PDE- and ODE-constrained optimization problems with integer-valued control functions are often computationally intractable using the first-discretize-then-optimize approach. This is mainly because computational complexity generally increases exponentially with the number of integer variables, which increases quickly as control meshes are refined.

We discuss a method which avoids these issues for a class of problems with a single binary-valued control function by reformulating the original problem as an optimization problem over the sigma-algebra of Lebesgue-measurable sets. By reformulating the problem in terms of set-valued variables, we can transfer much of the theory of continuous nonlinear programming to mixed-integer problems, which we demonstrate by developing a trust-region steepest descent algorithm.

In addition, we address issues of precision and convergence, as well as possible extensions and theoretical limitations of our approach.