TY - JOUR
T1 - A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design
AU - Wu , Mengyue
AU - Yuan , Jianhua
AU - Zhang , Jianxin
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1540
EP - 1561
PY - 2023
DA - 2023/10
SN - 15
DO - http://doi.org/10.4208/aamm.OA-2022-0003
UR - https://global-sci.org/intro/article_detail/aamm/22051.html
KW - PDE constrained optimization, fractional programming, linear approximation, finite element method, photonic band gap.
AB - <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>