Second order optimality conditions for the \(L_ 1\)-minimization problem
type: Item
instance of: scholarly article
title: Second order optimality conditions for the \(L_ 1\)-minimization problem
zbMATH Open document ID: 0566.49013
DOI: 10.1007/BF01442198
published in: Applied Mathematics and Optimization
publication date: 1985-01-01T00:00:00Z
review text: Second order necessary and second order sufficient conditions are obtained for the unconstrained minimization of a functional \(f(x)=\int^{b}_{a}| g(x,t)| dt\). They are expressed in terms of the directional derivative, f'(x;h), and the curved second directional derivative, \[ f''(x:h,z)=\lim_{\lambda \to 0+}\lambda^{- 2}[f(x+\lambda h+\lambda^ 2z)-f(x)-\lambda f'(x;h)]. \] These derivatives are obtained explicitly for the given f(\(\cdot)\). An example is calculated.
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 58C20
zbMATH DE Number: 3903411
zbMATH Keywords: Second order necessary and second order sufficient conditions
zbMATH Keywords: unconstrained minimization
zbMATH Keywords: directional derivative
author: Aharon Ben-Tal
author: J. Zowe
reviewed by: B. D. Craven
MaRDI profile type: Publication
cites work: A unified theory of first and second order conditions for extremum problems in topological vector spaces
cites work: Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems
cites work: Technical note: Directional derivatives in nonsmooth optimization
cites work: Q5544244
cites work: Nonlinear \(L_1\) approximation of smooth functions
full work available at URL: https://doi.org/10.1007/bf01442198
OpenAlex ID: W2998406152
description: scientific article; zbMATH DE number 3903411