Multi-Objective Reliability-Based Optimization with Stochastic Metamodels

Abstract

This paper addresses continuous optimization problems with multiple objectives and parameter uncertainty defined by probability distributions. First, a reliability-based formulation is proposed, defining the nondeterministic Pareto set as the minimal solutions such that user-defined probabilities of nondominance and constraint satisfaction are guaranteed. The formulation can be incorporated with minor modifications in a multiobjective evolutionary algorithm (here: the nondominated sorting genetic algorithm-II). Then, in the perspective of applying the method to large-scale structural engineering problems—for which the computational effort devoted to the optimization algorithm itself is negligible in comparison with the simulation—the second part of the study is concerned with the need to reduce the number of function evaluations while avoiding modification of the simulation code. Therefore, nonintrusive stochastic metamodels are developed in two steps. First, for a given sampling of the deterministic variables, a preliminary decomposition of the random responses (objectives and constraints) is performed through polynomial chaos expansion (PCE), allowing a representation of the responses by a limited set of coefficients. Then, a metamodel is carried out by kriging interpolation of the PCE coefficients with respect to the deterministic variables. The method has been tested successfully on seven analytical test cases and on the 10-bar truss benchmark, demonstrating the potential of the proposed approach to provide reliability-based Pareto solutions at a reasonable computational cost.

Sequential improvement for robust optimization using an uncertainty measure for radial basis functions

“…Some researchers choose to combine these measures in a single objective, for example by minimizing μ Y + kσ Y , as is done in this study. Many other robust optimization objectives can be found in literature, such as Pareto front determination (Coelho and Bouillard 2011;Nishida et al 2013), quantile optimization (Rhein et al 2014), minimization of the worst case scenario (Marzat et al 2013) and minimization of the signal-to-noise ratio (Taguchi and Phadke 1984;Leon et al 1987).…”

Automatic Design of Scheduling Policies for Dynamic Multi-objective Job Shop Scheduling via Cooperative Coevolution Genetic Programming

“…As compared to the literature related to probabilistic formulation of the Pareto dominance, the problem in our work has some special features. First, our problem needs to be handled with a probabilistic formulation rather than a robustness formulation (uncertainties defined by intervals) since the uncertainties in our problem come from a probability distributions [76]. Second, it is computationally expensive to evaluate the rules; and therefore it is very time consuming to perform a large number of simulation replications.…”

“…Finally, our methods try to evolve rules in tree form which are very different from vectors of design variables in most studies in the literature. Because of these features, analytical or reliability approaches which depend on some assumptions on the uncertainties [76], [77] cannot be used, and the Pareto dominance under uncertainty in our work can only be determined via the pure sampling technique [76]. In order to examine the Pareto dominance of our evolved scheduling policies against the existing ones, we need to use statistical tests and introduce the notion of statistical Pareto dominance as opposed to probabilistic Pareto dominance [78], [79].…”

Single-objective and Multi-objective Robust Optimization of Airfoils using Adjoint Solutions

“…The research related to the multi-objective method has led to several extensions of the classical Pareto front concept. 6,8,9,10,12,23,19 IV. Optimization Under Uncertainty of a NACA 0012 Airfoil…”