On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient

Understand

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.