Generalized invex sets and preinvex functions on Riemannian manifolds
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)].
scientific article; zbMATH DE number 6103930
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