Martin Schmidt

Mixed-Integer Linear Reformulations of Pessimistic Bilinear Bilevel Infima

Categories (Mixed) Integer Linear Programming, Bilevel Optimization, Nonlinear Optimization Tags , ,

We study pessimistic bilevel optimization with bilinear objectives and linear constraints. Assuming a compact polyhedral leader domain, nonempty follower solution sets, and a finite pessimistic infimum, we derive mixed-integer linear optimization formulations that attain this infimum, even when the original problem has no optimal solution. This setting has applications such as pricing, security and decision-focused … Read more

Are Single-Leader-Multi-Follower Problems Particular Cases of Bilevel Optimization?

Categories Bilevel Optimization, Game Theory, Global Optimization Tags , , ,

For more than 20 years, single-leader multi-follower (SLMF) problems have been treated as a more difficult class of problems in comparison to bilevel optimization. In this work we challenge this common belief, studying whether SLMF problems can be reformulated as equivalent bilevel optimization problems. We consider both optimistic and pessimistic formulations. For the optimistic case … Read more

When do Mixed-Integer Games Admit Rational Equilibria?

Categories (Mixed) Integer Linear Programming, Game Theory Tags , , , ,

We consider mixed-integer linear-quadratic generalized Nash equilibrium problems, i.e., games in which each player solves a mixed-integer program subject to linear constraints in her own and rivals’ strategies as well as an objective which is quadratic in her own strategies and bilinear in her own and rivals’ strategies. For this class of games, we study … Read more

Calmness of the Solution-Set Mapping for Linear Bilevel and Pricing Problems

Categories Bilevel Optimization Tags , , , , ,

We study linear bilevel and pricing problems in which the upper- and lower-level constraints’ right-hand sides are perturbed. In this setting, it is an important question, also for the validity of numerical solution schemes, if the solution-set mapping of the parametric bilevel problem is calm at the zero-perturbation. We provide the complete picture both for … Read more

Multi-Leader Single-Follower Power-Market Modeling: The Impact of DC Market-Clearing on AC Feasibility

Categories Bilevel Optimization, Energy Tags , , , ,

We study the impact of DC power flow modeling in multi-leader single-follower market models on the AC feasibility of the market outcome. To this end, we consider strategically bidding power producers that are connected to an electricity network and a market-clearing executed by an ISO. The focus is on a pay-as-bid electricity market in which … Read more

A Successive Proximal DC Penalty Method with an Application to Mathematical Programs with Complementarity Constraints

Categories Bilevel Optimization, Constrained Nonlinear Optimization, Global Optimization Tags , , , ,

We develop a successive, proximal difference-of-convex (DC) function penalty method for solving DC programs with DC constraints. The proposed approach relies on a DC penalty function that measures the violation of constraints and leads to a penalty reformulation sharing the same solution set as the original problem. The resulting penalty problem is a DC program … Read more

Branch-and-Cut for Mixed-Integer Linear Decision-Dependent Robust Optimization

Categories Bilevel Optimization, Branch and Cut Algorithms, Robust Optimization Tags , , , , ,

Decision-dependent robust optimization (DDRO) problems are usually tackled by reformulating them using a strong-duality theorem for the uncertainty set model. If the uncertainty set is, however, described by a mixed-integer linear model, dualization techniques cannot be applied and the literature on solution methods is very scarce. In this paper, we exploit the equivalence of DDRO … Read more

Chance-Constrained Linear Complementarity Problems

Categories Complementarity and Variational Inequalities, Energy, Stochastic Programming Tags , , , , ,

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 … Read more

Potential-Based Flows – An Overview

Categories Energy, Network Optimization

Potential-based flows provide an algebraic way to model static physical flows in networks, for example, in gas, water, and lossless DC power networks. The flow on an arc in the network depends on the difference of the potentials at its end-nodes, possibly in a nonlinear way. Potential-based flows have several nice properties like uniqueness and … Read more

A One-Extra Player Reduction of GNEPs to NEPs

Categories Convex Optimization, Game Theory, Integer Programming Tags , , , ,

It is common opinion that generalized Nash equilibrium problems are harder than Nash equilibrium problems. In this work, we show that by adding a new player, it is possible to reduce many generalized problems to standard equilibrium problems. The reduction holds for linear problems and smooth convex problems verifying a Slater-type condition. We also derive … Read more