Existence of Solutions in Bi-level Stochastic Linear Programming with Integer Variables

TL;DR

This paper investigates the existence of solutions in bi-level stochastic linear programming with integer variables, focusing on the impact of integrality constraints and risk measures on solution properties.

Contribution

It provides sufficient conditions for the existence of solutions and analyzes the stability of the problem under perturbations of the probability measure.

Findings

Sufficient conditions for Hölder continuity of the leader's risk functional.

Existence of solutions under finite lower level feasible sets.

Joint continuity of the objective with respect to decisions and probability measures.

Abstract

The addition of lower level integrality constraints to a bi-level linear program is known to result in significantly weaker analytical properties. Most notably, the upper level goal function in the optimistic setting lacks lower semicontinuity and the existence of an optimal solution cannot be guaranteed under standard assumptions. In this paper, we study a setting where the right-hand side of the lower level constraint system is affected by the leader's choice as well as the realization of some random vector. Assuming that only the follower decides under complete information, we employ a convex risk measure to assess the upper level outcome. Confining the analysis to the cases where the lower level feasible set is finite, we provide sufficient conditions for H\"older continuity of the leader's risk functional and draw conclusions about the existence of optimal solutions. Finally, we examine qualitative stability with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the leader's decision and the underlying probability measure.