Alleviating Search Bias in Bayesian Evolutionary Optimization With Many Heterogeneous Objectives

Multiobjective optimization problems whose objectives have different evaluation costs are commonly seen in the real world. Such problems are now known as multiobjective optimization problems with heterogeneous objectives (HE-MOPs). So far, however, only a few studies have been reported on addressing HE-MOPs, and most of them focus on biobjective problems with one fast objective and one slow objective. In this work, we aim to deal with HE-MOPs having more than two black-box and heterogeneous objectives. To this end, we develop a multiobjective Bayesian evolutionary optimization (BEO) approach to HE-MOPs that can alleviate search biases resulting from the different numbers of function evaluations allowed for the cheap and expensive objectives, which is achieved by designing a new acquisition function that penalizes the search bias toward the fast objectives, thereby achieving a balance between convergence and diversity. In addition, to make the best use of the different amounts of training data while avoiding increasing the computational cost, an ensemble consisting of two Gaussian processes is constructed for each cheap objective, one trained on the data collected before the Bayesian optimization starts, and the other on those evaluated during the BEO. Empirical studies on widely used multi-/many-objective benchmark problems whose objectives are assumed to be heterogeneously expensive demonstrate that the proposed algorithm is able to find high-quality solutions for HE-MOPs compared with the state-of-the-art methods.