Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds

Abstract

In this paper, we consider a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MMPEC)). We introduce the generalized Guignard constraint qualification for (MMPEC) and employ it to derive Karush–Kuhn–Tucker (KKT)-type necessary optimality criteria. Further, we derive sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. The significance of the results deduced in the paper has been demonstrated by suitable non-trivial examples. The results deduced in this article generalize several well-known results in the literature to a more general space, that is, Hadamard manifolds, and extend them to a more general class of optimization problems. To the best of our knowledge, this is the first time that generalized Guignard constraint qualification and optimality conditions have been studied for (MMPEC) in manifold settings.

Pareto Efficiency Criteria and Duality for Multiobjective Fractional Programming Problems with Equilibrium Constraints on Hadamard Manifolds

“…The novelty and the contributions of the paper are two-fold. Firstly, the results explored in this article generalize the corresponding results presented by [7] for a broader category of optimization problems, that is, MFPPEC. Secondly, the results investigated by [6] are extended for MFPPEC class in the Hadamard manifolds setting by the results presented in this article.…”

“…MPECs have been extensively used to model various real-life problems appearing in several fields of science and technology, for instance, the hydro-economic river basin model [2], process engineering [3], traffic and telecommunications networks [4], cyber attacks in electricity market [5], etc. For further details and updated surveys of MPEC and its applications, we refer the readers to [6][7][8][9][10] and the references cited therein.…”

“…KKT-type criteria for optimality, as well as some duality models for multiobjective MPEC, were deduced by Singh and Mishra [6]. Besides this, optimality conditions for multiobjective MPEC on Hadamard manifolds have been studied by Treanţȃ et al [7]. However, multiobjective fractional programming problems with equilibrium constraints have not yet been studied in the framework of manifolds.…”

“…Motivated by the results presented in [6][7][8]23,25], a class of multiobjective fractional programming problems with equilibrium constraints MFPPEC is studied in this article, in the setting of Hadamard manifolds. First, we present GGCQ for MFPPEC in the setting of Hadamard manifolds.…”

Optimality Conditions and Duality for MultiobjectiveSemi-infinite Optimization Problems with SwitchingConstraints on Hadamard Manifolds

“…Further, in various őelds related to engineering, technology and science, several optimization problems arise which can be formulated in a more effective manner in the framework of Riemannian/Hadamard manifold, rather than on Euclidean space setting, see [6,40,52] and the references cited therein. Extending as well as generalizing different techniques involved in optimization theory from the setting of Euclidean spaces to the setting of manifolds are associated with numerous crucial advantages, such as: (a) Several complicated constrained mathematical optimization problems can be conveniently converted into unconstrained problems, and thereby the complexity of the original problem can be reduced (see, [11,49] and the references cited therein). (b) Numerous non-convex programming problems can be converted into convex problems by utilizing Riemannian geometry framework (see, [37,38]).…”