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Journal Articles Optimization Year : 2015

A second-order globally convergent direct-search method and its worst-case complexity

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Abstract

Direct-search algorithms form one of the main classes of algorithms for smooth unconstrained derivative-free optimization, due to their simplicity and their well-established convergence results. They proceed by iteratively looking for improvement along some vectors or directions. In the presence of smoothness, first-order global convergence comes from the ability of the vectors to approximate the steepest descent direction, which can be quantified by a first-order criticality (cosine) measure. The use of a set of vectors with a positive cosine measure together with the imposition of a sufficient decrease condition to accept new iterates leads to a convergence result as well as a worst-case complexity bound. In this paper, we present a second-order study of a general class of direct-search methods. We start by proving a weak second-order convergence result related to a criticality measure defined along the directions used throughout the iterations. Extensions of this result to obtain a true second-order optimality one are discussed, one possibility being a method using approximate Hessian eigenvectors as directions (which is proved to be truly second-order globally convergent). Numerically guaranteeing such a convergence can be rather expensive to ensure, as it is indicated by the worst-case complexity analysis provided in this paper, but turns out to be appropriate for some pathological examples.
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Dates and versions

hal-01515985 , version 1 (28-04-2017)

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Serge Gratton, Clément Royer, Luis Vicente. A second-order globally convergent direct-search method and its worst-case complexity. Optimization, 2015, 65 (6), pp.1105-1128. ⟨10.1080/02331934.2015.1124271⟩. ⟨hal-01515985⟩
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