Surrogate-Assisted Bounding-Box Approach for Optimization Problems with Approximated Objectives

Mickael Rivier 1 Pietro Marco Congedo 2
1 CARDAMOM - Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
2 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : In this work, we present a novel framework to perform multi-objective optimization, when considering an error on the objective functions. In many engineering optimization problems, the computation of the objective functions are affected by an error arising from the model employed for the computation. For example, in the case of uncertainty-based optimization the objective functions are statistics of a performance of interest which is uncertain due to the variability of the system input variables. These estimated objectives are affected by an error, which can be modeled with a confidence interval. The framework proposed here is general and aims at dealing with any error affecting a given objective function. The strategy is based on the extension of the Bounding-Box concept to the Pareto optima, where the error can be regarded with the abstraction of an interval (in one-dimensional problems) or a Bounding-Box (in multi-dimensional problems) around the estimated value. This allows the computation of an approximated Pareto front, whose accuracy is strongly dependent on the acceptable computational cost. This approach is then supplemented by the construction of an evolutive surrogate model on the objective functions, iteratively refined during the optimization process. This allows ultimately to further reduce the computation cost of the Pareto front with approximations of the objective functions at a negligible cost. The validation of the proposed method is accomplished first by proving the mathematical convergence toward the true continuous Pareto front under some hypothesis. Secondly, a numerical algorithm is proposed and its performances are assessed on several numerical test-cases. Results are systematically compared to a simple Double-loop approach and to the classical Bounding-Box method.
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Mickael Rivier, Pietro Marco Congedo. Surrogate-Assisted Bounding-Box Approach for Optimization Problems with Approximated Objectives. [Research Report] RR-9155, Inria. 2018, pp.1-35. ⟨hal-01713043v1⟩

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