On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient

By Xingyu Zhou. 2018

138 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3ebb600a7a4279f1d1438f8aa8c60b90aa16ec8d

Citations in the corpus

• Applications of weak transport theory (2020, unknown): “In the subsequent proof, we will use as a black box a characterization of convex functions with 1-Lipschitz gradient in the spirit of cyclical monotonicity, see [55].”