On Lagrange duality theory for dynamics vaccination games

TL;DR

This paper develops a duality theory for a convex optimization problem related to vaccination management, establishing strong duality under less restrictive conditions and analyzing the associated dynamic vaccination game.

Contribution

It introduces an improved duality framework that relaxes previous assumptions and applies it to understand the structure of vaccination game models.

Findings

Established strong duality without requiring interiority of the convex cone

Proved the existence of Lagrange multipliers in the vaccination game context

Enhanced understanding of the dual problem's economic and strategic implications

Abstract

The authors study an infinite dimensional duality theory finalized to obtain the existence of a strong duality between a convex optimization problem connected with the management of vaccinations and its Lagrange dual. Specifically, the authors show the solvability of a dual problem using as basic tool an hypothesis known as Assumption S. Roughly speaking, it requires to show that a particular limit is nonnegative. This technique improves the previous strong duality results that need the nonemptyness of the interior of the convex ordering cone. The authors use the duality theory to analyze the dynamic vaccination game in order to obtain the existence of the Lagrange multipliers related to the problem and to better comprehend the meaning of the problem.