TY - INPR A1 - Krug, Richard A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems N2 - We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and which may result in simpler classes of optimization problems since not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research. KW - Gas transport networks KW - Mixed-integer nonlinear optimization KW - Alternating direction methods KW - Graph decomposition KW - Penalty methods Y1 - 2022 ER - TY - INPR A1 - Horländer, Andreas A1 - Schmidt, Martin T1 - A Penalty Branch-and-Bound Method for Mixed-Integer Quadratic Bilevel Problems N2 - We propose an algorithm for solving bilevel problems with mixed-integer convex-quadratic upper level as well as convex-quadratic and continuous lower level. The method is based on a classic branch-and-bound procedure, where branching is performed on the integer constraints and on the complementarity constraints resulting from the KKT reformulation of the lower-level problem. However, instead of branching on constraints as usual, suitably chosen penalty terms are added to the objective function in order to create new subproblems in the tree. We prove the correctness of the method and present its applicability by some first numerical results. KW - Bilevel optimization KW - Branch-and-bound KW - Penalty methods KW - Mixed-integer optimization Y1 - 2022 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method JF - INFORMS Journal on Computing N2 - Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, etc. Often, these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization one develops primal heuristics in order to obtain feasible points of good quality quickly or to enhance the search process of exact global methods. However, there are comparably few heuristics for bilevel problems. In this paper, we develop such a primal heuristic for bilevel problems with mixed-integer linear or quadratic upper level and linear or quadratic lower level. The heuristic is based on a penalty alternating direction method, which allows for a theoretical analysis. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem and extensively test the method on a test set of more than 2800 instances - which is one of the largest computational test sets ever used in bilevel programming. The study illustrates the very good performance of the proposed method, both in terms of running times and solution quality. This renders the method a suitable sub-routine in global bilevel solvers as well as a reasonable standalone approach. KW - Bilevel optimization KW - Mixed-integer bilevel optimization KW - Stationary points KW - Penalty methods KW - Alternating direction methods Y1 - 2019 U6 - https://doi.org/10.1287/ijoc.2019.0945 IS - 33 (1) SP - 198 EP - 215 ER -