Solving a PDE-constrained antenna design problem by mixed-integer programming

Abstract

This paper shows that a mixed integer PDE-constrained design optimization problem can be made amenable to be solved to optimality using standard mixed-integer programming solvers. The concrete problem that we consider is the optimal design of the topology of an antenna, a problem that keeps haunting the engineering community. Indeed, up to this point no exact method was known. Existing methods rely on iterative use of PDE-solvers, embedded in either gradient methods or metaheuristics. However, none of these methods come with performance guarantees. We break this barrier, showing that important antenna design problems can be solved optimally using a mixed integer linear program, plus one singleton second order cone constraint in two variables. The main contribution of the paper is to develop such a formulation. There are two major methodological steps to get there. The first is to show how a standard boundary element method (BEM), based on a discretized design space, translates into a mixed integer program at all. Approached intuitively, this yields a maximization problem with non-concave objective, which is hence not tractable. The second is to make the objective function linear, using a new Charnes-Cooper transform for mathematical programming over complex numbers. We evaluate our approach on the classic dipole antenna, reconfirming well-known experimental results, and apply it to a widely-used test case, the Yagi-Uda antenna. In a successor paper, we report on further computational enhancements using physics-informed cuts.