Duality for Composite Optimization Problem within the Framework of Abstract Convexity

TL;DR

This paper investigates duality theories for composite optimization problems in the context of abstract convexity, establishing conditions for zero duality gap and exploring the relationship between conjugate and Lagrange duality.

Contribution

It introduces new conditions for zero duality gap in conjugate and Lagrange duality within abstract convexity, connecting these dualities and providing relevant examples.

Findings

Conditions for zero duality gap in conjugate duality

Application of intersection property for Lagrange duality

Connection established between Lagrange and conjugate duality

Abstract

We study conjugate and Lagrange dualities for composite optimization problems within the framework of abstract convexity. We provide conditions for zero duality gap in conjugate duality. For Lagrange duality, intersection property is applied to obtain zero duality gap. Connection between Lagrange dual and conjugate dual is also established. Examples related to convex and weakly convex functions are given.