PSwarm: a hybrid solver for linearly constrained global derivative-free optimization

Abstract

PSwarm was developed originally for the global optimization of functions without derivatives and where the variables are within upper and lower bounds. The underlying algorithm used is a pattern search method, or more specifically, a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the (optional) search step of coordinate search, the algorithm incorporates a particle swarm scheme for dissemination of points in the feasible region, equipping the overall method with the capability of finding a global minimizer. Our extensive numerical experiments showed that the resulting algorithm is highly competitive with other global optimization methods based only on function values. PSwarm is extended in this paper to handle general linear constraints. The poll step now incorporates positive generators for the tangent cone of the approximated active constraints, including a provision for the degenerate case. The search step has also been adapted accordingly. In particular, the initial population for particle swarm used in the search step is computed by first inscribing an ellipsoid of maximum volume to the feasible set. We have again compared PSwarm with other solvers (including some designed for global optimization) and the results confirm its competitiveness in terms of efficiency and robustness.

Keywords:

• direct search
• linear constraints
• pattern search
• particle swarm
• derivative-free optimization
• global optimization

AMS Subject Classification :

• 90C26
• 90C30
• 90C56

Acknowledgements

Support for the first author was provided by Algoritmi Research Center and by FCT under grants POCI/MAT/59442/2004 and POCI/MAT/58957/2004. Support for the second author was provided by FCT under grants POCI/MAT/59442/2004 and PTDC/MAT/64838/2006.

Notes

The code in 35 is originally implemented in MATLAB. We rewrote it in C using the BLAS 4 and LAPACK 2 linear algebra packages, for our own usage in the C version of PSwarm.