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function [x, f, eflag, output, lambda] = qps_cplex(H, c, A, l, u, xmin, xmax, x0, opt)
%QPS_CPLEX Quadratic Program Solver based on CPLEX.
% [X, F, EXITFLAG, OUTPUT, LAMBDA] = ...
% QPS_CPLEX(H, C, A, L, U, XMIN, XMAX, X0, OPT)
% A wrapper function providing a MATPOWER standardized interface for using
% CPLEXQP or CPLEXLP to solve the following QP (quadratic programming)
% problem:
%
% min 1/2 X'*H*X + C'*X
% X
%
% subject to
%
% L <= A*X <= U (linear constraints)
% XMIN <= X <= XMAX (variable bounds)
%
% Inputs (all optional except H, C, A and L):
% H : matrix (possibly sparse) of quadratic cost coefficients
% C : vector of linear cost coefficients
% A, L, U : define the optional linear constraints. Default
% values for the elements of L and U are -Inf and Inf,
% respectively.
% XMIN, XMAX : optional lower and upper bounds on the
% X variables, defaults are -Inf and Inf, respectively.
% X0 : optional starting value of optimization vector X
% OPT : optional options structure with the following fields,
% all of which are also optional (default values shown in
% parentheses)
% verbose (0) - controls level of progress output displayed
% 0 = no progress output
% 1 = some progress output
% 2 = verbose progress output
% cplex_opt - options struct for CPLEX, value in
% verbose overrides these options
% PROBLEM : The inputs can alternatively be supplied in a single
% PROBLEM struct with fields corresponding to the input arguments
% described above: H, c, A, l, u, xmin, xmax, x0, opt
%
% Outputs:
% X : solution vector
% F : final objective function value
% EXITFLAG : CPLEXQP/CPLEXLP exit flag
% (see CPLEXQP and CPLEXLP documentation for details)
% OUTPUT : CPLEXQP/CPLEXLP output struct
% (see CPLEXQP and CPLEXLP documentation for details)
% LAMBDA : struct containing the Langrange and Kuhn-Tucker
% multipliers on the constraints, with fields:
% mu_l - lower (left-hand) limit on linear constraints
% mu_u - upper (right-hand) limit on linear constraints
% lower - lower bound on optimization variables
% upper - upper bound on optimization variables
%
% Note the calling syntax is almost identical to that of QUADPROG
% from MathWorks' Optimization Toolbox. The main difference is that
% the linear constraints are specified with A, L, U instead of
% A, B, Aeq, Beq.
%
% Calling syntax options:
% [x, f, exitflag, output, lambda] = ...
% qps_cplex(H, c, A, l, u, xmin, xmax, x0, opt)
%
% x = qps_cplex(H, c, A, l, u)
% x = qps_cplex(H, c, A, l, u, xmin, xmax)
% x = qps_cplex(H, c, A, l, u, xmin, xmax, x0)
% x = qps_cplex(H, c, A, l, u, xmin, xmax, x0, opt)
% x = qps_cplex(problem), where problem is a struct with fields:
% H, c, A, l, u, xmin, xmax, x0, opt
% all fields except 'c', 'A' and 'l' or 'u' are optional
% x = qps_cplex(...)
% [x, f] = qps_cplex(...)
% [x, f, exitflag] = qps_cplex(...)
% [x, f, exitflag, output] = qps_cplex(...)
% [x, f, exitflag, output, lambda] = qps_cplex(...)
%
%
% Example: (problem from from http://www.jmu.edu/docs/sasdoc/sashtml/iml/chap8/sect12.htm)
% H = [ 1003.1 4.3 6.3 5.9;
% 4.3 2.2 2.1 3.9;
% 6.3 2.1 3.5 4.8;
% 5.9 3.9 4.8 10 ];
% c = zeros(4,1);
% A = [ 1 1 1 1;
% 0.17 0.11 0.10 0.18 ];
% l = [1; 0.10];
% u = [1; Inf];
% xmin = zeros(4,1);
% x0 = [1; 0; 0; 1];
% opt = struct('verbose', 2);
% [x, f, s, out, lambda] = qps_cplex(H, c, A, l, u, xmin, [], x0, opt);
%
% See also CPLEXQP, CPLEXLP, CPLEX_OPTIONS.
% MATPOWER
% $Id: qps_cplex.m,v 1.8 2011/12/12 16:07:39 cvs Exp $
% by Ray Zimmerman, PSERC Cornell
% Copyright (c) 2010, 2011 by Power System Engineering Research Center (PSERC)
%
% This file is part of MATPOWER.
% See http://www.pserc.cornell.edu/matpower/ for more info.
%
% MATPOWER 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.
%
% MATPOWER 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 MATPOWER. If not, see <http://www.gnu.org/licenses/>.
%
% Additional permission under GNU GPL version 3 section 7
%
% If you modify MATPOWER, or any covered work, to interface with
% other modules (such as MATLAB code and MEX-files) available in a
% MATLAB(R) or comparable environment containing parts covered
% under other licensing terms, the licensors of MATPOWER grant
% you additional permission to convey the resulting work.
%% check for CPLEX
% if ~have_fcn('cplexqp')
% error('qps_cplex: requires the Matlab interface for CPLEX');
% end
%%----- input argument handling -----
%% gather inputs
if nargin == 1 && isstruct(H) %% problem struct
p = H;
if isfield(p, 'opt'), opt = p.opt; else, opt = []; end
if isfield(p, 'x0'), x0 = p.x0; else, x0 = []; end
if isfield(p, 'xmax'), xmax = p.xmax; else, xmax = []; end
if isfield(p, 'xmin'), xmin = p.xmin; else, xmin = []; end
if isfield(p, 'u'), u = p.u; else, u = []; end
if isfield(p, 'l'), l = p.l; else, l = []; end
if isfield(p, 'A'), A = p.A; else, A = []; end
if isfield(p, 'c'), c = p.c; else, c = []; end
if isfield(p, 'H'), H = p.H; else, H = []; end
else %% individual args
if nargin < 9
opt = [];
if nargin < 8
x0 = [];
if nargin < 7
xmax = [];
if nargin < 6
xmin = [];
end
end
end
end
end
%% define nx, set default values for missing optional inputs
if isempty(H) || ~any(any(H))
if isempty(A) && isempty(xmin) && isempty(xmax)
error('qps_cplex: LP problem must include constraints or variable bounds');
else
if ~isempty(A)
nx = size(A, 2);
elseif ~isempty(xmin)
nx = length(xmin);
else % if ~isempty(xmax)
nx = length(xmax);
end
end
else
nx = size(H, 1);
end
if isempty(c)
c = zeros(nx, 1);
end
if ~isempty(A) && (isempty(l) || all(l == -Inf)) && ...
(isempty(u) || all(u == Inf))
A = sparse(0,nx); %% no limits => no linear constraints
end
nA = size(A, 1); %% number of original linear constraints
if isempty(u) %% By default, linear inequalities are ...
u = Inf * ones(nA, 1); %% ... unbounded above and ...
end
if isempty(l)
l = -Inf * ones(nA, 1); %% ... unbounded below.
end
if isempty(xmin) %% By default, optimization variables are ...
xmin = -Inf * ones(nx, 1); %% ... unbounded below and ...
end
if isempty(xmax)
xmax = Inf * ones(nx, 1); %% ... unbounded above.
end
if isempty(x0)
x0 = zeros(nx, 1);
end
%% default options
if ~isempty(opt) && isfield(opt, 'verbose') && ~isempty(opt.verbose)
verbose = opt.verbose;
else
verbose = 0;
end
% if ~isempty(opt) && isfield(opt, 'max_it') && ~isempty(opt.max_it)
% max_it = opt.max_it;
% else
% max_it = 0;
% end
%% split up linear constraints
ieq = find( abs(u-l) <= eps ); %% equality
igt = find( u >= 1e10 & l > -1e10 ); %% greater than, unbounded above
ilt = find( l <= -1e10 & u < 1e10 ); %% less than, unbounded below
ibx = find( (abs(u-l) > eps) & (u < 1e10) & (l > -1e10) );
Ae = A(ieq, :);
be = u(ieq);
Ai = [ A(ilt, :); -A(igt, :); A(ibx, :); -A(ibx, :) ];
bi = [ u(ilt); -l(igt); u(ibx); -l(ibx)];
%% grab some dimensions
nlt = length(ilt); %% number of upper bounded linear inequalities
ngt = length(igt); %% number of lower bounded linear inequalities
nbx = length(ibx); %% number of doubly bounded linear inequalities
%% set up options struct for CPLEX
if ~isempty(opt) && isfield(opt, 'cplex_opt') && ~isempty(opt.cplex_opt)
cplex_opt = cplex_options(opt.cplex_opt);
else
cplex_opt = cplex_options;
end
cplex = Cplex('null');
vstr = cplex.getVersion;
[s,e,tE,m,t] = regexp(vstr, '(\d+\.\d+)\.');
vnum = str2num(t{1}{1});
vrb = max([0 verbose-1]);
cplex_opt.barrier.display = vrb;
cplex_opt.conflict.display = vrb;
cplex_opt.mip.display = vrb;
cplex_opt.sifting.display = vrb;
cplex_opt.simplex.display = vrb;
cplex_opt.tune.display = vrb;
if vrb && vnum > 12.2
cplex_opt.diagnostics = 'on';
end
% if max_it
% cplex_opt. %% not sure what to set here
% end
if isempty(Ai) && isempty(Ae)
unconstrained = 1;
Ae = sparse(1, nx);
be = 0;
else
unconstrained = 0;
end
%% call the solver
if verbose
methods = {
'default',
'primal simplex',
'dual simplex',
'network simplex',
'barrier',
'sifting',
'concurrent'
};
end
if isempty(H) || ~any(any(H))
if verbose
fprintf('CPLEX Version %s -- %s LP solver\n', ...
vstr, methods{cplex_opt.lpmethod+1});
end
[x, f, eflag, output, lam] = ...
cplexlp(c, Ai, bi, Ae, be, xmin, xmax, x0, cplex_opt);
else
if verbose
fprintf('CPLEX Version %s -- %s QP solver\n', ...
vstr, methods{cplex_opt.qpmethod+1});
end
%% ensure H is numerically symmetric
if ~isequal(H, H')
H = (H + H')/2;
end
[x, f, eflag, output, lam] = ...
cplexqp(H, c, Ai, bi, Ae, be, xmin, xmax, x0, cplex_opt);
end
%% check for empty results (in case optimization failed)
if isempty(x)
x = NaN(nx, 1);
end
if isempty(f)
f = NaN;
end
if isempty(lam)
lam.ineqlin = NaN(length(bi), 1);
lam.eqlin = NaN(length(be), 1);
lam.lower = NaN(nx, 1);
lam.upper = NaN(nx, 1);
mu_l = NaN(nA, 1);
mu_u = NaN(nA, 1);
else
mu_l = zeros(nA, 1);
mu_u = zeros(nA, 1);
end
if unconstrained
lam.eqlin = [];
end
%% negate prices depending on version
if vnum < 12.3
lam.eqlin = -lam.eqlin;
lam.ineqlin = -lam.ineqlin;
end
%% repackage lambdas
kl = find(lam.eqlin < 0); %% lower bound binding
ku = find(lam.eqlin > 0); %% upper bound binding
mu_l(ieq(kl)) = -lam.eqlin(kl);
mu_l(igt) = lam.ineqlin(nlt+(1:ngt));
mu_l(ibx) = lam.ineqlin(nlt+ngt+nbx+(1:nbx));
mu_u(ieq(ku)) = lam.eqlin(ku);
mu_u(ilt) = lam.ineqlin(1:nlt);
mu_u(ibx) = lam.ineqlin(nlt+ngt+(1:nbx));
lambda = struct( ...
'mu_l', mu_l, ...
'mu_u', mu_u, ...
'lower', lam.lower, ...
'upper', lam.upper ...
);