Co-Optimization Free Lunches: Tractability of Optimal Black-Box Algorithms for Maximizing Expected Utility

Co-optimization problems often involve settings in which the quality (utility) of a potential solution is dependent on the scenario within which it is evaluated, and many such scenarios exist. Maximizing expected utility is simply the goal of finding the potential solution whose expected utility value over all possible scenarios is best. Such problems are often approached using coevolutionary algorithms. We are interested in the design of generally well-performing black-box algorithms for this problem, that is, algorithms which have access to the utility function only via input–output queries. We research this matter by focusing on three main questions: 1) are some algorithms strictly better than others when judged in aggregation over all possible instances of the problem? that is, is there “free lunch”? 2) do optimal algorithms exist? and 3) if so, do they have a tractable implementation? For a specific expected-utility maximization context, involving several assumptions and performance choices, we answer all three questions affirmatively and concretely: we provide examples of free lunch; we describe the general operation of optimal algorithms; we characterize situations when this operation has a very simple and efficient implementation, situations when the computational cost can be significantly reduced, and situations when tractability of optimal algorithms might be out of reach.