arXiv subjects

A. L. Custódio

Publications and source records attributed to A. L. Custódio.

3 recordsLinked to original sources

arXiv2021

Parallel Strategies for Direct Multisearch

Direct Multisearch (DMS) is a Derivative-free Optimization class of algorithms suited for computing approximations to the complete Pareto front of a given Multiobjective Optimization problem. It has a well-supported convergence analysis and simple implementations present a good numerical performance, both in academic test sets and in real applications. Recently, this numerical performance was improved with the definition of a search step based on the minimization of quadratic polynomial models, corresponding to the algorithm BoostDMS. In this work, we propose and numerically evaluate the performance of parallelization strategies for this solver, applied to the search and to the poll steps. The final parallelized version not only considerably decreases the computational time required for solving a Multiobjective Optimization problem, but also increases the quality of the computed approximation to the Pareto front. Extensive numerical results will be reported in an academic test set and in a chemical engineering application.

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arXiv2021

Using first-order information in Direct Multisearch for multiobjective optimization

Derivatives are an important tool for single-objective optimization. In fact, it is commonly accepted that derivative-based methods present a better performance than derivative-free optimization approaches. In this work, we will show that the same does not apply to multiobjective derivative-based optimization, when the goal is to compute an approximation to the complete Pareto front of a given problem. The competitiveness of Direct MultiSearch (DMS), a robust and efficient derivative-free optimization algorithm, will be stated for derivative-based multiobjective optimization problems. We will then assess the potential enrichment of adding first-order information to the DMS framework. Derivatives will be used to prune the positive spanning sets considered at the poll step of the algorithm, highlighting the role that ascent directions, that conform to the geometry of the nearby feasible region, can have. Both variants of DMS show to be competitive against a state-of-art derivative-based algorithm. Moreover, for reasonable small budgets of function evaluations, the new variant is not only competitive with the derivative-based solver but also with the original implementation of DMS.

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arXiv2020

Worst-case Complexity Bounds of Directional Direct-search Methods for Multiobjective Optimization

Direct Multisearch 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 to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.

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