Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds

Journal of Applied Mathematics, Mar 2014

We introduce a class of functions called geodesic -preinvex and geodesic -invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudo -preinvex and geodesic quasi/pseudo -invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic -preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic -invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.

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Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds

Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds Sheng-lan Chen,1,2 Nan-Jing Huang,1 and Donal O'Regan3,4 1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2School of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland 4Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Received 10 January 2014; Accepted 21 February 2014; Published 26 March 2014 Academic Editor: Xian-Jun Long Copyright © 2014 Sheng-lan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce a class of functions called geodesic -preinvex and geodesic -invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudo -preinvex and geodesic quasi/pseudo -invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic -preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic -invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming. 1. Introduction Convex functions play an important role in optimization theory and there are several classes of functions given in the literature with the goal to weaken the limitations of convexity in mathematical programming. Generalized convex functions, labelled as -vex functions, were introduced by Bector and Singh [1]. In 1981, Hanson [2] introduced the concept of invexity and proved that the Kuhn-Tucker conditions are sufficient for optimality of a nonlinear programming problem under invexity conditions. Preinvex functions were defined by Ben-Israel and Mond [3], and, in [4], Weir and Mond showed how and where preinvex functions could replace convex functions in multiple objective optimization problem. These functions were further generalized to pseudo/quasi -vex, -invex, and pseudo/quasi -invex functions by Bector et al. [5] and to -preinvex by Suneja et al. [6]. In [5], Bector et al. obtained sufficient optimality criteria and duality results for a nonlinear programming problem involving -vex and -invex functions. There are also many papers in the literature concerning the generalization of convexity in connection with sufficiency and duality in optimization problems (see, e.g., [7–12] and the references therein). A manifold is not a linear space and extensions of concepts and techniques from linear spaces to Riemannian manifolds are natural. In the literature many authors studied generalized convex functions and many results in convex analysis and optimization theory were extended to Riemannian manifolds (see [13–28] and the references therein). Rapcsák [27] and Udriste [28] considered a generalization of convexity called geodesic convexity. In this setting the linear space is replaced by a Riemannian manifold and the line segment by a geodesic. Pini [22] introduced the notion of invex function on Riemannian manifolds, while Mititelu [24] investigated its generalization. The concepts of geodesic invex sets, geodesic invex, and preinvex functions on Riemannian manifolds were defined by Barani and Pouryayevali [17]. They established the relation between geodesic invexity and preinvexity of functions, and they also obtained results concerning extremum points of a nonsmooth geodesic preinvex function by using the proximal subdifferential. Subsequently, Agarwal et al. [20] proposed and discussed geodesic -preinvexity on Riemannian manifolds, which generalized the corresponding results studied by Barani and Pouryayevali [17]. A new concept of geodesic roughly -invexity and its generalization on Hadamard manifolds were introduced by Zhou and Huang [26]. They studied the properties of these functions and they established sufficient optimality conditions and duality in nonlinear programming problems. In this paper, we introduce a class of geodesic -preinvex and -invex functions on Riemannian manifolds and extend them to geodesic quasi/pseudo -preinvex and geodesic quasi/pseudo -invex functions. We discuss the links among these functions under suitable assumptions. By applying the proximal subdifferential, we relax the smoothness condition and study the question of global minima for geodesic -preinvex functions on Riemannian manifolds. As applications, we investigate a multiobjective programming problem involvin (...truncated)


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Sheng-lan Chen, Nan-Jing Huang, Donal O'Regan. Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds, Journal of Applied Mathematics, 2014, 2014, DOI: 10.1155/2014/524698