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function [x, f, eflag, output, lambda] = qps_ot(H, c, A, l, u, xmin, xmax, x0, opt) %QPS_OT Quadratic Program Solver based on QUADPROG/LINPROG. % [X, F, EXITFLAG, OUTPUT, LAMBDA] = ... % QPS_OT(H, C, A, L, U, XMIN, XMAX, X0, OPT) % A wrapper function providing a MATPOWER standardized interface for using % QUADPROG or LINPROG from the Optimization Toolbox 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 % max_it (0) - maximum number of iterations allowed % 0 = use algorithm default % ot_opt - options struct for QUADPROG/LINPROG, values in % verbose and max_it override 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 : QUADPROG/LINPROG exit flag % (see QUADPROG and LINPROG documentation for details) % OUTPUT : QUADPROG/LINPROG output struct % (see QUADPROG and LINPROG 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_ot(H, c, A, l, u, xmin, xmax, x0, opt) % % x = qps_ot(H, c, A, l, u) % x = qps_ot(H, c, A, l, u, xmin, xmax) % x = qps_ot(H, c, A, l, u, xmin, xmax, x0) % x = qps_ot(H, c, A, l, u, xmin, xmax, x0, opt) % x = qps_ot(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_ot(...) % [x, f] = qps_ot(...) % [x, f, exitflag] = qps_ot(...) % [x, f, exitflag, output] = qps_ot(...) % [x, f, exitflag, output, lambda] = qps_ot(...) % % % 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_ot(H, c, A, l, u, xmin, [], x0, opt); % % See also QUADPROG, LINPROG. % MATPOWER % $Id: qps_ot.m,v 1.13 2011/09/09 15:26:09 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 Optimization Toolbox % if ~have_fcn('quadprog') % error('qps_ot: requires the Optimization Toolbox'); % 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_ot: 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 if ~isempty(opt) && isfield(opt, 'ot_opt') && ~isempty(opt.ot_opt) ot_opt = opt.ot_opt; else if isempty(H) || ~any(any(H)) ot_opt = optimset('linprog'); else ot_opt = optimset('quadprog'); if have_fcn('quadprog_ls') ot_opt = optimset(ot_opt, 'Algorithm', 'interior-point-convex'); else ot_opt = optimset(ot_opt, 'LargeScale', 'off'); end end end if max_it ot_opt = optimset(ot_opt, 'MaxIter', max_it); end if verbose > 1 ot_opt = optimset(ot_opt, 'Display', 'iter'); %% seems to be same as 'final' elseif verbose == 1 ot_opt = optimset(ot_opt, 'Display', 'final'); else ot_opt = optimset(ot_opt, 'Display', 'off'); end %% call the solver if isempty(H) || ~any(any(H)) [x, f, eflag, output, lam] = ... linprog(c, Ai, bi, Ae, be, xmin, xmax, x0, ot_opt); else [x, f, eflag, output, lam] = ... quadprog(H, c, Ai, bi, Ae, be, xmin, xmax, x0, ot_opt); end %% repackage lambdas kl = find(lam.eqlin < 0); %% lower bound binding ku = find(lam.eqlin > 0); %% upper bound binding mu_l = zeros(nA, 1); 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 = zeros(nA, 1); 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(1:nx), ... 'upper', lam.upper(1:nx) ... );