QPS_MIPS Quadratic Program Solver based on MIPS.
[X, F, EXITFLAG, OUTPUT, LAMBDA] = ...
QPS_MIPS(H, C, A, L, U, XMIN, XMAX, X0, OPT)
[X, F, EXITFLAG, OUTPUT, LAMBDA] = QPS_MIPS(PROBLEM)
A wrapper function providing a MATPOWER standardized interface for using
the MATPOWER Interior Point Solver (MIPS) 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 MIPS options struct (see MIPS for details)
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 = first order optimality conditions satisfied
0 = maximum number of iterations reached
-1 = numerically failed
OUTPUT : output struct with the following fields:
iterations - number of iterations performed
hist - struct array with trajectories of the following:
feascond, gradcond, compcond, costcond, gamma,
stepsize, obj, alphap, alphad
message - exit message
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_mips(H, c, A, l, u, xmin, xmax, x0, opt)
x = qps_mips(H, c, A, l, u)
x = qps_mips(H, c, A, l, u, xmin, xmax)
x = qps_mips(H, c, A, l, u, xmin, xmax, x0)
x = qps_mips(H, c, A, l, u, xmin, xmax, x0, opt)
x = qps_mips(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_mips(...)
[x, f] = qps_mips(...)
[x, f, exitflag] = qps_mips(...)
[x, f, exitflag, output] = qps_mips(...)
[x, f, exitflag, output, lambda] = qps_mips(...)
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_mips(H, c, A, l, u, xmin, [], x0, opt);
See also MIPS.0001 function [x, f, eflag, output, lambda] = qps_mips(H, c, A, l, u, xmin, xmax, x0, opt) 0002 %QPS_MIPS Quadratic Program Solver based on MIPS. 0003 % [X, F, EXITFLAG, OUTPUT, LAMBDA] = ... 0004 % QPS_MIPS(H, C, A, L, U, XMIN, XMAX, X0, OPT) 0005 % [X, F, EXITFLAG, OUTPUT, LAMBDA] = QPS_MIPS(PROBLEM) 0006 % A wrapper function providing a MATPOWER standardized interface for using 0007 % the MATPOWER Interior Point Solver (MIPS) to solve the following QP 0008 % (quadratic programming) problem: 0009 % 0010 % min 1/2 X'*H*X + C'*X 0011 % X 0012 % 0013 % subject to 0014 % 0015 % L <= A*X <= U (linear constraints) 0016 % XMIN <= X <= XMAX (variable bounds) 0017 % 0018 % Inputs (all optional except H, C, A and L): 0019 % H : matrix (possibly sparse) of quadratic cost coefficients 0020 % C : vector of linear cost coefficients 0021 % A, L, U : define the optional linear constraints. Default 0022 % values for the elements of L and U are -Inf and Inf, 0023 % respectively. 0024 % XMIN, XMAX : optional lower and upper bounds on the 0025 % X variables, defaults are -Inf and Inf, respectively. 0026 % X0 : optional starting value of optimization vector X 0027 % OPT : optional MIPS options struct (see MIPS for details) 0028 % PROBLEM : The inputs can alternatively be supplied in a single 0029 % PROBLEM struct with fields corresponding to the input arguments 0030 % described above: H, c, A, l, u, xmin, xmax, x0, opt 0031 % 0032 % Outputs: 0033 % X : solution vector 0034 % F : final objective function value 0035 % EXITFLAG : exit flag 0036 % 1 = first order optimality conditions satisfied 0037 % 0 = maximum number of iterations reached 0038 % -1 = numerically failed 0039 % OUTPUT : output struct with the following fields: 0040 % iterations - number of iterations performed 0041 % hist - struct array with trajectories of the following: 0042 % feascond, gradcond, compcond, costcond, gamma, 0043 % stepsize, obj, alphap, alphad 0044 % message - exit message 0045 % LAMBDA : struct containing the Langrange and Kuhn-Tucker 0046 % multipliers on the constraints, with fields: 0047 % mu_l - lower (left-hand) limit on linear constraints 0048 % mu_u - upper (right-hand) limit on linear constraints 0049 % lower - lower bound on optimization variables 0050 % upper - upper bound on optimization variables 0051 % 0052 % Note the calling syntax is almost identical to that of QUADPROG 0053 % from MathWorks' Optimization Toolbox. The main difference is that 0054 % the linear constraints are specified with A, L, U instead of 0055 % A, B, Aeq, Beq. 0056 % 0057 % Calling syntax options: 0058 % [x, f, exitflag, output, lambda] = ... 0059 % qps_mips(H, c, A, l, u, xmin, xmax, x0, opt) 0060 % 0061 % x = qps_mips(H, c, A, l, u) 0062 % x = qps_mips(H, c, A, l, u, xmin, xmax) 0063 % x = qps_mips(H, c, A, l, u, xmin, xmax, x0) 0064 % x = qps_mips(H, c, A, l, u, xmin, xmax, x0, opt) 0065 % x = qps_mips(problem), where problem is a struct with fields: 0066 % H, c, A, l, u, xmin, xmax, x0, opt 0067 % all fields except 'c', 'A' and 'l' or 'u' are optional 0068 % x = qps_mips(...) 0069 % [x, f] = qps_mips(...) 0070 % [x, f, exitflag] = qps_mips(...) 0071 % [x, f, exitflag, output] = qps_mips(...) 0072 % [x, f, exitflag, output, lambda] = qps_mips(...) 0073 % 0074 % Example: (problem from from https://v8doc.sas.com/sashtml/iml/chap8/sect12.htm) 0075 % H = [ 1003.1 4.3 6.3 5.9; 0076 % 4.3 2.2 2.1 3.9; 0077 % 6.3 2.1 3.5 4.8; 0078 % 5.9 3.9 4.8 10 ]; 0079 % c = zeros(4,1); 0080 % A = [ 1 1 1 1; 0081 % 0.17 0.11 0.10 0.18 ]; 0082 % l = [1; 0.10]; 0083 % u = [1; Inf]; 0084 % xmin = zeros(4,1); 0085 % x0 = [1; 0; 0; 1]; 0086 % opt = struct('verbose', 2); 0087 % [x, f, s, out, lambda] = qps_mips(H, c, A, l, u, xmin, [], x0, opt); 0088 % 0089 % See also MIPS. 0090 0091 % MIPS 0092 % Copyright (c) 2010-2020, Power Systems Engineering Research Center (PSERC) 0093 % by Ray Zimmerman, PSERC Cornell 0094 % 0095 % This file is part of MIPS. 0096 % Covered by the 3-clause BSD License (see LICENSE file for details). 0097 % See https://github.com/MATPOWER/mips for more info. 0098 0099 %%----- input argument handling ----- 0100 %% gather inputs 0101 if nargin == 1 && isstruct(H) %% problem struct 0102 p = H; 0103 else %% individual args 0104 p = struct('H', H, 'c', c, 'A', A, 'l', l, 'u', u); 0105 if nargin > 5 0106 p.xmin = xmin; 0107 if nargin > 6 0108 p.xmax = xmax; 0109 if nargin > 7 0110 p.x0 = x0; 0111 if nargin > 8 0112 p.opt = opt; 0113 end 0114 end 0115 end 0116 end 0117 end 0118 0119 %% define nx, set default values for H and c 0120 if ~isfield(p, 'H') || isempty(p.H) || ~any(any(p.H)) 0121 if (~isfield(p, 'A') || isempty(p.A)) && ... 0122 (~isfield(p, 'xmin') || isempty(p.xmin)) && ... 0123 (~isfield(p, 'xmax') || isempty(p.xmax)) 0124 error('qps_mips: LP problem must include constraints or variable bounds'); 0125 else 0126 if isfield(p, 'A') && ~isempty(p.A) 0127 nx = size(p.A, 2); 0128 elseif isfield(p, 'xmin') && ~isempty(p.xmin) 0129 nx = length(p.xmin); 0130 else % if isfield(p, 'xmax') && ~isempty(p.xmax) 0131 nx = length(p.xmax); 0132 end 0133 end 0134 p.H = sparse(nx, nx); 0135 else 0136 nx = size(p.H, 1); 0137 end 0138 if ~isfield(p, 'c') || isempty(p.c) 0139 p.c = zeros(nx, 1); 0140 end 0141 if ~isfield(p, 'x0') || isempty(p.x0) 0142 p.x0 = zeros(nx, 1); 0143 end 0144 0145 %%----- run optimization ----- 0146 p.f_fcn = @(x)qp_f(x, p.H, p.c); 0147 [x, f, eflag, output, lambda] = mips(p); 0148 0149 %%----- objective function ----- 0150 function [f, df, d2f] = qp_f(x, H, c) 0151 f = 0.5 * x' * H * x + c' * x; 0152 if nargout > 1 0153 df = H * x + c; 0154 if nargout > 2 0155 d2f = H; 0156 end 0157 end