Hypodifferentials of nonsmooth convex functions and their applications to nonsmooth convex optimization
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910894886223872 |
|---|---|
| author | Dolgopolik, M. V. |
| author_facet | Dolgopolik, M. V. |
| contents | 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 $\mathcal{O}(1/k^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13464 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hypodifferentials of nonsmooth convex functions and their applications to nonsmooth convex optimization Dolgopolik, M. V. Optimization and Control 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 $\mathcal{O}(1/k^2)$. |
| title | Hypodifferentials of nonsmooth convex functions and their applications to nonsmooth convex optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2303.13464 |