Chance-constrained linear complementarity problems
Authors
- Henrion, René
ORCID: 0000-0001-5572-7213 - Schmidt, Martin
2020 Mathematics Subject Classification
- 90C33 90C15 90C17
Keywords
- Linear complementarity problems, chance constraints, existence, convexity, optimality conditions, energy markets
DOI
Abstract
We study linear complementarity problems (LCPs) under uncertainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relaxations, 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 convexity 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.
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