Generalized invex sets and preinvex functions on Riemannian manifolds
type: Item
instance of: scholarly article
title: Generalized invex sets and preinvex functions on Riemannian manifolds
zbMATH Open document ID: 1263.53032
author: Ravi P. Agarwal
author: Akhlad Iqbal
author: Shahid Ali
author: I. Ahmad
published in: Taiwanese Journal of Mathematics
publication date: 2012-11-09T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2382
review text: This paper introduces a notion of geodesic \(\alpha\)-invex subset of a Riemannian manifold, which generalizes \textit{M. A. Hanson}'s notion of invexity [J. Math. Anal. Appl. 80, 545--550 (1981; Zbl 0463.90080)]. Invexity may be seen as a weakened convexity condition and has applications to optimization [\textit{V. Jeyakumar}, Methods Oper. Res. 55, 109--125 (1985; Zbl 0566.90086)]. After a brief introduction which puts this work into the context of pre-existing literature, the authors recall geometric preliminaries in \S 2. The definitions of geodesic \(\alpha\)-invex set, \(\alpha\)-invex function, geodesic \(\alpha\)-preinvex function, and examples of Cartan-Hadamard manifolds are presented in \S 3. Proposition 3.1 shows that every lower section of a geodesic \(\alpha\)-preinvex function is a geodesic \(\alpha\)-invex set. In \S 4, a geometric property (P) and a related condition (C) are defined and used in Theorems 4.1 and 4.2 to relate the notions of geodesic \(\alpha\)-preinvex function and \(\alpha\)-invex function. In \S 5, the authors prove Theorem 5.1 which shows that if a geodesic \(\alpha\)-preinvex function on an open geodesic \(\alpha\)-invex subset \(S\) of a Riemannian manifold has a local minimum, then it is a global minimum on \(S\). Theorem 5.2 gives an estimate for the proximal subdifferential of lower semicontinuous geodesic \(\alpha\)-preinvex functions. This is applied in Corollary 5.1 to show that if \(0\) is contained in the proximal subdifferential of a lower semicontinuous geodesic \(\alpha\)-preinvex function at a point of an open geodesic \(\alpha\)-invex set \(S\), then the function has a global minimum on \(S\) at that point. The last section of the paper presents a mean value inequality and mean value theorem for Cartan-Hadamard manifolds which extend the results of \textit{T. Antczak} [Nonlinear Anal., Theory Methods Appl. 60, No. 8, A, 1473--1484 (2005; Zbl 1100.26005)].
reviewed by: Julie Rowlett
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 32F17
Mathematics Subject Classification ID: 49Q99
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 6103930
zbMATH Keywords: geodesic \(\alpha\)-invex set
zbMATH Keywords: \(\alpha\)-invex function
zbMATH Keywords: geodesic \(\alpha\)-preinvex function
zbMATH Keywords: invexity
zbMATH Keywords: Cartan-Hadamard manifold
MaRDI profile type: Publication
OpenAlex ID: W4249123670
Wikidata QID: Q115242957
DOI: 10.11650/TWJM/1500406792
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description: scientific article; zbMATH DE number 6103930