Set-Valued Steepest Descent for Binary Topology and Control Optimization
Abstract: 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 that 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.