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...

Full description

Saved in:
Bibliographic Details
Published in:Optimization Vol. 68; no. 2-3; pp. 691 - 711
Main Author: Khan, Kamil A.
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 04.03.2019
Taylor & Francis LLC
Subjects:
convex underestimators
Convexity
Domains
Euclidean geometry
Euclidean space
Global optimization
Mathematical programming
Parametric optimization
Pareto optimum
Quotients
smoothing methods
ISSN:0233-1934, 1029-4945
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2018.1534108