t_mips

T_MIPS  Tests of MIPS NLP solver.

0001 function t_mips(quiet)
0002 %T_MIPS  Tests of MIPS NLP solver.
0003 
0004 %   MIPS
0005 %   Copyright (c) 2010-2018, Power Systems Engineering Research Center (PSERC)
0006 %   by Ray Zimmerman, PSERC Cornell
0007 %
0008 %   This file is part of MIPS.
0009 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0010 %   See https://github.com/MATPOWER/mips for more info.
0011 
0012 if nargin < 1
0013     quiet = 0;
0014 end
0015 
0016 num_tests = 60;
0017 
0018 t_begin(num_tests, quiet);
0019 
0020 t = 'unconstrained banana function : ';
0021 %% from MATLAB Optimization Toolbox's bandem.m
0022 f_fcn = @(x)f2(x);
0023 x0 = [-1.9; 2];
0024 % [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], [], [], [], [], struct('verbose', 2));
0025 [x, f, s, out, lam] = mips(f_fcn, x0);
0026 t_is(s, 1, 13, [t 'success']);
0027 t_is(x, [1; 1], 13, [t 'x']);
0028 t_is(f, 0, 13, [t 'f']);
0029 t_is(out.hist(end).compcond, 0, 6, [t 'compcond']);
0030 t_ok(isempty(lam.mu_l), [t 'lam.mu_l']);
0031 t_ok(isempty(lam.mu_u), [t 'lam.mu_u']);
0032 t_is(lam.lower, zeros(size(x)), 13, [t 'lam.lower']);
0033 t_is(lam.upper, zeros(size(x)), 13, [t 'lam.upper']);
0034 
0035 t = 'unconstrained 3-d quadratic : ';
0036 %% from http://www.akiti.ca/QuadProgEx0Constr.html
0037 f_fcn = @(x)f3(x);
0038 x0 = [0; 0; 0];
0039 % [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], [], [], [], [], struct('verbose', 2));
0040 [x, f, s, out, lam] = mips(f_fcn, x0);
0041 t_is(s, 1, 13, [t 'success']);
0042 t_is(x, [3; 5; 7], 13, [t 'x']);
0043 t_is(f, -244, 13, [t 'f']);
0044 t_is(out.hist(end).compcond, 0, 6, [t 'compcond']);
0045 t_ok(isempty(lam.mu_l), [t 'lam.mu_l']);
0046 t_ok(isempty(lam.mu_u), [t 'lam.mu_u']);
0047 t_is(lam.lower, zeros(size(x)), 13, [t 'lam.lower']);
0048 t_is(lam.upper, zeros(size(x)), 13, [t 'lam.upper']);
0049 
0050 t = 'constrained 4-d QP : ';
0051 %% from https://v8doc.sas.com/sashtml/iml/chap8/sect12.htm
0052 f_fcn = @(x)f4(x);
0053 x0 = [1; 0; 0; 1];
0054 A = [   1       1       1       1;
0055         0.17    0.11    0.10    0.18    ];
0056 l = [1; 0.10];
0057 u = [1; Inf];
0058 xmin = zeros(4,1);
0059 % [x, f, s, out, lam] = mips(f_fcn, x0, A, l, u, xmin, [], [], [], struct('verbose', 2));
0060 [x, f, s, out, lam] = mips(f_fcn, x0, A, l, u, xmin);
0061 t_is(s, 1, 13, [t 'success']);
0062 t_is(x, [0; 2.8; 0.2; 0]/3, 6, [t 'x']);
0063 t_is(f, 3.29/3, 6, [t 'f']);
0064 t_is(out.hist(end).compcond, 0, 6, [t 'compcond']);
0065 t_is(lam.mu_l, [6.58;0]/3, 6, [t 'lam.mu_l']);
0066 t_is(lam.mu_u, [0;0], 13, [t 'lam.mu_u']);
0067 t_is(lam.lower, [2.24;0;0;1.7667], 4, [t 'lam.lower']);
0068 t_is(lam.upper, zeros(size(x)), 13, [t 'lam.upper']);
0069 
0070 % %% check with quadprog (for dev testing only)
0071 % H = [   1003.1  4.3     6.3     5.9;
0072 %         4.3     2.2     2.1     3.9;
0073 %         6.3     2.1     3.5     4.8;
0074 %         5.9     3.9     4.8     10  ];
0075 % c = zeros(4,1);
0076 % [x, f, s, out, lam] = quadprog(H,c,-A(2,:), -0.10, A(1,:), 1, xmin);
0077 % t_is(s, 1, 13, [t 'success']);
0078 % t_is(x, [0; 2.8; 0.2; 0]/3, 6, [t 'x']);
0079 % t_is(f, 3.29/3, 6, [t 'f']);
0080 % t_is(lam.eqlin, -6.58/3, 6, [t 'lam.eqlin']);
0081 % t_is(lam.ineqlin, 0, 13, [t 'lam.ineqlin']);
0082 % t_is(lam.lower, [2.24;0;0;1.7667], 4, [t 'lam.lower']);
0083 % t_is(lam.upper, [0;0;0;0], 13, [t 'lam.upper']);
0084 
0085 t = 'constrained 2-d nonlinear : ';
0086 %% from https://en.wikipedia.org/wiki/Nonlinear_programming#2-dimensional_example
0087 f_fcn = @(x)f5(x);
0088 gh_fcn = @(x)gh5(x);
0089 hess_fcn = @(x, lam, cost_mult)hess5(x, lam, cost_mult);
0090 x0 = [1.1; 0];
0091 xmin = zeros(2, 1);
0092 % xmax = 3 * ones(2, 1);
0093 % [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], xmin, [], gh_fcn, hess_fcn, struct('verbose', 2));
0094 [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], xmin, [], gh_fcn, hess_fcn);
0095 t_is(s, 1, 13, [t 'success']);
0096 t_is(x, [1; 1], 6, [t 'x']);
0097 t_is(f, -2, 6, [t 'f']);
0098 t_is(out.hist(end).compcond, 0, 6, [t 'compcond']);
0099 t_is(lam.ineqnonlin, [0;0.5], 6, [t 'lam.ineqnonlin']);
0100 t_ok(isempty(lam.mu_l), [t 'lam.mu_l']);
0101 t_ok(isempty(lam.mu_u), [t 'lam.mu_u']);
0102 t_is(lam.lower, zeros(size(x)), 13, [t 'lam.lower']);
0103 t_is(lam.upper, zeros(size(x)), 13, [t 'lam.upper']);
0104 % %% check with fmincon (for dev testing only)
0105 % % fmoptions = optimset('Algorithm', 'interior-point');
0106 % % [x, f, s, out, lam] = fmincon(f_fcn, x0, [], [], [], [], xmin, [], gh_fcn, fmoptions);
0107 % [x, f, s, out, lam] = fmincon(f_fcn, x0, [], [], [], [], [], [], gh_fcn);
0108 % t_is(s, 1, 13, [t 'success']);
0109 % t_is(x, [1; 1], 4, [t 'x']);
0110 % t_is(f, -2, 6, [t 'f']);
0111 % t_is(lam.ineqnonlin, [0;0.5], 6, [t 'lam.ineqnonlin']);
0112 
0113 t = 'constrained 3-d nonlinear : ';
0114 %% from https://en.wikipedia.org/wiki/Nonlinear_programming#3-dimensional_example
0115 f_fcn = @(x)f6(x);
0116 gh_fcn = @(x)gh6(x);
0117 hess_fcn = @(x, lam, cost_mult)hess6(x, lam, cost_mult);
0118 x0 = [1; 1; 0];
0119 % [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], [], [], gh_fcn, hess_fcn, struct('verbose', 2, 'comptol', 1e-9));
0120 [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], [], [], gh_fcn, hess_fcn);
0121 t_is(s, 1, 13, [t 'success']);
0122 t_is(x, [1.58113883; 2.23606798; 1.58113883], 6, [t 'x']);
0123 t_is(f, -5*sqrt(2), 6, [t 'f']);
0124 t_is(out.hist(end).compcond, 0, 6, [t 'compcond']);
0125 t_is(lam.ineqnonlin, [0;sqrt(2)/2], 7, [t 'lam.ineqnonlin']);
0126 t_ok(isempty(lam.mu_l), [t 'lam.mu_l']);
0127 t_ok(isempty(lam.mu_u), [t 'lam.mu_u']);
0128 t_is(lam.lower, zeros(size(x)), 13, [t 'lam.lower']);
0129 t_is(lam.upper, zeros(size(x)), 13, [t 'lam.upper']);
0130 % %% check with fmincon (for dev testing only)
0131 % % fmoptions = optimset('Algorithm', 'interior-point');
0132 % % [x, f, s, out, lam] = fmincon(f_fcn, x0, [], [], [], [], xmin, [], gh_fcn, fmoptions);
0133 % [x, f, s, out, lam] = fmincon(f_fcn, x0, [], [], [], [], [], [], gh_fcn);
0134 % t_is(s, 1, 13, [t 'success']);
0135 % t_is(x, [1.58113883; 2.23606798; 1.58113883], 4, [t 'x']);
0136 % t_is(f, -5*sqrt(2), 8, [t 'f']);
0137 % t_is(lam.ineqnonlin, [0;sqrt(2)/2], 8, [t 'lam.ineqnonlin']);
0138 
0139 t = 'constrained 3-d nonlinear (struct) : ';
0140 p = struct('f_fcn', f_fcn, 'x0', x0, 'gh_fcn', gh_fcn, 'hess_fcn', hess_fcn);
0141 [x, f, s, out, lam] = mips(p);
0142 t_is(s, 1, 13, [t 'success']);
0143 t_is(x, [1.58113883; 2.23606798; 1.58113883], 6, [t 'x']);
0144 t_is(f, -5*sqrt(2), 6, [t 'f']);
0145 t_is(out.hist(end).compcond, 0, 6, [t 'compcond']);
0146 t_is(lam.ineqnonlin, [0;sqrt(2)/2], 7, [t 'lam.ineqnonlin']);
0147 t_ok(isempty(lam.mu_l), [t 'lam.mu_l']);
0148 t_ok(isempty(lam.mu_u), [t 'lam.mu_u']);
0149 t_is(lam.lower, zeros(size(x)), 13, [t 'lam.lower']);
0150 t_is(lam.upper, zeros(size(x)), 13, [t 'lam.upper']);
0151 
0152 t = 'constrained 4-d nonlinear : ';
0153 %% Hock & Schittkowski test problem #71
0154 f_fcn = @(x)f7(x);
0155 gh_fcn = @(x)gh7(x);
0156 hess_fcn = @(x, lam, sigma)hess7(x, lam, sigma);
0157 x0 = [1; 5; 5; 1];
0158 xmin = ones(4, 1);
0159 xmax = 5 * xmin;
0160 % [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], xmin, xmax, gh_fcn, hess_fcn, struct('verbose', 2, 'comptol', 1e-9));
0161 [x, f, s, out, lam] = mips(f_fcn, x0, [], [], [], xmin, xmax, gh_fcn, hess_fcn);
0162 t_is(s, 1, 13, [t 'success']);
0163 t_is(x, [1; 4.7429994; 3.8211503; 1.3794082], 6, [t 'x']);
0164 t_is(f, 17.0140173, 6, [t 'f']);
0165 t_is(lam.eqnonlin, 0.1614686, 5, [t 'lam.eqnonlin']);
0166 t_is(lam.ineqnonlin, 0.55229366, 5, [t 'lam.ineqnonlin']);
0167 t_ok(isempty(lam.mu_l), [t 'lam.mu_l']);
0168 t_ok(isempty(lam.mu_u), [t 'lam.mu_u']);
0169 t_is(lam.lower, [1.08787121024; 0; 0; 0], 5, [t 'lam.lower']);
0170 t_is(lam.upper, zeros(size(x)), 7, [t 'lam.upper']);
0171 
0172 t_end;
0173 
0174 
0175 % %%-----  eg99 : linearly constrained fmincon example, mips can't solve  -----
0176 % function [f, df, d2f] = eg99(x)
0177 % f = -x(1)*x(2)*x(3);
0178 % df = -[ x(2)*x(3);
0179 %         x(1)*x(3);
0180 %         x(1)*x(2)   ];
0181 % d2f = -[    0       x(3)    x(2);
0182 %             x(3)    0       x(1);
0183 %             x(2)    x(1)    0   ];
0184 % end
0185 %
0186 % x0 = [10;10;10];
0187 % A = [1 2 2];
0188 % l = 0;
0189 % u = 72;
0190 % fmoptions = optimset('Display', 'testing');
0191 % fmoptions = optimset(fmoptions, 'Algorithm', 'interior-point');
0192 % [x, f, s, out, lam] = fmincon(f_fcn, x0, [-A; A], [-l; u], [], [], [], [], [], fmoptions);
0193 % t_is(x, [24; 12; 12], 13, t);
0194 % t_is(f, -3456, 13, t);
0195 
0196 
0197 %% unconstrained banana function
0198 %% from MATLAB Optimization Toolbox's bandem.m
0199 function [f, df, d2f] = f2(x)
0200     a = 100;
0201     f = a*(x(2)-x(1)^2)^2+(1-x(1))^2;
0202     df = [  4*a*(x(1)^3 - x(1)*x(2)) + 2*x(1)-2;
0203             2*a*(x(2) - x(1)^2)                     ];
0204     d2f = 4*a*[ 3*x(1)^2 - x(2) + 1/(2*a),  -x(1);
0205                 -x(1)                       1/2       ];
0206 
0207 
0208 %% unconstrained 3-d quadratic
0209 %% from http://www.akiti.ca/QuadProgEx0Constr.html
0210 function [f, df, d2f] = f3(x)
0211     H = [5 -2 -1; -2 4 3; -1 3 5];
0212     c = [2; -35; -47];
0213     f = 1/2 * x'*H*x + c'*x + 5;
0214     df = H*x + c;
0215     d2f = H;
0216 
0217 
0218 %% constrained 4-d QP
0219 %% from https://v8doc.sas.com/sashtml/iml/chap8/sect12.htm
0220 function [f, df, d2f] = f4(x)
0221     H = [   1003.1  4.3     6.3     5.9;
0222             4.3     2.2     2.1     3.9;
0223             6.3     2.1     3.5     4.8;
0224             5.9     3.9     4.8     10  ];
0225     c = zeros(4,1);
0226     f = 1/2 * x'*H*x + c'*x;
0227     df = H*x + c;
0228     d2f = H;
0229 
0230 
0231 %% constrained 2-d nonlinear
0232 %% from https://en.wikipedia.org/wiki/Nonlinear_programming#2-dimensional_example
0233 function [f, df, d2f] = f5(x)
0234     c = -[1; 1];
0235     f = c'*x;
0236     df = c;
0237     d2f = zeros(2,2);
0238 
0239 function [h, g, dh, dg] = gh5(x)
0240     h = [ -1 -1; 1 1] * x.^2 + [1; -2];
0241     dh = 2 * [-x(1) x(1); -x(2) x(2)];
0242     g = []; dg = [];
0243 
0244 function Lxx = hess5(x, lam, cost_mult)
0245     mu = lam.ineqnonlin;
0246     Lxx = 2*[-1 1]*mu*eye(2);
0247 
0248 
0249 %% constrained 3-d nonlinear
0250 %% from https://en.wikipedia.org/wiki/Nonlinear_programming#3-dimensional_example
0251 function [f, df, d2f] = f6(x)
0252     f = -x(1)*x(2) - x(2)*x(3);
0253     df = -[x(2); x(1)+x(3); x(2)];
0254     d2f = -[0 1 0; 1 0 1; 0 1 0];
0255 
0256 function [h, g, dh, dg] = gh6(x)
0257     h = [ 1 -1 1; 1 1 1] * x.^2 + [-2; -10];
0258     dh = 2 * [x(1) x(1); -x(2) x(2); x(3) x(3)];
0259     g = []; dg = [];
0260 
0261 function Lxx = hess6(x, lam, cost_mult)
0262     if nargin < 3, cost_mult = 1; end
0263     mu = lam.ineqnonlin;
0264     Lxx = cost_mult * [0 -1 0; -1 0 -1; 0 -1 0] + ...
0265             [2*[1 1]*mu 0 0; 0 2*[-1 1]*mu 0; 0 0 2*[1 1]*mu];
0266 
0267 
0268 %% constrained 4-d nonlinear
0269 %% Hock & Schittkowski test problem #71
0270 function [f, df, d2f] = f7(x)
0271     f = x(1)*x(4)*sum(x(1:3)) + x(3);
0272     df = [ x(1)*x(4) + x(4)*sum(x(1:3));
0273            x(1)*x(4);
0274            x(1)*x(4) + 1;
0275            x(1)*sum(x(1:3)) ];
0276     d2f = sparse([ 2*x(4)        x(4)   x(4)  2*x(1)+x(2)+x(3);
0277               x(4)               0      0     x(1);
0278               x(4)               0      0     x(1);
0279               2*x(1)+x(2)+x(3)  x(1)  x(1)    0
0280         ]);
0281 
0282 function [h, g, dh, dg] = gh7(x)
0283     g = sum(x.^2) - 40;
0284     h = -prod(x) + 25;
0285     dg = 2*x;
0286     dh = -prod(x)./x;
0287 
0288 function Lxx = hess7(x, lam, sigma)
0289     if nargin < 3, sigma = 1; end
0290     lambda = lam.eqnonlin;
0291     mu     = lam.ineqnonlin;
0292     [f, df, d2f] = f7(x);
0293     Lxx = sigma * d2f + lambda*2*speye(4) - ...
0294        mu*sparse([      0     x(3)*x(4) x(2)*x(4) x(2)*x(3);
0295                     x(3)*x(4)     0     x(1)*x(4) x(1)*x(3);
0296                     x(2)*x(4) x(1)*x(4)     0     x(1)*x(2);
0297                     x(2)*x(3) x(1)*x(3) x(1)*x(2)     0  ]);