Whitney differentiability of optimal-value functions for bound-constrained convex programming problems

In the spirit of the Whitney Extension Theorem, consider a function on a compact subset of Euclidean space to be 'Whitney-differentiable' if it is a restriction of a continuously Fréchet-differentiable function with an open domain. Whitney-differentiable functions have been shown to have u...

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Vydáno v:Optimization Ročník 68; číslo 2-3; s. 691 - 711
Hlavní autor: Khan, Kamil A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia Taylor & Francis 04.03.2019
Taylor & Francis LLC
Témata:
convex underestimators
Convexity
Domains
Euclidean geometry
Euclidean space
Global optimization
Mathematical programming
Parametric optimization
Pareto optimum
Quotients
smoothing methods
ISSN:0233-1934, 1029-4945
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Shrnutí:In the spirit of the Whitney Extension Theorem, consider a function on a compact subset of Euclidean space to be 'Whitney-differentiable' if it is a restriction of a continuously Fréchet-differentiable function with an open domain. Whitney-differentiable functions have been shown to have useful (yet possibly nonunique) derivatives and calculus properties even on the boundaries of their domains. This article shows that optimal-value functions for bound-constrained convex programmes with Whitney-differentiable objective functions are themselves Whitney-differentiable, even when the linear-independence constraint qualification is not satisfied. This result extends classic sensitivity results for convex programmes, and generalizes recent work. As an application, sufficient conditions are presented for generating continuously differentiable convex underestimators of nonconvex functions for use in methods for deterministic global optimization in the multivariate McCormick framework. In particular, the main result is applied to generate Whitney-differentiable convex underestimators for quotients of functions with known Whitney-differentiable relaxations.
Bibliografie:ObjectType-Article-1
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2018.1534108