11 research outputs found
Roughly geodesic B - r-preinvex functions on Cartan Hadamard manifolds
In this article, we introduce a new class of functions called roughly geodesic B????r????preinvex on a Hadamard manifold and establish some properties of roughly geodesic B - r-preinvex functions on Hadamard manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under roughly geodesic B-r- preinvexity on Hadamard manifolds. The results presented in this paper extend and generalize the results appeared in the literature
Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions
The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush-Kuhn-Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash's critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces
Generalized geodesic convex functions on Riemannian manifolds
In the present paper, we introduce the generalized geodesic convex functions on Riemannian manifolds and present some of their properties. Based on these properties, the generalized geodesic star-shaped functions are established. Results obtained in this paper may inspire future research in convex analysis on manifolds
Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds
The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker
optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the
addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical
examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient
conditions for weakly efficient Pareto points to the constrained vector optimization problem are
presented. The results described in this article generalize results obtained by Gong (2008) andWei
and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed
spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of
Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds
Efficiency for Vector Variational Quotient Problems with Curvilinear Integrals on Riemannian Manifolds via Geodesic Quasiinvexity
In the paper, we analyze the necessary efficiency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufficient conditions of efficiency to the above variational problems. The efficiency sufficient conditions use of notions of the geodesic invex set and of (strictly, monotonic) ( ρ , b)-geodesic quasiinvex functions
Invex Programs: First Order Algorithms and Their Convergence
Invex programs are a special kind of non-convex problems which attain global
minima at every stationary point. While classical first-order gradient descent
methods can solve them, they converge very slowly. In this paper, we propose
new first-order algorithms to solve the general class of invex problems. We
identify sufficient conditions for convergence of our algorithms and provide
rates of convergence. Furthermore, we go beyond unconstrained problems and
provide a novel projected gradient method for constrained invex programs with
convergence rate guarantees. We compare and contrast our results with existing
first-order algorithms for a variety of unconstrained and constrained invex
problems. To the best of our knowledge, our proposed algorithm is the first
algorithm to solve constrained invex programs
A Note on -Convex Fuzzy Processes
We first define the concept of -convex fuzzy processes. Second, we present some basic properties of such processes