On saddle points in nonconvex semi-infinite programming
type: Item
instance of: scholarly article
title: On saddle points in nonconvex semi-infinite programming
zbMATH Open document ID: 1303.90109
author: Ralf Werner
author: Francisco Guerra Vazquez
author: Jan-Joachim Rückmann
published in: Journal of Global Optimization
publication date: 2013-01-03T00:00:00Z
review text: The Lagrangian of a nonconvex SIP (\(\min f(x) \) s.t. \(G(x,y)\geq 0\) for all \(y\in Y\), \(Y\) compact) w.r.t. the reduction approach is -- roughly spoken -- convexified by using the \(p\)-th power method of \textit{D. Li} and \textit{X. L. Sun} [J. Optimization Theory Appl. 104, No. 1, 109--120 (2000; Zbl 0960.90070)]. The main assumptions are the EMFCQ and the ESSOSC at the local solution \(\bar x\) of the SIP, and the SSOSC for each solution \(y\) of the lower level problem (LL(x): \(\min G(x,y)\) s.t. \(y\in Y\)). \(Y\) is given by a finite number of equality and inequality constraints with over all valid LICQ. The functions \(f\) and \(G\) have the not much restricted structure \(G(x,y):=g(x,y)-b(y)\), where \(g(x,y)\geq 0\), \(b(y)>0\) and \(f(x)\geq 0\) for each \(y\in Y\) and \(x\) in some neighborhood \(U(\bar x)\). Now the main steps of the tricky construction of the local saddle point property for the SIP: Each solution \(y^{B_j}\) of LL(x) with valid SSOSC defines a smooth function \(x\rightarrow y^{B_j}(x)\) over the reduced KKT-conditions with active inequalities (index set \(B_j\) arbitrary between the one of SSOSC and of KKT) as equalities by using the implicit function theorem. The corresponding Lagrangian \(L_p^B(x,\mu):=f(x)^p+\sum_{j=1}^s \mu_j(b(y^{B_j}(x))^p-g(x,y^{B_j}(x))^p) \) has a positively definite Hessian w.r.t. \(x\) at \(\bar x\) and hence is then convex in \(x\) in some neighborhood \(U(\bar x)\) for all KKT-multipliers \(\mu_j\) and for all sufficiently large \(p>0\) which yields a intermediate local saddle point statement for all possible \(L_p^B\). Now the existence theorem of \textit{M. Kojima} [in: Analysis and computation of fixed points, Proc. Symp., Univ. Wis. 1979, 93--138 (1980; Zbl 0478.90062)] for parametric behavior of local solutions of LL(x) and the resulting reduction approach of SIP with finite \(C^{1,1}\) constraints \(G(x,y^j(x))\geq 0\), \(j=1,\dots,s\), is used. Since by this theorem for each \(x\in U(\bar x)\…
reviewed by: Armin Hoffmann
Mathematics Subject Classification ID: 90C34
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 6121380
zbMATH Keywords: semi-infinite programming (SIP)
zbMATH Keywords: Lagrangian function
zbMATH Keywords: reduction approach
zbMATH Keywords: (partial) \(p\)-power formulation
zbMATH Keywords: local convexification
zbMATH Keywords: duality
zbMATH Keywords: local saddle point
zbMATH Keywords: extended Mangasarian Fromovitz constraint qualification (EMFCQ)
zbMATH Keywords: extended strong second-order sufficient condition (ESSOSC)
zbMATH Keywords: strong second-order sufficient condition (SSOSC)
zbMATH Keywords: linear independent constraint qualification
zbMATH Keywords: primal dual method
MaRDI profile type: Publication
full work available at URL: s10898-011-9753-7
OpenAlex ID: W2022364027
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DOI: 10.1007/S10898-011-9753-7
description: scientific article; zbMATH DE number 6121380