% initial iterate
%%-----------------------------------------
%%
if (nargin <= 5) | (isempty(X0) | isempty(y0) | isempty(Z0));
par.startpoint = 1;
[X0,y0,Z0] = infeaspt(blk,At,C,b);
else
par.startpoint = 2;
if ~iscell(X0); X0 = {X0}; end;
if ~iscell(Z0); Z0 = {Z0}; end;
y0 = real(y0);
if (length(y0) ~= length(b));
error('sdpt3: length of b and y0 not compatible');
end
[X0,Z0] = validate_startpoint(blk,X0,Z0,par.spdensity,iscmp);
end
%%
%%-----------------------------------------
%% detect unrestricted blocks in linear blocks
%%-----------------------------------------
%%
user_supplied_schurfun = 0;
for p = 1:size(blk,1)
if ~isempty(par.schurfun{p}); user_supplied_schurfun = 1; end
end
if (user_supplied_schurfun == 0)
[blk2,At2,C2,ublkinfo,parbarrier2,X02,Z02] = ...
detect_ublk(blk,At,C,parbarrier,X0,Z0,par.printlevel);
else
blk2 = blk; At2 = At; C2 = C;
parbarrier2 = parbarrier; X02 = X0; Z02 = Z0;
ublkinfo = cell(size(blk2,1),1);
end
ublksize = blkdim(4);
for p = 1:size(ublkinfo,1)
ublksize = ublksize + length(ublkinfo{p});
end
%%
%%-----------------------------------------
%% detect diagonal blocks in semidefinite blocks
%%-----------------------------------------
%%
if (user_supplied_schurfun==0)
[blk3,At3,C3,diagblkinfo,diagblkchange,parbarrier3,X03,Z03] = ...
detect_lblk(blk2,At2,C2,b,parbarrier2,X02,Z02,par.printlevel);
else
blk3 = blk2; At3 = At2; C3 = C2;
parbarrier3 = parbarrier2; X03 = X02; Z03 = Z02;
diagblkchange = 0;
diagblkinfo = cell(size(blk3,1),1);
end
%%
%%-----------------------------------------
%% main solver
%% model = 1: use sqlp
%% = 2: use HSDsqlp
%%-----------------------------------------
%%
if (par.vers == 0);
if blkdim(1); par.vers = 1; else; par.vers = 2; end
end
par.blkdim = blkdim;
par.ublksize = ublksize;
if (exist_analytic_term);
model = 1;
else
if (model == 0)
if (ublksize > 0); model = 2; else; model = 1; end
end
end
if (model==1)
[obj,X3,y,Z3,info,runhist] = ...
sqlpmain(blk3,At3,C3,b,par,parbarrier3,X03,y0,Z03);
elseif (model==2)
if (nargin <= 5) | (isempty(X0) | isempty(y0) | isempty(Z0));
if (max([ops(At3,'norm'),ops(C3,'norm'),norm(b)]) > 1e2)
[X03,y03,Z03] = infeaspt(blk3,At3,C3,b,1);
else
[X03,y03,Z03] = infeaspt(blk3,At3,C3,b,2,1);
end
end
[obj,X3,y,Z3,info,runhist] = ...
HSDsqlpmain(blk3,At3,C3,b,par,X03,y0,Z03);
end
info.model = model;
%%
%%-----------------------------------------
%% recover semidefinite blocks from linear blocks
%%-----------------------------------------
%%
if any(diagblkchange)
X2 = cell(size(blk2,1),1); Z2 = cell(size(blk2,1),1);
count = 0;
for p = 1:size(blk2,1)
pblk = blk2(p,:);
n = sum(pblk{2});
blkno = diagblkinfo{p,1};
idxdiag = diagblkinfo{p,2};
idxnondiag = diagblkinfo{p,3};
if ~isempty(idxdiag)
len = length(idxdiag);
Xtmp = [idxdiag,idxdiag,X3{end}(count+[1:len]); n, n, 0];
Ztmp = [idxdiag,idxdiag,Z3{end}(count+[1:len]); n, n, 0];
if ~isempty(idxnondiag)
[ii,jj,vv] = find(X3{blkno});
Xtmp = [Xtmp; idxnondiag(ii),idxnondiag(jj),vv];
[ii,jj,vv] = find(Z3{blkno});
Ztmp = [Ztmp; idxnondiag(ii),idxnondiag(jj),vv];
end
X2{p} = spconvert(Xtmp);
Z2{p} = spconvert(Ztmp);
count = count + len;
else
X2(p) = X3(blkno); Z2(p) = Z3(blkno);
end
end
else
X2 = X3; Z2 = Z3;
end
%%
%%-----------------------------------------
%% recover linear block from unrestricted block
%%-----------------------------------------
%%
numblk = size(blk,1);
numblknew = numblk;
X = cell(numblk,1); Z = cell(numblk,1);
for p = 1:numblk
n = blk{p,2};
if isempty(ublkinfo{p,1})
X{p} = X2{p}; Z{p} = Z2{p};
else
Xtmp = zeros(n,1); Ztmp = zeros(n,1);
Xtmp(ublkinfo{p,1}) = max(0,X2{p});
Xtmp(ublkinfo{p,2}) = max(0,-X2{p});
Ztmp(ublkinfo{p,1}) = max(0,Z2{p});
Ztmp(ublkinfo{p,2}) = max(0,-Z2{p});
if ~isempty(ublkinfo{p,3})
numblknew = numblknew + 1;
Xtmp(ublkinfo{p,3}) = X2{numblknew};
Ztmp(ublkinfo{p,3}) = Z2{numblknew};
end
X{p} = Xtmp; Z{p} = Ztmp;
end
end
%%
%%-----------------------------------------
%% recover complex solution
%%-----------------------------------------
%%
if (iscmp)
for p = 1:numblk
pblk = blk(p,:);
n = sum(pblk{2})/2;
if strcmp(pblk{1},'s');
X{p} = X{p}(1:n,1:n) + sqrt(-1)*X{p}(n+[1:n],1:n);
Z{p} = Z{p}(1:n,1:n) + sqrt(-1)*Z{p}(n+[1:n],1:n);
X{p} = 0.5*(X{p}+X{p}');
Z{p} = 0.5*(Z{p}+Z{p}');
end
end
end
if (isemptyAtb)
X = X(1:end-1); Z = Z(1:end-1);
end
%%*****************************************************************************