Bayesian optimization approach of general bi-level problems.

Real-life problems including transportation, planning and management often involve several decision makers whose actions depend on the interaction between each other. When involving two decision makers, such problems are classified as bi-level optimization problems. In terms of mathematical programming, a bi-level program can be described as two nested problems where the second decision problem is part of the first problem's constraints. Bi-level problems are NP-hard even if the two levels are linear. Since each solution implies the resolution of the second level to optimality, efficient algorithms at the first level are mandatory. In this work we propose BOBP, a Bayesian Optimization algorithm to solve Bi-level Problems, in order to limit the number of evaluations at the first level by extracting knowledge from the solutions which have been solved at the second level. Bayesian optimization for hyper parameter tuning has been intensively used in supervised learning (e.g., neural networks). Indeed, hyper parameter tuning problems can be considered as bi-level optimization problems where two levels of optimization are involved as well. The advantage of the bayesian approach to tackle multi-level problems over the BLEAQ algorithm, which is a reference in evolutionary bi-level optimization, is empirically demonstrated on a set of bi-level benchmarks.