Nested evolutionary algorithms for computationally expensive bilevel optimization problems: Variants and their systematic analysis

Bilevel optimization problems involve a hierarchical model where an upper level optimization problem is solved with a constraint on the optimality of a nested lower level problem. The use of evolutionary algorithms (EAs) and other metaheuristics has been gaining attention to solve bilevel problems, especially when they contain nonlinear/black-box objective(s) and/or constraint(s). However, EAs typically operate in a nested mode wherein a lower level optimization is executed for each upper level solution. Evidently, this process requires excessive number of function evaluations, which might become untenable if the underlying functions are computationally expensive. In order to reduce this expense, the use of approximations (also referred to as surrogates or meta models) has been suggested previously. However, the previous works have focused only on the use of surrogates for the lower level problem, whereas the computational expense of the upper level problem has not been considered. In this paper, we aim to make two contributions to address this research gap. The first is to introduce an improved nested EA which uses surrogate-assisted search at both levels in order to solve bilevel problems using limited number of function evaluations. The second is the revelation and a systematic investigation of a previously overlooked aspect of bilevel search - that the objective/constraints at the upper and lower levels may involve different computational expense. Consideration of this aspect can help in deciding a suitable strategy, i.e., in which level is the use of surrogates most appropriate for the given problem. Towards this end, four different nested strategies - with surrogate at either level, none or at both levels, are compared under various experimental settings. Numerical experiments are presented on a wide range of problems to demonstrate the efficacy and utility of the proposed contributions.