New results on constraint qualifications for nonlinear extremum problems and extensions
type: Item
instance of: scholarly article
title: New results on constraint qualifications for nonlinear extremum problems and extensions
zbMATH Open document ID: 1327.90309
author: Lei Guo
author: Guihua Lin
author: Jin Zhang
published in: Journal of Optimization Theory and Applications
publication date: 2014-12-12T00:00:00Z
review text: The authors compare different new constraint qualifications for nonlinear optimization problems with equality and inequality constraints according to \[ \min f(x)\quad\text{s.t.}\quad g(x)\leq 0,\;h(x)= 0, \] where all functions \(f: \mathbb{R}^n\to\mathbb{R}\), \(g: \mathbb{R}^n\to\mathbb{R}^p\), \(h:\mathbb{R}^n\to\mathbb{R}^q\) are assumed to be continuously differentiable. The discussed constraint qualifications are modifications of the well-known Mangasarian-Fromovitz constraint qualification: \(*\) the constant rank Mangasarian-Fromovitz constraint qualification (CRMFCQ), \(*\) the relaxed constant positive linear dependence condition (RCPLD), \(*\) the constant rank of the subspace component condition (CRSC), \(*\) the relaxed Mangasarian-Fromovitz constraint qualification (RMFCQ). It is shown that \[ \text{CRMFCQ}\Longrightarrow \text{RCPLD} \Longrightarrow\text{CRSC/RMFCQ} \] (especially CRSC and RMFCQ are equivalent), that CRMFCQ is well-posed (i.e. this condition persists also in some feasible neighborhood) and that all the mentioned conditions imply the existence of local error bounds for the feasible set (and hence they imply the Abady constraint qualification). The results are extended to optimization problems with equilibrium constraints. It is shown that also here the associated RCPLD implies the existence of local error bounds for the mixed complementarity system.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 6379728
zbMATH Keywords: nonlinear extremum problem
zbMATH Keywords: constraint qualification
zbMATH Keywords: error bound
zbMATH Keywords: mathematical programs with equilibrium constraints
MaRDI profile type: Publication
full work available at URL: s10957-013-0510-6
OpenAlex ID: W2048843365
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DOI: 10.1007/S10957-013-0510-6
description: scientific article; zbMATH DE number 6379728