Characterization of Maximizers in A Non-Convex Geometric Optimization Problem With Application to Optical Wireless Power Transfer Systems

TL;DR

This paper analyzes a complex non-convex geometric optimization problem from optical wireless power transfer, characterizing its maximizers using bifurcation theory and identifying conditions for unique global optimizers.

Contribution

It introduces a novel application of bifurcation theory to characterize maximizers in a non-convex geometric optimization problem, providing explicit conditions and counts of maximizers.

Findings

Existence of a critical inter-plane distance for unique global optimizer

Number of maximizers depends on bifurcation branches and inter-plane distance

Numerical simulations confirm theoretical bifurcation analysis

Abstract

This research studies a non-convex geometric optimization problem arising from the field of optical wireless power transfer. In the considered optimization problem, the cost function is a sum of negatively and fractionally powered distances from given points arbitrarily located in a plane to another point belonging to a different plane. Therefore, it is a strongly nonlinear and non-convex programming, hence posing a challenge on the characterization of its optimizer set, especially its set of global optimizers. To tackle this challenge, the bifurcation theory is employed to investigate the continuation and bifurcation structures of the Hessian matrix of the cost function. As such, two main results are derived. First, there is a critical distance between the two considered planes such that beyond which a unique global optimizer exists. Second, the exact number of maximizers is locally derived by the number of bifurcation branches determined via one-dimensional isotropic subgroups of a Lie group acting on $\mathbb{R}^2$, when the inter-plane distance is smaller than the above-mentioned critical distance. Consequently, numerical simulations and computations of bifurcation points are carried out for various configurations of the given points, whose results confirm the derived theoretical outcomes.