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Low-cost methods for constrained multi-objective optimization under uncertainty

Mickael Rivier 1
1 PLATON - Uncertainty Quantification in Scientific Computing and Engineering
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : Optimization Under Uncertainty is a fundamental axis of research in many companies nowadays, due to both the evergrowing computational power available and the need for efficiency, reliability and cost optimality. Among others, some challenges are the formulation of a suitable metric for the optimization problem of interest and the search for an ideal trade-off between computational cost and accuracy in the case of problems involving complex and expensive numerical solvers. The targeted class of problem here is constrained multiobjective optimization where fitness functions are uncertainty-driven metrics, such as statistical moments or quantiles. This thesis relies on two main ideas. First, the accuracy for approximating the objectives and constraints at a given design should be driven by the probability for this design of being non-dominated. This choice permits to reduce the effort for evaluating designs which are unlikely to be optimal. To this extent, we introduce the concept of probabilistic dominance for constrained multi-objective optimization under uncertainty through the computation of the so-called Pareto-Optimal Probability (POP). Secondly, these approximated evaluations and their associated errors can be used to construct a predictive representation of the objectives and constraints over the whole design space to accelerate the optimization process. Overall, the approximation of different uncertainty-based metrics with tunable accuracy and the use of a Surrogate-Assisting strategy are the main ingredients of the proposed algorithm, called SAMATA. This approach is flexible in terms of metrics formulations and reveals very parsimonious. Note that this algorithm is applicable with generic optimization methods. This thesis then explores the influence of the error distribution on the algorithm performance. We first make a simplifying and conservative assumption by considering a Uniform distribution of the error. In this case, the proposed formulation yields a Bounding-Box approach, where the estimation error can be regarded with the abstraction of an interval (in one-dimensional problems) or a product of intervals (in multi-dimensional problems) around the estimated value, naturally allowing for the computation of an approximated Pareto front. This approach is then supplemented by a Surrogate-Assisting strategy that directly estimates the objective and constraint values. Under some hypotheses, we study the convergence properties of the method in terms of the distance between the approximated Pareto front and the true continuous one. Secondly, we propose to compute non-parametric approximations of the error distributions with a sampling-based technique. We propose a first algorithm relying on an approximation scheme with controlled accuracy for drawing large-scale Gaussian random field realizations in the coupled space between design and uncertain parameters. It notably permits a sharper computation of the POP and detects possible correlations between the different objectives and constraints. Joint realizations can be drawn on multiple designs in order to generate Surrogate-Assisting models of the objectives and constraints. Since the construction of a Gaussian random field can be hard in the context of high dimensionality or non-parametric inputs, we also propose a KDE-based Surrogate-Assisting model as an extension of the classical heteroscedastic Gaussian process, with the capability to take as input disjoint objective and constraint realizations. We assess the proposed variants on several analytical uncertainty-based optimization test-cases with respect to an a priori metamodel approach by computing a probabilistic modified Hausdorff distance to the exact Pareto optimal set. The method is then employed on several engineering applications: the two-bar truss, a thermal protection system for atmospheric reentry and the blades of an Organic Rankine Cycle turbine.
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https://hal.inria.fr/tel-03084593
Contributor : Mickaël Rivier <>
Submitted on : Monday, December 21, 2020 - 11:27:19 AM
Last modification on : Monday, January 18, 2021 - 11:56:02 AM

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Mickael Rivier. Low-cost methods for constrained multi-objective optimization under uncertainty. Statistics [math.ST]. Institut Polytechnique de Paris, 2020. English. ⟨tel-03084593⟩

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