{"ID":2895924,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.07377","arxiv_id":"2507.07377","title":"Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2507.07377","url_pdf":"https://arxiv.org/pdf/2507.07377v3","authors":"[\"V. S. T. Long\",\"B. S. Mordukhovich\",\"N. M. Nam\",\"L. White\"]","published":"2025-07-10T02:12:21Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}