TY - INPR A1 - Heitsch, Holger A1 - Henrion, René A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - On Convex Lower-Level Black-Box Constraints in Bilevel Optimization with an Application to Gas Market Models with Chance Constraints N2 - Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage. KW - Bilevel optimization KW - Black-box constraints KW - Chance constraints KW - Cutting planes KW - European gas market Y1 - 2021 ER - TY - JOUR A1 - Farshbaf-Shaker, M. Hassan A1 - Henrion, Rene A1 - Hömberg, Dietmar T1 - Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization JF - Set-Valued and Variational Analysis N2 - Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints. KW - Chance constraints KW - Probabilistic constraints KW - PDE constrained optimization Y1 - 2018 U6 - https://doi.org/doi:10.1007/s11228-017-0452-5 ER - TY - JOUR A1 - Henrion, René A1 - Schmidt, Martin T1 - Chance-Constrained Linear Complementarity Problems N2 - We study linear complementarity problems (LCPs) under uncer- tainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relax- ations, which naturally leads to optimization problems in which the relaxation parameters are minimized for given probability levels. We focus on these optimization problems and first study the continuity of the related probability functions and the compactness of the feasible sets. This leads to existence results for both types of models: one with a joint chance constraint and one with separate chance constraints for both uncertainty-affected conditions of the LCP. For both, we prove the differentiability of all probability functions and derive respective gradient formulae. For the separate case, we prove con- vexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small case study regarding a Cournot oligopoly among energy producers, we finally illustrate the applicability of our theoretical findings. KW - Linear complementarity problems KW - Chance constraints KW - Existence KW - Convexity KW - Optimality conditions Y1 - 2026 ER -