Optimality conditions in mathematical programming and composite optimization
type: Item
instance of: scholarly article
title: Optimality conditions in mathematical programming and composite optimization
zbMATH Open document ID: 0828.90137
DOI: 10.1007/BF01582222
author: Jean-Paul Penot
published in: Mathematical Programming. Series A. Series B
publication date: 1995-02-23T00:00:00Z
review text: In a very clear manner and with all important proofs new second-order necessary optimality conditions are derived for the problems \(({\mathcal M})\): \(j(x)\to \inf\), s.t. \(x\in B\cap k^{- 1}(C)\), and \(({\mathcal C})\): \(f(x):= g(h(x))\to \inf\), s.t. \(x\in D\) where \(j: X\to \mathbb{R}\), \(k: X\to Z\) are twice differentiable at some point \(a\) of \(F:= B\cap k^{- 1}(C)\), \(B\) and \(C\) being closed convex subsets of the Banach spaces \(X\), and \(Z\), resp. and where \(h: X\to Y\) is twice differentiable, \(g: Y\to \mathbb{R}\) is a closed proper convex function, \(Y\) is a Banach space and \(D\) is a closed convex subset of \(X\). At first the author derives a general second-order optimality condition for the problem \(({\mathcal P})\): \(f(x)\to \inf\), s.t. \(x\in F\) where \(f: X\to \mathbb{R}\) defines a continuous linear map \(f'(a): X\to \mathbb{R}\) and a continuous bilinear map \(f''(a): X\times X\to \mathbb{R}\) such that \(\lim_{(t, u)\to (0, x)} t^{- 2}[f(a+ tu)- f(a)- tf'(a) u- 0.5t^2 f''(a)uu]= 0\) for all \(x\in X\) and where the feasible set \(F\) belongs to an arbitrary topological vector space \(X\). Using the contingent cone \(F'(a)\) to \(F\) at \(a\in F\) and the (superior) second-order tangent set \(F''(a, v)\) to \(F\) at \(a\in F\) in the direction \(v\in X\) defined by \(F''(a, v):= \limsup_{t\to 0+} 2t^{- 2}(F- a- tv)\) the following second-order condition holds: If \(a\) is a local solution of \(({\mathcal P})\) then (i) \(f'(a)v\geq 0\) for each \(u\in F'(a)\) and (ii) \((a)vv+ \lim_{(t, u)\to (0, x), a+ tu\in F} f'(a)t^{- 1}(u- v)\geq 0\) for each \(v\in F'(a)\cap \ker f'(a)\). Under metrical regularity for the mapping \(k\) (some Ljusternik-like condition) and some Zowe/Kurcyusz-like regularity condition of \(k'\) w.r.t. \(B'(a)\) and \(C'(k'(a))\) a second-order condition of Lagrange type is derived from the above geometrical condition for \(({\mathcal M})\). Further the equivalence of the problems \(({\mathcal M})\) and …
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 707412
zbMATH Keywords: second-order necessary optimality conditions
zbMATH Keywords: contingent cone
reviewed by: Armin Hoffmann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 707412