Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap
type: Item
instance of: scholarly article
title: Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap
zbMATH Open document ID: 1395.90237
author: William Echegaray
author: Fernando Flores-Bazán
author: Eladio Ocaña
author: Fabián Flores-Bazan
published in: Journal of Global Optimization
publication date: 2017-12-14T00:00:00Z
review text: This paper deals with cone-constrained optimization problems of the form \[ \mu :=\inf_{x\in K}f\left( x\right), \tag{1} \] where \(f:X\longrightarrow \mathbb{R\cup }\left\{ +\infty \right\}\), \( K=\left\{ x\in C:g\left( x\right) \in -P\right\}\), with \(C\subset X\), \( P\subset Y\), \(g:X\longrightarrow Y\), \(X,Y\) normed spaces (even though the results could be straightforwardly extended to locally convex Hausdorff topological vector spaces by using nets instead of sequences), \(P\) is a convex cone, and \(K\neq \emptyset\). The Lagrange dual of (1) is \[ \upsilon :=\sup_{\lambda \in P^{\ast }}\inf_{x\in C}\left\{ f\left( x\right) +\left\langle \lambda ,g\left( x\right) \right\rangle \right\}. \tag{2} \] The paper provides known and new results guaranteeing zero-duality gap together with the solvability of either (1) or (2), situations described as primal or dual strong duality (primal strong duality is also called reverse strong duality in the literature). The authors appeal to a perturbational scheme based on the following parametric problem (to be interpreted as a perturbation of (1)): \[ \psi \left( a\right) :=\inf_{x\in K(a)}f\left( x\right) , \tag{3} \] where \(K(a)=\left\{ x\in C:g\left( x\right) \in a-P\right\}\). So, (1) is embedded into (3) as \(K(0)=K\) and \(\psi \left( 0\right) =\mu\). The main results are Theorem 4.4, characterizing the primal strong duality in terms of the closedness of the set \(\left\{ \left( f\left( x\right) ,g\left( x\right) \right) :x\in C\right\} +\mathbb{R}_{+}\times P\) (expression depending on the data) relative to vertical half-lines contained in the line \(\mathbb{R}\times \left\{ 0\right\} \subset \mathbb{R}\times Y\) (or, equivalently, \(\overline{\psi }\left( 0\right) =\psi \left( 0\right) \) together with the solvability of (1), by Proposition 2.2), and Theorem 4.2, characterizing the dual strong duality by the condition that \(\partial \psi \left( 0\right) \neq \emptyset\). The last two sections provide, thanks to…
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 90C20
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 90C25
zbMATH DE Number: 6818683
zbMATH Keywords: zero duality gap
zbMATH Keywords: strong duality
zbMATH Keywords: linear fractional programming
zbMATH Keywords: quasiconvex programming
zbMATH Keywords: quadratic programming
reviewed by: Miguel Angel Goberna
MaRDI profile type: Publication
full work available at URL: s10898-017-0542-9
OpenAlex ID: W2725387418
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DOI: 10.1007/S10898-017-0542-9
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