Necessary Optimality Conditions for Integrated Learning and Optimization Problem in Contextual Optimization

TL;DR

This paper establishes the first-order necessary optimality conditions for integrated learning and optimization problems in contextual optimization, addressing a gap in the theoretical understanding of stochastic bilevel programs.

Contribution

It derives novel optimality conditions for ILO problems using Mordukhovich subdifferentials and sensitivity analysis, applicable to both convex and nonconvex lower-level problems.

Findings

Derived optimality conditions for convex lower-level problems.

Extended conditions to nonconvex lower-level problems under stochastic partial calmness.

Applied conditions to existing ILO problems to facilitate algorithm design.

Abstract

Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality of downstream decisions. This framework represents a new class of stochastic bilevel programs, which are extensively utilized in the literature of operations research and management science, yet remain underexplored from the perspective of optimization theory. In this paper, we fill the gap. Specifically, we derive the first-order necessary optimality conditions in terms of Mordukhovich limiting subdifferentials. To this end, we formulate the bilevel program as a two-stage stochastic program with variational inequality constraints when the lower-level decision-making problem is convex, and establish an optimality condition via sensitivity analysis of the second-stage value function. In the case where the lower level optimization problem is nonconvex, we adopt the value function approach in the literature of bilevel programs and derive the first-order necessary conditions under stochastic partial calmness conditions. The derived optimality conditions are applied to several existing ILO problems in the literature. These conditions may be used for the design of gradient-based algorithms for solving ILO problems.