Bridging k-sum and CVaR optimization in MILP

Abstract

Mixed-Integer Linear Programming models often optimize the sum, or average, of different outcomes. In a deterministic setting, each outcome may be associated with an agent, for example a customer, an employee, or a time period. In a stochastic setting, each outcome may be associated with a discrete scenario. The average approach optimizes the overall efficiency of the solution, but neglects the possible unfair distribution of outcomes among agents or the risk of very bad scenarios. In this paper, we exploit the analogies of the two settings to derive a common optimization paradigm bridging the gap between k-sum optimization in the deterministic setting and Conditional Value-at-Risk optimization in the stochastic setting. We show that the proposed paradigm satisfies properties that make it an attractive criterion. To illustrate the proposed paradigm, we apply it to the multidimensional knapsack problem and the p-median/p-center problem.