A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design (Advances in Applied Mathematics and Mechanics, 2023)
Based on a subspace method and a linear approximation method, a convex algorithm is designed to solve a kind of non-convex PDE constrained fractionaloptimization problem in this paper. This PDE constrained problem is an infinite-dimensional Hermitian eigenvalue optimization problem with non-convex and lowregularity. Usually, such a continuous optimization problem can be transformed into alarge-scale discrete optimization problem by using the finite element methods. We usea subspace technique to reduce the scale of discrete problem, which is really effectiveto deal with the large-scale problem. To overcome the difficulties caused by the lowregularity and non-convexity, we creatively introduce several new artificial variablesto transform the non-convex problem into a convex linear semidefinite programming.By introducing linear approximation vectors, this linear semidefinite programmingcan be approximated by a very simple linear relaxation problem. Moreover, we theoretically prove this approximation. Our proposed algorithm is used to optimize thephotonic band gaps of two-dimensional Gallium Arsenide-based photonic crystals asan application. The results of numerical examples show the effectiveness of our proposed algorithm, while they also provide several optimized photonic crystal structureswith a desired wide-band-gap. In addition, our proposed algorithm provides a technical way for solving a kind of PDE constrained fractional optimization problems with ageneralized eigenvalue constraint.