Estimates of Generalized Hessians for Optimal Value Functions in Mathematical Programming

We consider the optimal value function of a parametric optimization problem. A large number of publications have been dedicated to the study of continuity and differentiability properties of the function. However, the differentiability aspect of works in the current literature has mostly been limite...

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Vydáno v:Set-valued and variational analysis Ročník 30; číslo 3; s. 847 - 871
Hlavní autor: Zemkoho, Alain B.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Dordrecht Springer Netherlands 01.09.2022
Springer Nature B.V
Témata:
Algorithms
Analysis
Derivatives
Estimates
Functions (mathematics)
Lagrange multiplier
Mathematical analysis
Mathematical programming
Mathematics
Mathematics and Statistics
Newton methods
Optimization
Original Research
Robustness (mathematics)
90C31
Parametric optimization
Optimal value function
Sensitivity analysis
90C30
Generalized Hessian
Second order subdifferential
Coderivative
ISSN:1877-0533, 1877-0541
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Popis
Shrnutí:We consider the optimal value function of a parametric optimization problem. A large number of publications have been dedicated to the study of continuity and differentiability properties of the function. However, the differentiability aspect of works in the current literature has mostly been limited to first order analysis, with focus on estimates of its directional derivatives and subdifferentials, given that the function is typically nonsmooth. With the progress made in the last two to three decades in major subfields of optimization such as robust, minmax, semi-infinite and bilevel optimization, and their connection to the optimal value function, there is a need for a second order analysis of the generalized differentiability properties of this function. This could enable the development of robust solution algorithms, such as the Newton method. The main goal of this paper is to provide estimates of the generalized Hessian for the optimal value function. Our results are based on two handy tools from parametric optimization, namely the optimal solution and Lagrange multiplier mappings, for which completely detailed estimates of their generalized derivatives are either well-known or can easily be obtained.
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ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-021-00623-y