As part of the colloquium of the Algorithmic Optimization Research Training Group, the following lecture will take place on
Monday, September 8, 2025,
at 4:00 p.m.,
Lecture Hall 9:
Continuity and Lipschitzianity of expected value under decision-dependent uncertainty with moving support
Dr. Anton Svensson Graan, University of Chile
We address the problem of stochastic optimization with decision-dependent uncertainty, a class of problems where the probability distribution of the uncertain parameters is influenced by the decision-maker's actions. We explore some of the properties of the corresponding expected value function for a specific construction by means of the moving support of the distributions, modeled by a set-valued map S, and a density function. This construction is motivated by the Bayesian approach to bilevel programming, where the (optimal) response of a follower is modeled as the uncertainty, drawn from the moving set of optimal responses, which depends on the leader's decision.
Our main contribution is to establish sufficient conditions for the continuity and Lipschitz continuity of the expected value function, the first condition ensuring well-posedness, while the second is paramount for the application of first-order optimization methods. These conditions are framed in two cases: when S has compact convex images with nonempty interior, and when S is the parametric solution of a fully linear programming problem.