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

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 infinitedimensional 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.