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On the optimal order of worst case complexity of direct search

M. Dodangeh(dodangeh***at***mat.uc.pt)
L. N. Vicente(lnv***at***mat.uc.pt)
Z. Zhang(zaikun.zhang***at***irit.fr)

Abstract: The worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. Assuming that the objective function is smooth, it is now known that such methods require at most O(n^2 epsilon^{-2}) function evaluations to compute a gradient of norm below epsilon in (0,1), where n is the dimension of the problem. Such a maximal effort is reduced to O(n^2 epsilon^{-1}) if the function is convex. The factor n^2 has been derived using the positive spanning set formed by the coordinate vectors and their negatives at all iterations. In this paper, we prove that such a factor of n^2 is optimal in these worst case complexity bounds, in the sense that no other positive spanning set will yield a better order of n. The proof is based on an observation that reveals the connection between cosine measure in positive spanning and sphere covering.

Keywords: Direct search, worst case complexity, optimal order, sphere covering, positive spanning set, cosine measure.

Category 1: Convex and Nonsmooth Optimization

Category 2: Nonlinear Optimization

Citation: Preprint 14-38, Dept. Mathematics, Univ. Coimbra.

Download: [PDF]

Entry Submitted: 11/19/2014
Entry Accepted: 11/21/2014
Entry Last Modified: 11/19/2014

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