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function [x, f, eflag, output, lambda] = qps_gurobi(H, c, A, l, u, xmin, xmax, x0, opt) %QPS_GUROBI Quadratic Program Solver based on GUROBI. % [X, F, EXITFLAG, OUTPUT, LAMBDA] = ... % QPS_GUROBI(H, C, A, L, U, XMIN, XMAX, X0, OPT) % A wrapper function providing a MATPOWER standardized interface for using % GUROBI_MEX 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 % grb_opt - options struct for GUROBI, 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 : GUROBI_MEX exit flag % 1 = converged % 0 or negative values = negative of GUROBI_MEX exit flag % (see GUROBI_MEX documentation for details) % OUTPUT : GUROBI_MEX output struct % (see GUROBI_MEX 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_gurobi(H, c, A, l, u, xmin, xmax, x0, opt) % % x = qps_gurobi(H, c, A, l, u) % x = qps_gurobi(H, c, A, l, u, xmin, xmax) % x = qps_gurobi(H, c, A, l, u, xmin, xmax, x0) % x = qps_gurobi(H, c, A, l, u, xmin, xmax, x0, opt) % x = qps_gurobi(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_gurobi(...) % [x, f] = qps_gurobi(...) % [x, f, exitflag] = qps_gurobi(...) % [x, f, exitflag, output] = qps_gurobi(...) % [x, f, exitflag, output, lambda] = qps_gurobi(...) % % % 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_gurobi(H, c, A, l, u, xmin, [], x0, opt); % % See also GUROBI_MEX. % MATPOWER % $Id: qps_gurobi.m,v 1.3 2011/09/09 15:27:52 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. %%----- 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_gurobi: 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 %% set up options struct for Gurobi if ~isempty(opt) && isfield(opt, 'grb_opt') && ~isempty(opt.grb_opt) g_opt = gurobi_options(opt.grb_opt); else g_opt = gurobi_options; end g_opt.Display = min(verbose, 3); if verbose g_opt.DisplayInterval = 1; else g_opt.DisplayInterval = Inf; end if ~issparse(A) A = sparse(A); 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) ); %% 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 neq = length(ieq); %% number of equalities niq = nlt+ngt+2*nbx; %% number of inequalities AA = [ A(ieq, :); A(ilt, :); -A(igt, :); A(ibx, :); -A(ibx, :) ]; bb = [ u(ieq); u(ilt); -l(igt); u(ibx); -l(ibx) ]; contypes = char([ double('=')*ones(1,neq) double('<')*ones(1,niq) ]); %% call the solver if isempty(H) || ~any(any(H)) lpqp = 'LP'; else lpqp = 'QP'; [rr, cc, vv] = find(H); g_opt.QP.qrow = int32(rr' - 1); g_opt.QP.qcol = int32(cc' - 1); g_opt.QP.qval = 0.5 * vv'; end if verbose methods = { 'primal simplex', 'dual simplex', 'interior point', 'concurrent', 'deterministic concurrent' }; fprintf('Gurobi Version %s -- %s %s solver\n', ... '<unknown>', methods{g_opt.Method+1}, lpqp); end [x, f, eflag, output, lambda] = ... gurobi_mex(c', 1, AA, bb, contypes, xmin, xmax, 'C', g_opt); pi = lambda.Pi; rc = lambda.RC; output.flag = eflag; if eflag == 2 eflag = 1; %% optimal solution found else eflag = -eflag; %% failed somehow end %% check for empty results (in case optimization failed) if isempty(x) x = NaN(nx, 1); lam.lower = NaN(nx, 1); lam.upper = NaN(nx, 1); else lam.lower = zeros(nx, 1); lam.upper = zeros(nx, 1); end if isempty(f) f = NaN; end if isempty(pi) pi = NaN(length(bb), 1); end kl = find(rc > 0); %% lower bound binding ku = find(rc < 0); %% upper bound binding lam.lower(kl) = rc(kl); lam.upper(ku) = -rc(ku); lam.eqlin = pi(1:neq); lam.ineqlin = pi(neq+(1:niq)); mu_l = zeros(nA, 1); mu_u = zeros(nA, 1); %% 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 ... );