Strong Duality of Linear Optimisation Problems over Measure Spaces

TL;DR

This paper establishes strong duality results for specific classes of linear optimization problems over measure spaces, with applications in robust risk management and option pricing.

Contribution

It provides new strong duality theorems for measure-based linear optimization problems with $L^p$ densities and semi-infinite constraints.

Findings

Strong duality holds for measures with $L^p$ densities.

Semi-infinite problems with integral bounds also exhibit strong duality.

Applications include robust risk management and model-free option pricing.

Abstract

In this work we present two particular cases of the general duality result for linear optimisation problems over signed measures with infinitely many constraints in the form of integrals of functions with respect to the decision variables (the measure in question) for which strong duality holds. In the first case the optimisation problems are over measures with $L^p$ density functions with $1 < p < \infty$. In the second case we consider a semi-infinite optimisation problem where finitely many constraints are given in form of bounds on integrals. The latter case has a particular importance in practice where the model can be applied in robust risk management and model-free option pricing.