|
5289
|
1 ## Copyright (C) 2000, 2001, 2004, 2005 Gabriele Pannocchia. |
|
|
2 ## |
|
|
3 ## This file is part of Octave. |
|
|
4 ## |
|
|
5 ## Octave is free software; you can redistribute it and/or modify it |
|
|
6 ## under the terms of the GNU General Public License as published by |
|
|
7 ## the Free Software Foundation; either version 2, or (at your option) |
|
|
8 ## any later version. |
|
|
9 ## |
|
|
10 ## Octave is distributed in the hope that it will be useful, but |
|
|
11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
|
|
12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
|
|
13 ## General Public License for more details. |
|
|
14 ## |
|
|
15 ## You should have received a copy of the GNU General Public License |
|
|
16 ## along with Octave; see the file COPYING. If not, write to the Free |
|
5307
|
17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
|
|
18 ## 02110-1301, USA. |
|
5289
|
19 |
|
|
20 ## -*- texinfo -*- |
|
|
21 ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, @var{Ain}, @var{A_ub}) |
|
|
22 ## Solve the quadratic program |
|
|
23 ## @ifinfo |
|
|
24 ## |
|
|
25 ## @example |
|
|
26 ## min 0.5 x'*H*x + x'*q |
|
|
27 ## x |
|
|
28 ## @end example |
|
|
29 ## |
|
|
30 ## @end ifinfo |
|
|
31 ## @iftex |
|
|
32 ## @tex |
|
|
33 ## @end tex |
|
|
34 ## @end iftex |
|
|
35 ## subject to |
|
|
36 ## @ifinfo |
|
|
37 ## |
|
|
38 ## @example |
|
|
39 ## A*x = b |
|
|
40 ## lb <= x <= ub |
|
|
41 ## A_lb <= Ain*x <= A_ub |
|
|
42 ## @end example |
|
|
43 ## @end ifinfo |
|
|
44 ## @iftex |
|
|
45 ## @tex |
|
|
46 ## @end tex |
|
|
47 ## @end iftex |
|
|
48 ## |
|
|
49 ## @noindent |
|
|
50 ## using a null-space active-set method. |
|
|
51 ## |
|
|
52 ## Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, |
|
|
53 ## @var{A_ub}) may be set to the empty matrix (@code{[]}) if not |
|
|
54 ## present. If the initial guess is feasible the algorithm is faster. |
|
|
55 ## |
|
|
56 ## The value @var{info} is a structure with the following fields: |
|
|
57 ## @table @code |
|
|
58 ## @item solveiter |
|
|
59 ## The number of iterations required to find the solution. |
|
|
60 ## @item info |
|
|
61 ## An integer indicating the status of the solution, as follows: |
|
|
62 ## @table @asis |
|
|
63 ## @item 0 |
|
|
64 ## The problem is feasible and convex. Global solution found. |
|
|
65 ## @item 1 |
|
|
66 ## The problem is not convex. Local solution found. |
|
|
67 ## @item 2 |
|
|
68 ## The problem is not convex and unbounded. |
|
|
69 ## @item 3 |
|
|
70 ## Maximum number of iterations reached. |
|
|
71 ## @item 6 |
|
|
72 ## The problem is infeasible. |
|
|
73 ## @end table |
|
|
74 ## @end table |
|
|
75 ## @end deftypefn |
|
|
76 |
|
|
77 function [x, obj, INFO, lambda] = qp (x0, H, q, A, b, lb, ub, A_lb, A_in, A_ub) |
|
|
78 |
|
|
79 if (nargin == 5 || nargin == 7 || nargin == 10) |
|
|
80 |
|
|
81 ## Checking the quadratic penalty |
|
|
82 n = issquare (H); |
|
|
83 if (n == 0) |
|
|
84 error ("qp: quadratic penalty matrix not square"); |
|
|
85 endif |
|
|
86 |
|
|
87 n1 = issymmetric (H); |
|
|
88 if (n1 == 0) |
|
|
89 ## warning ("qp: quadratic penalty matrix not symmetric"); |
|
|
90 H = (H + H')/2; |
|
|
91 endif |
|
|
92 |
|
|
93 ## Checking the initial guess (if empty it is resized to the |
|
|
94 ## right dimension and filled with 0) |
|
|
95 if (isempty (x0)) |
|
|
96 x0 = zeros (n, 1); |
|
|
97 elseif (length (x0) != n) |
|
|
98 error ("qp: the initial guess has incorrect length"); |
|
|
99 endif |
|
|
100 |
|
|
101 ## Linear penalty. |
|
|
102 if (length (q) != n) |
|
|
103 error ("qp: the linear term has incorrect length"); |
|
|
104 endif |
|
|
105 |
|
|
106 ## Equality constraint matrices |
|
|
107 if (isempty (A) || isempty(b)) |
|
|
108 n_eq = 0; |
|
|
109 A = zeros (n_eq, n); |
|
|
110 b = zeros (n_eq, 1); |
|
|
111 else |
|
|
112 [n_eq, n1] = size (A); |
|
|
113 if (n1 != n) |
|
|
114 error ("qp: equality constraint matrix has incorrect column dimension"); |
|
|
115 endif |
|
|
116 if (length (b) != n_eq) |
|
|
117 error ("qp: equality constraint matrix and vector have inconsistent dimension"); |
|
|
118 endif |
|
|
119 endif |
|
|
120 |
|
|
121 ## Bound constraints |
|
|
122 Ain = zeros (0, n); |
|
|
123 bin = zeros (0, 1); |
|
|
124 n_in = 0; |
|
|
125 if (nargin > 5) |
|
|
126 if (! isempty (lb)) |
|
|
127 if (length(lb) != n) |
|
|
128 error ("qp: lower bound has incorrect length"); |
|
|
129 else |
|
|
130 Ain = [Ain; eye(n)]; |
|
|
131 bin = [bin; lb]; |
|
|
132 endif |
|
|
133 endif |
|
|
134 |
|
|
135 if (! isempty (ub)) |
|
|
136 if (length (ub) != n) |
|
|
137 error ("qp: upper bound has incorrect length"); |
|
|
138 else |
|
|
139 Ain = [Ain; -eye(n)]; |
|
|
140 bin = [bin; -ub]; |
|
|
141 endif |
|
|
142 endif |
|
|
143 endif |
|
|
144 |
|
|
145 ## Inequality constraints |
|
|
146 if (nargin > 7) |
|
|
147 [dimA_in, n1] = size (A_in); |
|
|
148 if (n1 != n) |
|
|
149 error ("qp: inequality constraint matrix has incorrect column dimension"); |
|
|
150 else |
|
|
151 if (! isempty (A_lb)) |
|
|
152 if (length (A_lb) != dimA_in) |
|
|
153 error ("qp: inequality constraint matrix and lower bound vector inconsistent"); |
|
|
154 else |
|
|
155 Ain = [Ain; A_in]; |
|
|
156 bin = [bin; A_lb]; |
|
|
157 endif |
|
|
158 endif |
|
|
159 if (! isempty (A_ub)) |
|
|
160 if (length (A_ub) != dimA_in) |
|
|
161 error ("qp: inequality constraint matrix and upper bound vector inconsistent"); |
|
|
162 else |
|
|
163 Ain = [Ain; -A_in]; |
|
|
164 bin = [bin; -A_ub]; |
|
|
165 endif |
|
|
166 endif |
|
|
167 endif |
|
|
168 endif |
|
|
169 |
|
|
170 ## Now we should have the following QP: |
|
|
171 ## |
|
|
172 ## min_x 0.5*x'*H*x + x'*q |
|
|
173 ## s.t. A*x = b |
|
|
174 ## Ain*x >= bin |
|
|
175 |
|
|
176 ## Discard inequality constraints that have -Inf bounds since those |
|
|
177 ## will never be active. |
|
|
178 idx = isinf (bin) & bin < 0; |
|
|
179 bin(idx) = []; |
|
|
180 Ain(idx,:) = []; |
|
|
181 |
|
5310
|
182 n_in = length (bin); |
|
|
183 |
|
5289
|
184 ## Check if the initial guess is feasible. |
|
|
185 rtol = sqrt (eps); |
|
|
186 |
|
|
187 eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+norm (b))); |
|
|
188 in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+norm (bin)))); |
|
|
189 |
|
|
190 info = 0; |
|
|
191 if (eq_infeasible || in_infeasible) |
|
|
192 ## The initial guess is not feasible. |
|
|
193 ## First define xbar that is feasible with respect to the equality |
|
|
194 ## constraints. |
|
|
195 if (eq_infeasible) |
|
|
196 if (rank (A) < n_eq) |
|
|
197 error ("qp: equality constraint matrix must be full row rank") |
|
|
198 endif |
|
|
199 xbar = pinv (A) * b; |
|
|
200 else |
|
|
201 xbar = x0; |
|
|
202 endif |
|
|
203 |
|
|
204 ## Check if xbar is feasible with respect to the inequality |
|
|
205 ## constraints also. |
|
|
206 if (n_in > 0) |
|
|
207 res = Ain * xbar - bin; |
|
|
208 if (any (res < -rtol * (1 + norm (bin)))) |
|
|
209 ## xbar is not feasible with respect to the inequality |
|
|
210 ## constraints. Compute a step in the null space of the |
|
|
211 ## equality constraints, by solving a QP. If the slack is |
|
|
212 ## small, we have a feasible initial guess. Otherwise, the |
|
|
213 ## problem is infeasible. |
|
|
214 if (n_eq > 0) |
|
|
215 Z = null (A); |
|
|
216 if (isempty (Z)) |
|
|
217 ## The problem is infeasible because A is square and full |
|
|
218 ## rank, but xbar is not feasible. |
|
|
219 info = 6; |
|
|
220 endif |
|
|
221 endif |
|
|
222 |
|
|
223 if (info != 6) |
|
|
224 ## Solve an LP with additional slack variables to find |
|
|
225 ## a feasible starting point. |
|
|
226 gamma = eye (n_in); |
|
|
227 if (n_eq > 0) |
|
|
228 Atmp = [Ain*Z, gamma]; |
|
|
229 btmp = -res; |
|
|
230 else |
|
|
231 Atmp = [Ain, gamma]; |
|
|
232 btmp = bin; |
|
|
233 endif |
|
|
234 ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; |
|
|
235 lb = [-Inf*ones(n-n_eq,1); zeros(n_in,1)]; |
|
|
236 ub = []; |
|
|
237 ctype = repmat ("L", n_in, 1); |
|
|
238 [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); |
|
|
239 |
|
|
240 if ((status == 180 || status == 181 || status == 151) |
|
|
241 && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) |
|
|
242 ## We found a feasible starting point |
|
|
243 if (n_eq > 0) |
|
5306
|
244 x0 = xbar + Z*P(1:n-n_eq); |
|
5289
|
245 else |
|
|
246 x0 = P(1:n); |
|
|
247 endif |
|
|
248 else |
|
|
249 ## The problem is infeasible |
|
|
250 info = 6; |
|
|
251 endif |
|
|
252 endif |
|
|
253 else |
|
|
254 ## xbar is feasible. We use it a starting point. |
|
|
255 x0 = xbar; |
|
|
256 endif |
|
|
257 else |
|
|
258 ## xbar is feasible. We use it a starting point. |
|
|
259 x0 = xbar; |
|
|
260 endif |
|
|
261 endif |
|
|
262 |
|
|
263 if (info == 0) |
|
|
264 ## The initial (or computed) guess is feasible. |
|
|
265 ## We call the solver. |
|
|
266 maxit = 100; |
|
|
267 [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); |
|
|
268 else |
|
|
269 iter = 0; |
|
|
270 x = x0; |
|
|
271 lambda = []; |
|
|
272 endif |
|
|
273 obj = 0.5 * x' * H * x + q' * x; |
|
|
274 INFO.solveiter = iter; |
|
|
275 INFO.info = info; |
|
|
276 |
|
|
277 else |
|
|
278 usage ("[x, obj, info, lambda] = qp (x0, H, q, A, b, lb, ub, A_lb, Ain, A_ub)"); |
|
|
279 endif |
|
|
280 |
|
|
281 endfunction |
