Efficiently Computing the Quasiconcave Envelope with Incomplete Information

Abstract

In this paper, we study the approximation of an unknown quasiconcave function based on limited partial information. Available information includes lower bounds on the values of the target function at a specified set of points, as well as some functional properties including monotonicity, Lipschitz continuity, ranking, and permutation invariance. We consider the class of admissible quasiconcave functions that dominate these lower bounds and satisfy these functional properties. We then compute the smallest quasiconcave function among the class of admissible quasiconcave functions. Specifically, we show how to efficiently compute the quasiconcave envelope (QCoE) of a data sample of points, subject to the additional functional properties. The solution procedure takes two steps. First, a value problem is solved to determine the values of the QCoE on the given data sample. Second, an interpolation problem is solved to compute the values of the QCoE on other points. Both the value problem and the interpolation problem introduce some theoretical and computational challenges, as they are non-convex and large-scale. The MILP reformulations of both problems require an exponential number of linear programs (LPs) to be solved in the worst-case. As our main contribution, we solve the value problem with only a polynomial number of LPs, and then solve the interpolation problem for any candidate point with only a logarithmic number of LPs. Some preliminary numerical tests show that the proposed approach is efficient and proper.