On the optimal order of worst case complexity of direct search

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. For smooth unconstrained optimization, it is now known that such methods require at most O(n2 ϵ−2 ) function evaluations to compute a gradient of norm below ϵ ∈ (0, 1), where n is the dimension of the problem. Such a maximal effort is reduced to O(n2 ϵ−1 ) if the function is convex. The factor n2 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 n2 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.

• Type: Journal
• Year: 2016
• Where: OL
• Authors: M. Dodangeh, Luís N. Vicente, Z. Zhang
• Keywords: OL 2016