  Multi-objective optimization problems whose objectives have different
evaluation costs are commonly seen in the real world. Such problems are now
known as multi-objective optimization problems with heterogeneous objectives
(HE-MOPs). So far, however, only a few studies have been reported to address
HE-MOPs, and most of them focus on bi-objective 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 multi-objective Bayesian evolutionary optimization approach to
HE-MOPs by exploiting the different data sets on the cheap and expensive
objectives in HE-MOPs to alleviate the search bias caused by the heterogeneous
evaluation costs for evaluating different objectives. To make the best use of
two different training data sets, one with solutions evaluated on all
objectives and the other with those only evaluated on the fast objectives, two
separate Gaussian process models are constructed. In addition, a new
acquisition function that mitigates search bias towards the fast objectives is
suggested, thereby achieving a balance between convergence and diversity. We
demonstrate the effectiveness of the proposed algorithm by testing it on widely
used multi-/many-objective benchmark problems whose objectives are assumed to
be heterogeneously expensive.
