A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design

Abstract

Based on a subspace method and a linear approximation method, a convex algorithm is designed to solve a kind of non-convex PDE constrained fractional optimization problem in this paper. This PDE constrained problem is an infinite-dimensional Hermitian eigenvalue optimization problem with non-convex and low regularity. Usually, such a continuous optimization problem can be transformed into a large-scale discrete optimization problem by using the finite element methods. We use a subspace technique to reduce the scale of discrete problem, which is really effective to deal with the large-scale problem. To overcome the difficulties caused by the low regularity and non-convexity, we creatively introduce several new artificial variables to transform the non-convex problem into a convex linear semidefinite programming. By introducing linear approximation vectors, this linear semidefinite programming can be approximated by a very simple linear relaxation problem. Moreover, we theoretically prove this approximation. Our proposed algorithm is used to optimize the photonic band gaps of two-dimensional Gallium Arsenide-based photonic crystals as an application. The results of numerical examples show the effectiveness of our proposed algorithm, while they also provide several optimized photonic crystal structures with 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 a generalized eigenvalue constraint.