Dual and bidual problems for a Lipschitz optimization problem based on quasi-conjugation
type: Item
instance of: scholarly article
title: Dual and bidual problems for a Lipschitz optimization problem based on quasi-conjugation
zbMATH Open document ID: 1228.90119
author: Syuuji Yamada
author: Tamaki Tanaka
author: Tetsuzo Tanino
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-03-14T00:00:00Z
review text: This paper deals with a Lipschitz optimization problem (LOP) constrained by linear functions in \(\mathbb R^{n}\) which is one of the most important problems in global optimization. The Lipschitz optimization problem is transformed into a certain problem (MP) constrained by a ball in \(\mathbb R^{n+1}\). Besides, for (MP), the authors define the dual problem (DP) and the bidual problem (BDP) by applying the quasi-conjugate and quasi-biconjugate functions defined by \textit{P. T. Thach} [J. Math. Anal. Appl. 159, No. 2, 299--322 (1991; Zbl 0734.90073)]. By the properties of quasi-conjugation, the objective functions of (DP) and (BDP) are quasi-convex. Despite there is no guarantee that the objective function of (MP) is quasi-convex, the authors prove that each optimal value of (MP) and (DP) coincides with the multiplication of the other by \(-1\), and that each optimal solution of them is computed by dividing the other by the square of its norm. Moreover, it is also shown that both sets of all globally optimal solutions of (MP) and (BDP) coincide. Furthermore, by using the quasi-convexity of the objective function of (DP), the authors propose the overestimation procedure and an outer approximation algorithm for (DP).
reviewed by: Samir Kumar Neogy
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 5865650
zbMATH Keywords: Lipschitz optimisation
zbMATH Keywords: duality
zbMATH Keywords: quasi-conjugate function
zbMATH Keywords: quasi-biconjugate functions
zbMATH Keywords: quasi-convexity
zbMATH Keywords: outer approximation method
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.12.046
OpenAlex ID: W2050712423
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DOI: 10.1016/J.JMAA.2010.12.046
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description: scientific article; zbMATH DE number 5865650