Thursday, September 28

Invited Lecture: Combinatorial Optimization under Nonlinear PDE-constraints

Time: 13:30 - 14:30
Room: L1, Building L
Chair: Stefania Bellavia, Università degli Studi di Firenze

We address optimal control problems containing semilinear elliptic PDE constraints as well as combinatorial constraints in the control variables, arising when modeling static diffusion processes controlled by discrete decisions. In the case of linear PDEs, such problems can be rewritten as (finite-dimensional) linear or quadratic integer programs. For the non-linear case, the standard solution approach is to directly discretize the entire problem, resulting however in huge non-convex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. We propose a new approach based on outer approximation, using an integer linear programming master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity and submodularity of the PDE solution operator in terms of the control variables, which we prove in the case of PDEs with a non-decreasing and convex nonlinear part. Our approach can handle general linear constraints on both control and state variables as well as tracking-type objective functions.

Christoph Buchheim
Technische Universität Dortmund