Mercurial > octave-nkf
view scripts/optimization/qp.m @ 17338:1c89599167a6
maint: End m-files with 1 blank line.
Simplifies automated grammarchecking script.
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scripts/ui/private/message_dialog.m: End m-files with 1 blank line.
author | Rik <rik@octave.org> |
---|---|
date | Wed, 28 Aug 2013 08:33:02 -0700 |
parents | 5d3a684236b0 |
children | d63878346099 |
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## Copyright (C) 2000-2012 Gabriele Pannocchia. ## ## This file is part of Octave. ## ## Octave 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. ## ## Octave 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 Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}) ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}) ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@var{x0}, @var{H}, @var{q}, @var{A}, @var{b}) ## @deftypefnx {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}) ## @deftypefnx {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{A_in}, @var{A_ub}) ## @deftypefnx {Function File} {[@var{x}, @var{obj}, @var{info}, @var{lambda}] =} qp (@dots{}, @var{options}) ## Solve the quadratic program ## @tex ## $$ ## \min_x {1 \over 2} x^T H x + x^T q ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## min 0.5 x'*H*x + x'*q ## x ## @end group ## @end example ## ## @end ifnottex ## subject to ## @tex ## $$ ## Ax = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} \leq A_{ub} ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## A*x = b ## lb <= x <= ub ## A_lb <= A_in*x <= A_ub ## @end group ## @end example ## ## @end ifnottex ## @noindent ## using a null-space active-set method. ## ## Any bound (@var{A}, @var{b}, @var{lb}, @var{ub}, @var{A_lb}, ## @var{A_ub}) may be set to the empty matrix (@code{[]}) if not ## present. If the initial guess is feasible the algorithm is faster. ## ## @table @var ## @item options ## An optional structure containing the following ## parameter(s) used to define the behavior of the solver. Missing elements ## in the structure take on default values, so you only need to set the ## elements that you wish to change from the default. ## ## @table @code ## @item MaxIter (default: 200) ## Maximum number of iterations. ## @end table ## @end table ## ## @table @var ## @item info ## Structure containing run-time information about the algorithm. The ## following fields are defined: ## ## @table @code ## @item solveiter ## The number of iterations required to find the solution. ## ## @item info ## An integer indicating the status of the solution. ## ## @table @asis ## @item 0 ## The problem is feasible and convex. Global solution found. ## ## @item 1 ## The problem is not convex. Local solution found. ## ## @item 2 ## The problem is not convex and unbounded. ## ## @item 3 ## Maximum number of iterations reached. ## ## @item 6 ## The problem is infeasible. ## @end table ## @end table ## @end table ## @end deftypefn ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. ## PKG_ADD: [~] = __all_opts__ ("qp"); function [x, obj, INFO, lambda] = qp (x0, H, varargin) nargs = nargin; if (nargin == 1 && ischar (x0) && strcmp (x0, 'defaults')) x = optimset ("MaxIter", 200); return; endif if (nargs > 2 && isstruct (varargin{end})) options = varargin{end}; nargs--; else options = struct (); endif if (nargs >= 3) q = varargin{1}; else q = []; endif if (nargs >= 5) A = varargin{2}; b = varargin{3}; else A = []; b = []; endif if (nargs >= 7) lb = varargin{4}; ub = varargin{5}; else lb = []; ub = []; endif if (nargs == 10) A_lb = varargin{6}; A_in = varargin{7}; A_ub = varargin{8}; else A_lb = []; A_in = []; A_ub = []; endif if (nargs == 2 || nargs == 3 || nargs == 5 || nargs == 7 || nargs == 10) maxit = optimget (options, "MaxIter", 200); ## Checking the quadratic penalty if (! issquare (H)) error ("qp: quadratic penalty matrix not square"); elseif (! ishermitian (H)) ## warning ("qp: quadratic penalty matrix not hermitian"); H = (H + H')/2; endif n = rows (H); ## Checking the initial guess (if empty it is resized to the ## right dimension and filled with 0) if (isempty (x0)) x0 = zeros (n, 1); elseif (numel (x0) != n) error ("qp: the initial guess has incorrect length"); endif ## Linear penalty. if (isempty (q)) q = zeros (n, 1); elseif (numel (q) != n) error ("qp: the linear term has incorrect length"); endif ## Equality constraint matrices if (isempty (A) || isempty (b)) A = zeros (0, n); b = zeros (0, 1); n_eq = 0; else [n_eq, n1] = size (A); if (n1 != n) error ("qp: equality constraint matrix has incorrect column dimension"); endif if (numel (b) != n_eq) error ("qp: equality constraint matrix and vector have inconsistent dimension"); endif endif ## Bound constraints Ain = zeros (0, n); bin = zeros (0, 1); n_in = 0; if (nargs > 5) if (! isempty (lb)) if (numel (lb) != n) error ("qp: lower bound has incorrect length"); elseif (isempty (ub)) Ain = [Ain; eye(n)]; bin = [bin; lb]; endif endif if (! isempty (ub)) if (numel (ub) != n) error ("qp: upper bound has incorrect length"); elseif (isempty (lb)) Ain = [Ain; -eye(n)]; bin = [bin; -ub]; endif endif if (! isempty (lb) && ! isempty (ub)) rtol = sqrt (eps); for i = 1:n if (abs (lb (i) - ub(i)) < rtol*(1 + max (abs (lb(i) + ub(i))))) ## These are actually an equality constraint tmprow = zeros (1,n); tmprow(i) = 1; A = [A;tmprow]; b = [b; 0.5*(lb(i) + ub(i))]; n_eq = n_eq + 1; else tmprow = zeros (1,n); tmprow(i) = 1; Ain = [Ain; tmprow; -tmprow]; bin = [bin; lb(i); -ub(i)]; n_in = n_in + 2; endif endfor endif endif ## Inequality constraints if (nargs > 7) [dimA_in, n1] = size (A_in); if (n1 != n) error ("qp: inequality constraint matrix has incorrect column dimension"); else if (! isempty (A_lb)) if (numel (A_lb) != dimA_in) error ("qp: inequality constraint matrix and lower bound vector inconsistent"); elseif (isempty (A_ub)) Ain = [Ain; A_in]; bin = [bin; A_lb]; endif endif if (! isempty (A_ub)) if (numel (A_ub) != dimA_in) error ("qp: inequality constraint matrix and upper bound vector inconsistent"); elseif (isempty (A_lb)) Ain = [Ain; -A_in]; bin = [bin; -A_ub]; endif endif if (! isempty (A_lb) && ! isempty (A_ub)) rtol = sqrt (eps); for i = 1:dimA_in if (abs (A_lb(i) - A_ub(i)) < rtol*(1 + max (abs (A_lb(i) + A_ub(i))))) ## These are actually an equality constraint tmprow = A_in(i,:); A = [A;tmprow]; b = [b; 0.5*(A_lb(i) + A_ub(i))]; n_eq = n_eq + 1; else tmprow = A_in(i,:); Ain = [Ain; tmprow; -tmprow]; bin = [bin; A_lb(i); -A_ub(i)]; n_in = n_in + 2; endif endfor endif endif endif ## Now we should have the following QP: ## ## min_x 0.5*x'*H*x + x'*q ## s.t. A*x = b ## Ain*x >= bin ## Discard inequality constraints that have -Inf bounds since those ## will never be active. idx = isinf (bin) & bin < 0; bin(idx) = []; Ain(idx,:) = []; n_in = numel (bin); ## Check if the initial guess is feasible. if (isa (x0, "single") || isa (H, "single") || isa (q, "single") || isa (A, "single") || isa (b, "single")) rtol = sqrt (eps ("single")); else rtol = sqrt (eps); endif eq_infeasible = (n_eq > 0 && norm (A*x0-b) > rtol*(1+abs (b))); in_infeasible = (n_in > 0 && any (Ain*x0-bin < -rtol*(1+abs (bin)))); info = 0; if (eq_infeasible || in_infeasible) ## The initial guess is not feasible. ## First define xbar that is feasible with respect to the equality ## constraints. if (eq_infeasible) if (rank (A) < n_eq) error ("qp: equality constraint matrix must be full row rank"); endif xbar = pinv (A) * b; else xbar = x0; endif ## Check if xbar is feasible with respect to the inequality ## constraints also. if (n_in > 0) res = Ain * xbar - bin; if (any (res < -rtol * (1 + abs (bin)))) ## xbar is not feasible with respect to the inequality ## constraints. Compute a step in the null space of the ## equality constraints, by solving a QP. If the slack is ## small, we have a feasible initial guess. Otherwise, the ## problem is infeasible. if (n_eq > 0) Z = null (A); if (isempty (Z)) ## The problem is infeasible because A is square and full ## rank, but xbar is not feasible. info = 6; endif endif if (info != 6) ## Solve an LP with additional slack variables to find ## a feasible starting point. gamma = eye (n_in); if (n_eq > 0) Atmp = [Ain*Z, gamma]; btmp = -res; else Atmp = [Ain, gamma]; btmp = bin; endif ctmp = [zeros(n-n_eq, 1); ones(n_in, 1)]; lb = [-Inf(n-n_eq,1); zeros(n_in,1)]; ub = []; ctype = repmat ("L", n_in, 1); [P, dummy, status] = glpk (ctmp, Atmp, btmp, lb, ub, ctype); if ((status == 180 || status == 181 || status == 151) && all (abs (P(n-n_eq+1:end)) < rtol * (1 + norm (btmp)))) ## We found a feasible starting point if (n_eq > 0) x0 = xbar + Z*P(1:n-n_eq); else x0 = P(1:n); endif else ## The problem is infeasible info = 6; endif endif else ## xbar is feasible. We use it a starting point. x0 = xbar; endif else ## xbar is feasible. We use it a starting point. x0 = xbar; endif endif if (info == 0) ## The initial (or computed) guess is feasible. ## We call the solver. [x, lambda, info, iter] = __qp__ (x0, H, q, A, b, Ain, bin, maxit); else iter = 0; x = x0; lambda = []; endif obj = 0.5 * x' * H * x + q' * x; INFO.solveiter = iter; INFO.info = info; else print_usage (); endif endfunction