Hypodifferentials of nonsmooth convex functions and their applications to nonsmooth convex optimization
M. V. Dolgopolik
TL;DR
The work develops a general theory of hypodifferentials for nonsmooth convex functions on Banach spaces, providing a complete characterization of hypodifferentiability, a rich calculus, and Lipschitz properties that extend gradient-like reasoning to nonsmooth settings. It then applies these tools to convex optimization, introducing the method of hypodifferential descent and an accelerated proximal variant, with convergence rates $O(1/k)$ and $O(1/k^2)$ respectively under appropriate assumptions. The results bridge a gap between smooth and nonsmooth optimization, offering gradient-like advantages for certain problem classes while highlighting increased per-iteration cost. Overall, the paper provides a rigorous foundation and practical algorithms for leveraging hypodifferentials in moderate-dimensional nonsmooth convex optimization.
Abstract
A hypodifferential is a compact family of affine mappings that defines a local max-type approximation of a nonsmooth convex function. We present a general theory of hypodifferentials of nonsmooth convex functions defined on a Banach space. In particular, we provide complete characterizations of hypodifferentiability and hypodifferentials of nonsmooth convex functions, derive calculus rules for hypodifferentials, and study the Lipschitz continuity/Lipschitz approximation property of hypodifferentials that can be viewed as a natural extension of the Lipschitz continuity of the gradient to the general nonsmooth setting. As an application of our theoretical results, we study the rate of convergence of several versions of the method of hypodifferential descent for nonsmooth convex optimization and present an accelerated version of this method having the faster rater of convergence $\mathcal{O}(1/k^2)$.