On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient

TL;DR

This paper offers a straightforward proof of the Fenchel duality linking strong convexity and Lipschitz continuous gradient, using equivalent convexity conditions applicable even to non-differentiable functions.

Contribution

It introduces a simple proof for Fenchel duality and identifies broader conditions related to strong convexity and Lipschitz gradients.

Findings

Established equivalent convexity conditions for general functions.

Derived direct conditions for strong convexity and Lipschitz continuous gradient.

Identified broader implications beyond the main duality result.

Abstract

We provide a simple proof for the Fenchel duality between strong convexity and Lipschitz continuous gradient. To this end, we first establish equivalent conditions of convexity for a general function that may not be differentiable. By utilizing these equivalent conditions, we can directly obtain equivalent conditions for strong convexity and Lipschitz continuous gradient. Based on these results, we can easily prove Fenchel duality. Beside this main result, we also identify several conditions that are implied by strong convexity or Lipschitz continuous gradient, but are not necessarily equivalent to them. This means that these conditions are more general than strong convexity or Lipschitz continuous gradient themselves.