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...ies of DFNsimple, its formal definition does not include a stopping condition. We note that this is in accordance with most of the papers concerning convergence of derivative-free methods; see, e.g., =-=[6, 10, 12, 13, 16, 44]-=-, among others. We refer the reader to section 4 for a practical stopping condition. In the following results we analyze the global convergence properties of Algorithm DFNsimple. In particular, in the...

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...cally Lipschitz function on Rn. If g ∈ ∂of(x), such that 〈g, x− x〉 > 0 then, f(x) ≤ f(x). Proof. See Aussel [1], Theorem 2.1. Proposition 2.5.3 Let f : Rn −→ R be a convex function. Then ∂of(x) coincides with the subdifferential at x in the sense of convex analysis, and fo(x, d) coincides with the directional derivative f ′(x, d) for each d. Proof. See Clarke [8], Proposition 2.2.7 2.6 Descent direction We are now able to introduce the definition of Pareto-Clarke critical point for locally Lipschitz functions on Rn, which will play a key role in our paper. Definition 2.6.1 (Custodio et al. [9], Definition 4.6) Let F = (F1, ..., Fm) T : Rn −→ Rm be locally Lipschitz on Rn. We say that x∗ ∈ Rn is a Pareto-Clarke critical point of F if, for all directions d ∈ Rn, there exists i0 = i0(d) ∈ {1, ...,m} such that F o io (x∗, d) ≥ 0. Definition 2.6.1 says essentially that there is no direction in Rn that is descent for all the objective functions (see, for instance, (Custodio et al. [9]). If a point is a Pareto minimizer (local or global), then it is necessarily a Pareto-Clarke critical point . Remark 2.6.1 Follows from the previous definition that, if a point x is not Pareto-Clarke criti...

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...ring feasibility with respect to the other constraints. Thus the detailed method requires the use of a derivative-free method for biobjective optimization problems such as the ones proposed in [9] or =-=[14]-=-. See [17] for interesting contextual examples in aerospace design. The paper is divided as follows: Section 2 presents the simple and detailed methods to analyze the sensitivity to constraints. Secti...

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...on function is to be used as a parameter in any DFO algorithm. Multi-objective optimization search techniques tend to iteratively improve their current approximation of the Pareto front. As stated in =-=[5]-=-, the quality of a Pareto front approximation can be measured in terms of two criteria. First, it is desired that points in a Pareto front approximation are as close as possible to the optimal Pareto ...

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... for evaluating the outcomes of multi-objective algorithms. In this paper, we consider five metrics to use in comparing the 14 Trong-Viet Ho et al. performances of our local search approaches: Purity =-=[20]-=-, Spread Metric [20], Hypervolume [21], Hypervolume-Star, and Binary -indicator [22]. These five metrics are very useful for assessing the quality of an approximation set or a Pareto front. In this p...

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...ameter. Polling consists of evaluating the function at points obtained by adding to the poll center a multiple (defined by the step size) of a set of directions. These directions 13 typically form a positive spanning set (in other words, a set of directions that spans Rn with nonnegative coefficients). In direct multisearch, constraints are handled using an extreme barrier function FΩ(x) = { F (x) if x ∈ Ω, (+∞, . . . ,+∞)> otherwise, which prevents adding infeasible points to the current list of nondominated points. Direct multisearch is described below following the algorithmic framework in [12]. A number of details are omitted and the reader is referred to [12] for a complete description. In particular, we omitted the two known globalization strategies (generation of points in integer lattices and imposition of sufficient decrease), under which the algorithm is known to be globally convergent, in the sense of yielding some form of convergence independently of the starting point or starting list. In practice, these globalization strategies amount to very minor modifications to the algorithm. Algorithm A.1 (Direct Multisearch for MOO) Initialization Choose x0 ∈ Ω with fi(x0) < +∞,∀i ∈...