Using simplex gradients of nonsmooth functions in direct search methods
Department of Mathematics, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugal
Department of Computational and Applied Mathematics, Rice University, MS 134, 6100 South Main Street, Houston, TX 77005-1892, USA
CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal
Email: alcustodio{at}fct.unl.pt
Email: dennis{at}rice.edu
Corresponding author. Email: lnv{at}mat.uc.pt
Received on 27 October 2006. Accepted for publication 2 May 2007.
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Abstract |
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It has been shown recently that the efficiency of direct search methods that use opportunistic polling in positive spanning directions can be improved significantly by reordering the poll directions according to descent indicators built from simplex gradients. The purpose of this paper is two-fold. First, we analyse the properties of simplex gradients of nonsmooth functions in the context of direct search methods like the generalized pattern search and the mesh adaptive direct search, for which there exists a convergence analysis in the nonsmooth setting. Our analysis does not require continuous differentiability and can be seen as an extension of the accuracy properties of simplex gradients known for smooth functions. Secondly, we test the use of simplex gradients when pattern search is applied to nonsmooth functions, confirming the merit of the poll ordering strategy for such problems.
Key Words: derivative-free optimization; simplex gradients; poisedness; nonsmooth analysis; generalized pattern search methods; mesh adaptive direct search
Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday.