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.