Name
qp_solve — linear quadratic programming solver builtin
Calling Sequence
[x [,iact [,iter [,f]]]]=qp_solve(Q,p1,C1,b,me)
Parameters
- Q
real positive definite symmetric matrix (dimension
n x n).- p
real (column) vector (dimension
n)- C
real matrix (dimension
(me + md) x n). This matrix may be dense or sparse.- b
RHS column vector (dimension
m=(me + md))- me
number of equality constraints (i.e.
x'*C(:,1:me) = b(1:me)')- x
optimal solution found.
- iact
vector, indicator of active constraints. The first non zero entries give the index of the active constraints
- iter
2x1 vector, first component gives the number of "main" iterations, the second one says how many constraints were deleted after they became active.
Description
This function requires Q to be symmetric positive
definite. If this hypothesis is not satisfied, one may use the contributed
quapro toolbox.
Examples
// Find x in R^6 such that:
// x'*C1 = b1 (3 equality constraints i.e me=3)
C1= [ 1,-1, 2;
-1, 0, 5;
1,-3, 3;
0,-4, 0;
3, 5, 1;
1, 6, 0];
b1=[1;2;3];
// x'*C2 >= b2 (2 inequality constraints)
C2= [ 0 ,1;
-1, 0;
0,-2;
-1,-1;
-2,-1;
1, 0];
b2=[ 1;-2.5];
// and minimize 0.5*x'*Q*x - p'*x with
p=[-1;-2;-3;-4;-5;-6]; Q=eye(6,6);
me=3;
[x,iact,iter,f]=qp_solve(Q,p,[C1 C2],[b1;b2],me)
// Only linear constraints (1 to 4) are active
See Also
optim, qld, qpsolveThe contributed toolbox "quapro" may also be of interest, in
particular for singular Q.
Memory requirements
Let r be
r=min(m,n)
Then the memory required by qp_solve during the computations is
2*n+r*(r+5)/2 + 2*m +1
Authors
- S. Steer
INRIA (Scilab interface)
- Berwin A. Turlach
School of Mathematics and Statistics (M019), The University of Western Australia, Crawley, AUSTRALIA (solver code)
References
Goldfarb, D. and Idnani, A. (1982). "Dual and Primal-Dual Methods for Solving Strictly Convex Quadratic Programs", in J.P. Hennart (ed.), Numerical Analysis, Proceedings, Cocoyoc, Mexico 1981, Vol. 909 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 226-239.
Goldfarb, D. and Idnani, A. (1983). "A numerically stable dual method for solving strictly convex quadratic programs", Mathematical Programming 27: 1-33.
QuadProg (Quadratic Programming Routines), Berwin A Turlach,http://www.maths.uwa.edu.au/~berwin/software/quadprog.html