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Worst-case complexity bounds of directional direct-search methods for multiobjective derivative-free optimization

A. L. Custódio (alcustodio***at***fct.unl.pt)
Y. Diouane (youssef.diouane***at***isae.fr)
R. Garmanjani (r.garmanjani***at***fct.unl.pt)
E. Riccietti (elisa.riccietti***at***enseeiht.fr)

Abstract: Direct Multisearch (DMS) is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that the DMS algorithm takes at most $\mathcal{O}(|L(\epsilon)|\epsilon^{-2m})$ iterations for driving a criticality measure below $\epsilon>0$ (here $m$ represents the number of components of the objective function and $|L(\epsilon)|$ the cardinality of the approximation to the Pareto front). We then focus on a particular instance of DMS, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound of $\mathcal{O}(\epsilon^{-2})$ for driving the same criticality measure below $\epsilon>0$.

Keywords: Multiobjective unconstrained optimization; Derivative-free optimization methods; Directional direct-search; Worst-case complexity; Nonconvex smooth optimization

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Unconstrained Optimization )


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Entry Submitted: 09/19/2019
Entry Accepted: 09/19/2019
Entry Last Modified: 08/04/2020

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