On Sion's Minimax Theorem, Compact QCQPs, and Wave Scattering Optimization

TL;DR

This paper explores the application of Sion's minimax theorem to compact QCQPs in wave scattering, proposing a dual perspective that could enhance global optimization methods in photonics, acoustics, and quantum mechanics.

Contribution

It introduces an alternative dual framework for QCQPs in wave scattering, linking minimax theory with practical inverse design algorithms for global optimality.

Findings

Sion's minimax theorem applies to certain wave scattering QCQPs.

A dual perspective offers new insights into optimization in wave phenomena.

Potential for algorithmic methods with guaranteed global optimality.

Abstract

In these notes, we examine certain implications of Sion's minimax theorem for compact quadratically constrained quadratic programs (QCQPs), particularly QCQPs arising in the context of optimizing wave scattering, in relation to Lagrangian duality. The discussion puts forward an alternative "dual" understanding of optimization for wave phenomena that anticipates the realization of algorithmic (inverse design) methods attaining a guaranteed degree of global optimality for common figures of merit appearing in applied photonics, acoustics, and quantum mechanics.