| Description |
QUADPROG2 - Convex Quadratic Programming Solver
Featuring the SOLVOPT freeware optimizer
New for version 1.1:
* Significant speed improvement
* Geometric Preconditioning
* Improved Error Checking
USAGE:
[x,v] = quadprog2(H,f,A,b)
[x,v] = quadprog2(H,f,A,b,guess)
[x,v,opt] = ...
Minimizes the function v = 0.5*x'*H*x + f*x
subject to the constraint A*x <= b.
Initial guess is optional.
("opt" returns SOLVOPT data for advanced use. Details are available in
the SOLVOPT documentation at the website identified below.)
Notes:
(1) For a problem with 100 variables and 300 constraints, you will
often get a result in under 5 seconds. However, sometimes
the optimizer has to work longer (see below) for difficult
optimizations. Alerts are provided. (Note: The calculation
time is more sensitive to the number of variables than it is
to the number of contraints.)
(2) Geometric preconditioning is undertaken for 10 or more
dimensions to greatly reduce calculation time. (With fewer
than 10 dimensions, there is negligible benefit, so the
preconditioning calculations are omitted.)
(3) Geometric preconditioning can impair the convergence of some
difficult optimizations. When this occurs, the optimization
is attempted again without the preconditioning.
(4) x and guess are column vectors. f is a row vector.
They will be converted if necessary.
(5) This m-file incorporates the SOLVOPT version 1.1 freeware
optimizer, which has been wholly reproduced, except for
a few slight modifications for convenience in parameter passing.
(6) SolvOpt is a general nonlinear local optimizer,
written by Alexei Kuntsevich & Franz Kappel, and
is available (as of this writing) as freeware from:
http://www.uni-graz.at/imawww/kuntsevich/solvopt/
(7) This Matlab function requires a convex QP problem
with a positive-definite symmetric matrix H.
This is a somewhat trivial application of
a general solver like SOLVOPT, but the use of precomputed
gradient vectors herein makes the solution fast enough
to warrant use.
(8) Any local solution of a convex QP is also a global solution.
Hence, your results will be globally optimal.
(9) Relative precision in the objective function is set to 1e-6.
(10) Absolute precision in constraint violation is 1e-6 or better.
(11) This program does not require the Optimization Toolbox
(12) ver 1.0: intial writing, Michael Kleder, June 2005
(13) ver 1.1: geometric preconditioning, Michael Kleder, July 2005
EXAMPLE:
% Convex QP with 100 variables and 300 constraints:
n = 100;
c = 300;
H = rand(n);
H=H*H';
f=rand(1,n);
A=rand(c,n)*2-1;
b=ones(c,1);
tic
x = quadprog2(H,f,A,b)
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