A class of nonlinear optimisation and applications

TL;DR

This paper introduces a new class of nonlinear optimization problems, establishes the existence of potential functions under mild conditions, and demonstrates their applications in optimal transportation and geometric optics.

Contribution

It presents a novel class of nonlinear optimization problems and links potential functions to Monge-Ampère type equations, expanding theoretical understanding and practical applications.

Findings

Existence of potential functions under mild assumptions

Potential functions solve a generalized Monge-Ampère equation

Applications demonstrated in optimal transportation and geometric optics

Abstract

In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Amp\`ere type equation. We also present some interesting applications in optimal transportation and geometric optics problems.