Journal of Convex Analysis 32 (2025), No. 4, 1255--1298
Copyright Heldermann Verlag 2025
Hypodifferentials of Nonsmooth Convex Functions and their Applications to Nonsmooth Convex Optimization
Maksim V. Dolgopolik
Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia
dmv@ipme.ru
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 rate of convergence O(1/k2).
Keywords: Hypodifferential, subdifferential, convex analysis, convex optimization, nonsmooth optimization, first-order method.
MSC: 90C25, 90C56, 68Q25.
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