Risk-Averse Models in Bilevel Stochastic Linear Programming

Abstract

We consider bilevel linear problems, where some parameters are stochastic, and the leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, which we evaluate based on some lawinvariant convex risk measure. A qualitative stability result under perturbations of the underlying probability distribution is presented. Moreover, for the expectation, the expected excess, and the upper semideviation, we establish Lipschitz continuity as well as sufficient conditions for differentiability. Finally, for finite discrete distributions, we reformulate the bilevel stochastic problems as standard bilevel problems and propose a regularization scheme for bilevel linear problems.

A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

“…The main result of the present work provides sufficient conditions, namely boundedness of the support and uniform boundedness of the Lebesgue density of the underlying probability measure, that ensure Lipschitz continuity of the gradient of the expectation functional. Moreover, we show that the assumptions of [1] are too weak to even guarantee local Lipschitz continuity of the gradient. By the main result, second-order necessary and sufficient optimality conditions can be formulated in terms of generalized Hessians.…”

“…By the main result, second-order necessary and sufficient optimality conditions can be formulated in terms of generalized Hessians. As part of the preparatory work for the proof of the main result, we in particular show that any region of strong stability in the sense of [1,Definition 4.1] is a finite union of polyhedral cones. This representation is of independent interest, as it may facilitate the calculation or estimation of gradients of the expectation functional and thus enhance gradient descent-based approaches.…”

“…The sequential nature of bi-level programming motivates a setting where the leader decides nonanticipatorily, while the follower can observe the realization of the randomness. A discussion of the related literature is provided in the recent [1]. A central result of [1] states that evaluating the leader's random outcome by taking the expectation leads to a continuously differentiable functional if the underlying probability measure is absolutely continuous w.r.t.…”

“…A discussion of the related literature is provided in the recent [1]. A central result of [1] states that evaluating the leader's random outcome by taking the expectation leads to a continuously differentiable functional if the underlying probability measure is absolutely continuous w.r.t. the Lebesgue Communicated by René Henrion.…”

“…This representation is of independent interest, as it may facilitate the calculation or estimation of gradients of the expectation functional and thus enhance gradient descent-based approaches. The paper is organized as follows: The model and related results of [1] are discussed in Sect. 2, while the main result and a variation with weaker assumptions are formulated in Sect.…”