Use of a Non-nested Formulation to Improve Search for Bilevel Optimization

Abstract

Bilevel optimization involves searching for the optimum of an upper level problem subject to optimality of a nested lower level problem. These are also referred to as the leader and follower problems, since the lower level problem is formulated based on the decision variables at the upper level. Most evolutionary algorithms designed to deal with such problems operate in a nested mode, which makes them computationally prohibitive in terms of the number of function evaluations. In the classical literature, one of the common ways of solving the problem has been to re-formulate it as a single-level problem using optimality measures (such as Karush-Kuhn-Tucker conditions) for lower level problem as complementary constraint(s). However, the mathematical properties such as linearity/convexity limits their application to more complex or black-box functions. In this study, we explore a non-nested strategy in the context of evolutionary algorithm. The constraints of the upper and lower level problems are considered together at a single-level while optimizing the upper level objective function. An additional constraint is formulated based on local exploration around the lower level decision vector, which reflects an estimate of its optimality. The approach is further enhanced through the use of periodic local search and selective “re-evaluation” of promising solutions. The proposed approach is implemented in a commonly used evolutionary algorithm framework and empirical results are shown for the SMD suite of test problems. A comparison is done with other established algorithms in the field such as BLEAQ, NBLEA, and BLMA to demonstrate the potential of the proposed approach.