@Article{AAMM-15-6,
author = {Wu, Mengyue and Yuan, Jianhua and Zhang, Jianxin},
title = {A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2023},
volume = {15},
number = {6},
pages = {1540--1561},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.OA-2022-0003},
url = {https://global-sci.com/article/72883/a-convex-approximation-for-a-pde-constrained-fractional-optimization-problem-with-an-application-to-photonic-crystal-design}
}