846 research outputs found
Equilibrium problems on Riemannian manifolds with applications
We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent
for the equilibrium problem on Hadamard manifolds.National Natural Science Foundation of ChinaNatural Science Foundation of Guizhou Province (China)Dirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa
Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities
This article has two objectives. Firstly, we use the vector variational-like inequalities
problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem
within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of
generalized approximate geodesic convex functions and illustrated them with examples. We see the
minimum requirements under which critical points, solutions of Stampacchia, and Minty weak
variational-like inequalities and local approximate weakly efficient solutions can be identified,
extending previous results from the literature for linear Euclidean spaces. Secondly, we show
an economical application, again using solutions of the variational problems to identify Stackelberg
equilibrium points on Hadamard manifolds and under geodesic convexity assumptions
Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions
In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results
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
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
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