On the Quantitative Solution Stability of Parameterized Set-Valued Inclusions

• Published in: Set-valued and variational analysis Vol. 29; no. 2; pp. 425 - 451
• Main Author: Uderzo, A.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2021; Springer Nature B.V
• Subjects: Analysis; Banach spaces; Inclusions; Mathematics; Mathematics and Statistics; Optimization; Parameterization; Robustness (mathematics); Stability; Optimal value function; Parametric optimization; Secondary: 49J52, 90C31, 90C48; Primary: 49J53; Set-valued inclusion; Solution mapping; Lipschitz semicontinuity; Calmness
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-020-00571-z

The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Consequences on the calmness of the optimal value function, in the context of parametric optimization, are explored. Some specific tools for the analysis of the sufficient conditions, in the case of set-valued inclusion with concave multifunction term, are provided in a Banach space setting.