Optimization with stochastic preferences based on a general class of scalarization functions

Abstract: It is of crucial importance to develop risk-averse models for multicriteria decision making under uncertainty. A major stream of the related literature studies optimization problems that feature multivariate stochastic benchmarking constraints. These problems typically involve a univariate stochastic preference relation, often based on stochastic dominance or a coherent risk measure such as conditional value-at-risk (CVaR), which is then extended to allow the comparison of random vectors by the use of a family of scalarization functions: All scalarized versions of the vector of the uncertain outcomes of a decision are required to be preferable to the corresponding scalarizations of the benchmark outcomes. While this line of research has been dedicated almost entirely to linear scalarizations, the corresponding deterministic literature uses a wide variety of scalarization functions that, among other advantages, offer a high degree of modeling flexibility. In this paper we aim to incorporate these scalarizations into a stochastic context by introducing the general class of min-biaffine functions. We study optimization problems in finite probability spaces with multivariate stochastic benchmarking constraints based on min-biaffine scalarizations. We develop duality results, optimality conditions, and a cut generation method to solve these problems. We also introduce a new characterization of the risk envelope of a coherent risk measure in terms of its Kusuoka representation as a tool towards proving the finite convergence of our solution method. The main computational challenge lies in solving cut generation subproblems; we develop several mixed-integer programming formulations by exploiting the min-affine structure and leveraging recent advances for solving similar problems with linear scalarizations. We conduct a computational study on a well-known homeland security budget allocation problem to examine the impact of the proposed scalarizations on optimal solutions, and illustrate the computational performance of our solution methods.