## Applied Math Colloquium: Dr Sven Leyffer

#### Argonne National Laboratory

Title: Mixed-Integer PDE-Constrained Optimization

Abstract: Many complex applications can be formulated as optimization problems constrained by partial differential equations (PDEs) with integer decision variables. This new class of problems, called mixed-integer PDE-constrained optimization (MIPDECO), must overcome the combinatorial challenge of integer decision variables combined with the numerical and computational complexity of PDE-constrained optimization. Examples of MIPDECOs include the remediation of contaminated sites and the maximization of oil recovery; the design of next-generation solar cells; the layout design of wind-farms; the design and control of gas networks; disaster recovery; and topology optimization.

We will present some emerging applications of mixed-integer PDE-constrained optimization, review existing approaches to solve these problems, and highlight their computational and mathematical challenges. We show how existing methods for solving mixed-integer optimization problems can be adapted to solve this new class of problems. In particular, we will present two classes of methods, survey their convergence properties, and illustrate their numerical behavior.