QPS_GLPK Linear Program Solver based on GLPK - GNU Linear Programming Kit.
[X, F, EXITFLAG, OUTPUT, LAMBDA] = ...
QPS_GLPK(H, C, A, L, U, XMIN, XMAX, X0, OPT)
A wrapper function providing a MATPOWER standardized interface for using
GLKP to solve the following LP (linear programming) problem:
min 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 : dummy matrix (possibly sparse) of quadratic cost coefficients
for QP problems, which GLPK does not handle
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 (NOT USED)
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
glpk_opt - options struct for GLPK, 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 : exit flag, 1 - optimal, <= 0 - infeasible, unbounded or other
OUTPUT : struct with errnum and status fields from GLPK output args
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 GLPK. 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_glpk(H, c, A, l, u, xmin, xmax, x0, opt)
x = qps_glpk(H, c, A, l, u)
x = qps_glpk(H, c, A, l, u, xmin, xmax)
x = qps_glpk(H, c, A, l, u, xmin, xmax, x0)
x = qps_glpk(H, c, A, l, u, xmin, xmax, x0, opt)
x = qps_glpk(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_glpk(...)
[x, f] = qps_glpk(...)
[x, f, exitflag] = qps_glpk(...)
[x, f, exitflag, output] = qps_glpk(...)
[x, f, exitflag, output, lambda] = qps_glpk(...)
Example: (problem from from https://v8doc.sas.com/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_glpk(H, c, A, l, u, xmin, [], x0, opt);
See also GLPK.0001 function [x, f, eflag, output, lambda] = qps_glpk(H, c, A, l, u, xmin, xmax, x0, opt) 0002 %QPS_GLPK Linear Program Solver based on GLPK - GNU Linear Programming Kit. 0003 % [X, F, EXITFLAG, OUTPUT, LAMBDA] = ... 0004 % QPS_GLPK(H, C, A, L, U, XMIN, XMAX, X0, OPT) 0005 % A wrapper function providing a MATPOWER standardized interface for using 0006 % GLKP to solve the following LP (linear programming) problem: 0007 % 0008 % min C'*X 0009 % X 0010 % 0011 % subject to 0012 % 0013 % L <= A*X <= U (linear constraints) 0014 % XMIN <= X <= XMAX (variable bounds) 0015 % 0016 % Inputs (all optional except H, C, A and L): 0017 % H : dummy matrix (possibly sparse) of quadratic cost coefficients 0018 % for QP problems, which GLPK does not handle 0019 % C : vector of linear cost coefficients 0020 % A, L, U : define the optional linear constraints. Default 0021 % values for the elements of L and U are -Inf and Inf, 0022 % respectively. 0023 % XMIN, XMAX : optional lower and upper bounds on the 0024 % X variables, defaults are -Inf and Inf, respectively. 0025 % X0 : optional starting value of optimization vector X (NOT USED) 0026 % OPT : optional options structure with the following fields, 0027 % all of which are also optional (default values shown in 0028 % parentheses) 0029 % verbose (0) - controls level of progress output displayed 0030 % 0 = no progress output 0031 % 1 = some progress output 0032 % 2 = verbose progress output 0033 % glpk_opt - options struct for GLPK, value in 0034 % verbose overrides these options 0035 % PROBLEM : The inputs can alternatively be supplied in a single 0036 % PROBLEM struct with fields corresponding to the input arguments 0037 % described above: H, c, A, l, u, xmin, xmax, x0, opt 0038 % 0039 % Outputs: 0040 % X : solution vector 0041 % F : final objective function value 0042 % EXITFLAG : exit flag, 1 - optimal, <= 0 - infeasible, unbounded or other 0043 % OUTPUT : struct with errnum and status fields from GLPK output args 0044 % LAMBDA : struct containing the Langrange and Kuhn-Tucker 0045 % multipliers on the constraints, with fields: 0046 % mu_l - lower (left-hand) limit on linear constraints 0047 % mu_u - upper (right-hand) limit on linear constraints 0048 % lower - lower bound on optimization variables 0049 % upper - upper bound on optimization variables 0050 % 0051 % Note the calling syntax is almost identical to that of GLPK. The main 0052 % difference is that the linear constraints are specified with A, L, U 0053 % instead of A, B, Aeq, Beq. 0054 % 0055 % Calling syntax options: 0056 % [x, f, exitflag, output, lambda] = ... 0057 % qps_glpk(H, c, A, l, u, xmin, xmax, x0, opt) 0058 % 0059 % x = qps_glpk(H, c, A, l, u) 0060 % x = qps_glpk(H, c, A, l, u, xmin, xmax) 0061 % x = qps_glpk(H, c, A, l, u, xmin, xmax, x0) 0062 % x = qps_glpk(H, c, A, l, u, xmin, xmax, x0, opt) 0063 % x = qps_glpk(problem), where problem is a struct with fields: 0064 % H, c, A, l, u, xmin, xmax, x0, opt 0065 % all fields except 'c', 'A' and 'l' or 'u' are optional 0066 % x = qps_glpk(...) 0067 % [x, f] = qps_glpk(...) 0068 % [x, f, exitflag] = qps_glpk(...) 0069 % [x, f, exitflag, output] = qps_glpk(...) 0070 % [x, f, exitflag, output, lambda] = qps_glpk(...) 0071 % 0072 % 0073 % Example: (problem from from https://v8doc.sas.com/sashtml/iml/chap8/sect12.htm) 0074 % H = [ 1003.1 4.3 6.3 5.9; 0075 % 4.3 2.2 2.1 3.9; 0076 % 6.3 2.1 3.5 4.8; 0077 % 5.9 3.9 4.8 10 ]; 0078 % c = zeros(4,1); 0079 % A = [ 1 1 1 1; 0080 % 0.17 0.11 0.10 0.18 ]; 0081 % l = [1; 0.10]; 0082 % u = [1; Inf]; 0083 % xmin = zeros(4,1); 0084 % x0 = [1; 0; 0; 1]; 0085 % opt = struct('verbose', 2); 0086 % [x, f, s, out, lambda] = qps_glpk(H, c, A, l, u, xmin, [], x0, opt); 0087 % 0088 % See also GLPK. 0089 0090 % MATPOWER 0091 % Copyright (c) 2010-2016, Power Systems Engineering Research Center (PSERC) 0092 % by Ray Zimmerman, PSERC Cornell 0093 % 0094 % This file is part of MATPOWER. 0095 % Covered by the 3-clause BSD License (see LICENSE file for details). 0096 % See https://matpower.org for more info. 0097 0098 %% check for Optimization Toolbox 0099 % if ~have_fcn('quadprog') 0100 % error('qps_glpk: requires the MEX interface to GLPK'); 0101 % end 0102 0103 %%----- input argument handling ----- 0104 %% gather inputs 0105 if nargin == 1 && isstruct(H) %% problem struct 0106 p = H; 0107 if isfield(p, 'opt'), opt = p.opt; else, opt = []; end 0108 if isfield(p, 'x0'), x0 = p.x0; else, x0 = []; end 0109 if isfield(p, 'xmax'), xmax = p.xmax; else, xmax = []; end 0110 if isfield(p, 'xmin'), xmin = p.xmin; else, xmin = []; end 0111 if isfield(p, 'u'), u = p.u; else, u = []; end 0112 if isfield(p, 'l'), l = p.l; else, l = []; end 0113 if isfield(p, 'A'), A = p.A; else, A = []; end 0114 if isfield(p, 'c'), c = p.c; else, c = []; end 0115 if isfield(p, 'H'), H = p.H; else, H = []; end 0116 else %% individual args 0117 if nargin < 9 0118 opt = []; 0119 if nargin < 8 0120 x0 = []; 0121 if nargin < 7 0122 xmax = []; 0123 if nargin < 6 0124 xmin = []; 0125 end 0126 end 0127 end 0128 end 0129 end 0130 0131 %% define nx, set default values for missing optional inputs 0132 if isempty(H) || ~any(any(H)) 0133 if isempty(A) && isempty(xmin) && isempty(xmax) 0134 error('qps_glpk: LP problem must include constraints or variable bounds'); 0135 else 0136 if ~isempty(A) 0137 nx = size(A, 2); 0138 elseif ~isempty(xmin) 0139 nx = length(xmin); 0140 else % if ~isempty(xmax) 0141 nx = length(xmax); 0142 end 0143 end 0144 else 0145 error('qps_glpk: GLPK handles only LP problems, not QP problems'); 0146 nx = size(H, 1); 0147 end 0148 if isempty(c) 0149 c = zeros(nx, 1); 0150 end 0151 if isempty(A) || (~isempty(A) && (isempty(l) || all(l == -Inf)) && ... 0152 (isempty(u) || all(u == Inf))) 0153 A = sparse(0,nx); %% no limits => no linear constraints 0154 end 0155 nA = size(A, 1); %% number of original linear constraints 0156 if isempty(u) %% By default, linear inequalities are ... 0157 u = Inf(nA, 1); %% ... unbounded above and ... 0158 end 0159 if isempty(l) 0160 l = -Inf(nA, 1); %% ... unbounded below. 0161 end 0162 if isempty(xmin) %% By default, optimization variables are ... 0163 xmin = -Inf(nx, 1); %% ... unbounded below and ... 0164 end 0165 if isempty(xmax) 0166 xmax = Inf(nx, 1); %% ... unbounded above. 0167 end 0168 if isempty(x0) 0169 x0 = zeros(nx, 1); 0170 end 0171 0172 %% default options 0173 if ~isempty(opt) && isfield(opt, 'verbose') && ~isempty(opt.verbose) 0174 verbose = opt.verbose; 0175 else 0176 verbose = 0; 0177 end 0178 0179 %% split up linear constraints 0180 ieq = find( abs(u-l) <= eps ); %% equality 0181 igt = find( u >= 1e10 & l > -1e10 ); %% greater than, unbounded above 0182 ilt = find( l <= -1e10 & u < 1e10 ); %% less than, unbounded below 0183 ibx = find( (abs(u-l) > eps) & (u < 1e10) & (l > -1e10) ); 0184 AA = [ A(ieq, :); A(ilt, :); -A(igt, :); A(ibx, :); -A(ibx, :) ]; 0185 bb = [ u(ieq); u(ilt); -l(igt); u(ibx); -l(ibx)]; 0186 0187 %% grab some dimensions 0188 nlt = length(ilt); %% number of upper bounded linear inequalities 0189 ngt = length(igt); %% number of lower bounded linear inequalities 0190 nbx = length(ibx); %% number of doubly bounded linear inequalities 0191 neq = length(ieq); %% number of equalities 0192 nie = nlt+ngt+2*nbx; %% number of inequalities 0193 0194 ctype = [repmat('S', neq, 1); repmat('U', nlt+ngt+2*nbx, 1)]; 0195 vtype = repmat('C', nx, 1); 0196 0197 %% set options struct for GLPK 0198 if ~isempty(opt) && isfield(opt, 'glpk_opt') && ~isempty(opt.glpk_opt) 0199 glpk_opt = glpk_options(opt.glpk_opt); 0200 else 0201 glpk_opt = glpk_options; 0202 end 0203 glpk_opt.msglev = verbose; 0204 0205 %% call the solver 0206 [x, f, errnum, extra] = ... 0207 glpk(c, AA, bb, xmin, xmax, ctype, vtype, 1, glpk_opt); 0208 0209 %% set exit flag 0210 if isfield(extra, 'status') %% status found in extra.status 0211 output.errnum = errnum; 0212 output.status = extra.status; 0213 eflag = -errnum; 0214 if eflag == 0 && extra.status == 5 0215 eflag = 1; 0216 end 0217 else %% status found in errnum 0218 output.errnum = []; 0219 output.status = errnum; 0220 if have_fcn('octave') 0221 if errnum == 180 || errnum == 151 || errnum == 171 0222 eflag = 1; 0223 else 0224 eflag = 0; 0225 end 0226 else 0227 if errnum == 5 0228 eflag = 1; 0229 else 0230 eflag = 0; 0231 end 0232 end 0233 end 0234 0235 %% repackage lambdas 0236 if isempty(extra) || ~isfield(extra, 'lambda') || isempty(extra.lambda) 0237 lambda = struct( ... 0238 'mu_l', zeros(nA, 1), ... 0239 'mu_u', zeros(nA, 1), ... 0240 'lower', zeros(nx, 1), ... 0241 'upper', zeros(nx, 1) ... 0242 ); 0243 else 0244 lam.eqlin = extra.lambda(1:neq); 0245 lam.ineqlin = extra.lambda(neq+(1:nie)); 0246 lam.lower = extra.redcosts; 0247 lam.upper = -extra.redcosts; 0248 lam.lower(lam.lower < 0) = 0; 0249 lam.upper(lam.upper < 0) = 0; 0250 kl = find(lam.eqlin > 0); %% lower bound binding 0251 ku = find(lam.eqlin < 0); %% upper bound binding 0252 0253 mu_l = zeros(nA, 1); 0254 mu_l(ieq(kl)) = lam.eqlin(kl); 0255 mu_l(igt) = -lam.ineqlin(nlt+(1:ngt)); 0256 mu_l(ibx) = -lam.ineqlin(nlt+ngt+nbx+(1:nbx)); 0257 0258 mu_u = zeros(nA, 1); 0259 mu_u(ieq(ku)) = -lam.eqlin(ku); 0260 mu_u(ilt) = -lam.ineqlin(1:nlt); 0261 mu_u(ibx) = -lam.ineqlin(nlt+ngt+(1:nbx)); 0262 0263 lambda = struct( ... 0264 'mu_l', mu_l, ... 0265 'mu_u', mu_u, ... 0266 'lower', lam.lower(1:nx), ... 0267 'upper', lam.upper(1:nx) ... 0268 ); 0269 end