Abstract

Bayesian optimization is a versatile numerical method to solve global optimization problems of high complexity at a reduced computational cost. The efficiency of Bayesian optimization relies on two key elements: a surrogate model and an acquisition function. The surrogate model is generated on a Gaussian process statistical framework and provides probabilistic information of the prediction. The acquisition function, which guides the optimization, uses the surrogate probabilistic information to balance the exploration and the exploitation of the design space. In the case of multi-objective problems, current implementations use acquisition functions such as the multi-objective expected improvement (MEI). The evaluation of MEI requires a surrogate model for each objective function. In order to expand the Pareto front, such implementations perform a multi-variate integral over an intricate hypervolume, which require high computational cost. The objective of this work is to introduce an efficient multi-objective Bayesian optimization method that avoids the need for multi-variate integration. The proposed approach employs the working principle of multi-objective traditional methods, e.g., weighted sum and min-max methods, which transform the multi-objective problem into a single-objective problem through a functional mapping of the objective functions. Since only one surrogate is trained, this approach has a low computational cost. The effectiveness of the proposed approach is demonstrated with the solution of four problems: (1) an unconstrained version of the Binh and Korn test problem (convex Pareto front), (2) the Fonseca and Fleming test problem (non-convex Pareto front), (3) a three-objective test problem and (4) the design optimization of a sandwich composite armor for blast mitigation. The optimization algorithm is implemented in MATLAB and the finite element simulations are performed in the explicit, nonlinear finite element analysis code LS-DYNA. The results are comparable (or superior) to the results of the MEI acquisition function.

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