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...s bound is achieved when using a quadratic function of the step size in the sufficient decrease condition, i.e., a function like ρ(t) = ct 2 , c > 0 — which corroborates previous numerical experience =-=[13]-=- where different functions of the form ρ(t) = ct p (with 2 = p > 1) where tested but leading to a worse performance. Once we allow new iterates to be accepted based simply on simple decrease (and the...

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...ar, given the type of random sampling used in ES, our work is inspired by direct-search methods for nonsmooth functions, where one must use a set of directions asymptotically dense in the unit sphere =-=[1, 20]-=-. Since the random sampling of ES will not likely provide us any integer lattice underlying structure for the iterates (like in MADS [1]), we will use a sufficient decrease condition (as opposed to ju...

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...HOMADS method. A different way to handle the constraints within MADS-type algorithms is proposed in [6], where the authors combine a filter-based strategy [17] with a progressive barrier approach. In =-=[46]-=-, the most general result for direct search methods of this type is given (for functions directionally Lipschitz) and, moreover, it is shown that integer lattices can be replaced by sufficient decreas...

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...‖} for all k ∈ K sufficiently large such that xk + σkak ∈ Ω, where, in the particular case of taking the weighted mean as the object of evaluation, one has ak = ∑µ i=1 ω i kd i k. 2.1 Asymptotic results when derivatives are unknown In this section we treat constraints as a pure black box in the sense that no information is assumed known about the constrained set Ω, rather than a yes/no answer to the question whether a given point is feasible. The following theorem is in the vein of those first established in [5] for simple decrease and Lipschitz continuous functions (and later generalized in [44] for sufficient decrease and directionally Lipschitz functions). 5 Theorem 2.1 Let x∗ ∈ Ω be the limit point of a convergent subsequence of unsuccessful iterates {xk}K for which limk∈K σk = 0. Assume that f is Lipschitz continuous near x∗ with constant ν > 0 and that THΩ (x) 6= ∅. Let ak = ∑µ i=1 ω i kd i k. Assume that the directions d i k’s and the weights ω i k’s are such that (i) σk‖ak‖ tends to zero when σk does, and (ii) ρ(σk)/(σk‖ak‖) also tends to zero. If d ∈ THΩ (x∗) is a refining direction associated with {ak/‖ak‖}K , then f◦(x∗; d) ≥ 0. If the set of refining directions associate...

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...ptimization where points outside bounds cannot be evaluated due to the inherent limitations of the underlying simulator. MADS relies on two basic ingredients to deliver certain convergence properties =-=[1,6,27]-=- for nonsmooth functions in the Clarke sense [10] and in the Rockafellar sense [23]. The first one is the restriction of the visited points to a discrete set (mesh). This restriction guarantees the ex...