Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915570951127040 |
|---|---|
| author | Long, V. S. T. Mordukhovich, B. S. Nam, N. M. White, L. |
| author_facet | Long, V. S. T. Mordukhovich, B. S. Nam, N. M. White, L. |
| contents | This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces are then established. We proceed by analyzing general properties of the optimal value function, including its domain, epigraph, strict epigraph, near convexity, semicontinuity, and Lipschitz-type continuity in both convex and nonconvex settings. Subsequently, we derive calculus rules and representation formulas for the $ε$-subdifferentials of the optimal value function and its Fenchel conjugate. We then develop a duality framework for constrained optimization problems with set-valued constraints using the Fenchel conjugate for set-valued mappings. This approach provides new perspectives on duality in generalized settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality Long, V. S. T. Mordukhovich, B. S. Nam, N. M. White, L. Optimization and Control 49J52, 46N10, 90C25, 90C31 This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces are then established. We proceed by analyzing general properties of the optimal value function, including its domain, epigraph, strict epigraph, near convexity, semicontinuity, and Lipschitz-type continuity in both convex and nonconvex settings. Subsequently, we derive calculus rules and representation formulas for the $ε$-subdifferentials of the optimal value function and its Fenchel conjugate. We then develop a duality framework for constrained optimization problems with set-valued constraints using the Fenchel conjugate for set-valued mappings. This approach provides new perspectives on duality in generalized settings. |
| title | Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality |
| topic | Optimization and Control 49J52, 46N10, 90C25, 90C31 |
| url | https://arxiv.org/abs/2507.07377 |