Plausible optima

We propose a framework and specific algorithms for screening a large (perhaps countably infinite) space of feasible solutions to generate a subset containing the optimal solution with high confidence. We attain this goal even when only a small fraction of the feasible solutions are simulated. To accomplish it we exploit structural information about the space of functions within which the true objective function lies, and then assess how compatible optimality is for each feasible solution with respect to the observed simulation outputs and the assumed function space. The result is a set of plausible optima. This approach can be viewed as a way to avoid slow simulation by leveraging fast optimization. Explicit formulations of the general approach are provided when the space of functions is either Lipschitz or convex. We establish both small- and large-sample properties of the approach, and provide two numerical examples.