Duality for convex infinite optimization on linear spaces

TL;DR

This paper derives a limiting formula for the conic Lagrangian dual of convex infinite optimization problems, correcting previous results and establishing strong duality under certain conditions.

Contribution

It provides a corrected and generalized duality formula for convex semi-infinite programs, including conditions for strong duality.

Findings

Established a limiting formula for the conic Lagrangian dual.

Corrected the classical duality result by Karney.

Proved strong sup-duality under the strong Slater condition.

Abstract

This note establishes a limiting formula for the conic Lagrangian dual of a convex infinite optimization problem, correcting the classical version of Karney [Math. Programming 27 (1983) 75-82] for convex semi-infinite programs. A reformulation of the convex infinite optimization problem with a single constraint leads to a limiting formula for the corresponding Lagrangian dual, called sup-dual, and also for the primal problem in the case when strong Slater condition holds, which also entails strong sup-duality.