Coderivatives in parametric optimization
type: Item
instance of: scholarly article
title: Coderivatives in parametric optimization
zbMATH Open document ID: 1079.90136
author: A. B. Levy
author: Boris S. Mordukhovich
published in: Mathematical Programming. Series A. Series B
publication date: 2004-07-01T00:00:00Z
review text: The paper is devoted to sensitivity analysis of constrained optimization problems and the main attention is paid to the study of Lipschitzian stability of solution maps in optimization problems (particularly stationary point multifunctions and stationary point-multiplier multifunctions in parametric mathematical programming). The basic tools involve derivative-like objects for set-valued mappings, called coderivatives, which give extensions of the classical adjoint derivative operator and play a role in characterizations of Lipschitzian behaviour and metric regularity for general multifunctions. The authors focus on a family of parametrized minimization problems where the sum of the parametrized objective function plus a parametrized function that incorporates the constraints (it is called the constraint function) is minimized. The constraint function is lower semicontinuous and extended real-valued. The authors use coderivatives to analyze stationary point multifunctions, which represent the different stationary points associated with each parameter value, by developing a general implicit mapping theorem in order to get an estimate for the coderivative of the stationary point multifunction in terms of the coderivative of the partial subgradient mapping corresponding to the constraint function. A general partial coderivative estimate in terms of the full coderivative is developed, which results in an estimate in terms of the second-order subdifferential of the constraint function. The paper focus attention to the estimation of such second-order sub\-differential, paying particular attention on the case when the constraint function is a special kind of composite function called ``strongly amenable'', which means a composite of a convex, lower semicontinuous function with a two times continuously differentiable mapping on a neighbourhood of the point \(x\) and the parameter \(w\) that fulfilled a constraint qualification which for nonlinear problems coincides with the…
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 2077961
zbMATH Keywords: parametric optimization
zbMATH Keywords: variational analysis
zbMATH Keywords: sensitivity
zbMATH Keywords: Lipschitzian stability
zbMATH Keywords: generalized differentiation
zbMATH Keywords: coderivatives
reviewed by: Francisco Guerra Vazquez
MaRDI profile type: Publication
full work available at URL: s10107-003-0452-0
OpenAlex ID: W2068025262
DOI: 10.1007/S10107-003-0452-0
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description: scientific article; zbMATH DE number 2077961