scholarly journals Finding Vector Critical Points on Hadamard Manifolds: Nonsmooth Case

2020 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana
Keyword(s):  
Vector Optimization ◽  
Critical Points ◽  
Optimization Problems ◽  
Equilibrium Points ◽  
Efficient Solutions ◽  
Hadamard Manifolds ◽  
Weakly Efficient Solutions ◽  
Economic Application ◽  
Payoff Functions ◽  
First Time

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.

scholarly journals Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 804
Author(s):  
Gabriel Ruiz-Garzón ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana
Keyword(s):  
Critical Points ◽  
Symmetric Spaces ◽  
Optimization Problems ◽  
Equilibrium Points ◽  
Efficient Solutions ◽  
Hadamard Manifolds ◽  
Weakly Efficient Solutions ◽  
Payoff Functions ◽  
Riemannian Symmetric Spaces ◽  
First Time

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.

scholarly journals Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2196
Author(s):  
Gabriel Ruiz-Garzón ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana ◽  
Beatriz Hernández-Jiménez
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Equilibrium Points ◽  
Variational Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Stackelberg Equilibrium ◽  
Hadamard Manifolds ◽  
Geodesic Convexity ◽  
Weakly Efficient Solutions

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.

Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Renying Zeng
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Lagrangian Multipliers ◽  
Weakly Efficient Solutions ◽  
Affine Maps ◽  
Convex Vector Optimization ◽  
Affine Set

Abstract In this paper, we introduce some definitions of generalized affine set-valued maps: affinelike, preaffinelike, nearaffinelike, and prenearaffinelike maps. We present examples to explain that our definitions of generalized affine maps are different from each other. We derive a theorem of alternative of Farkas–Minkowski type, discuss Lagrangian multipliers for constrained set-valued optimization problems, and obtain some optimality conditions for weakly efficient solutions.

scholarly journals Generalized Stampacchia Vector Variational-Like Inequalities and Vector Optimization Problems Involving Set-Valued Maps

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanfei Chai ◽  
Sanyang Liu ◽  
Guotao Wang
Keyword(s):  
Vector Optimization ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Gap Functions ◽  
Weak Formulation ◽  
Existence Of A Solution ◽  
Weakly Efficient Solutions ◽  
Set Valued Mapping ◽  
Finite Dimensional

We first obtain that subdifferentials of set-valued mapping from finite-dimensional spaces to finite-dimensional possess certain relaxed compactness. Then using this weak compactness, we establish gap functions for generalized Stampacchia vector variational-like inequalities which are defined by means of subdifferentials. Finally, an existence result of generalized weakly efficient solutions for vector optimization problem involving a subdifferentiable and preinvex set-valued mapping is established by exploiting the existence of a solution for the weak formulation of the generalized Stampacchia vector variational-like inequality via a Fan-KKM lemma.

scholarly journals On V-r-invexity and vector variational-like inequalities

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 1065-1073 ◽  
Author(s):  
S.K. Mishra ◽  
Vivek Laha
Keyword(s):  
Multiobjective Optimization ◽  
Vector Optimization ◽  
Critical Points ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Multiobjective Optimization Problems ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Weak Vector

In this paper, we consider the multiobjective optimization problems involving the differentiable V-r-invex vector valued functions. Under the assumption of V-r-invexity, we use the Stampacchia type vector variational-like inequalities as tool to solve the vector optimization problems. We establish equivalence among the vector critical points, the weak efficient solutions and the solutions of the Stampacchia type weak vector variational-like inequality problems using Gordan?s separation theorem under the V-r-invexity assumptions. These conditions are more general than those appearing in the literature.

scholarly journals Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

2020 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana ◽  
Beatriz Hernández-Jiménez
Keyword(s):  
Equilibrium Points ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Stackelberg Equilibrium ◽  
Hadamard Manifolds ◽  
Geodesic Convexity ◽  
Weakly Efficient Solutions ◽  
Minimum Requirements

This article has two objectives. Firstly, we will use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of Vector Optimization Problem within the novel field of the Hadamard manifolds. Previously, we will introduce the concepts of generalized approximate geodesic convex functions and illustrate them with examples. We will 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 will show an economical application, using again solutions of the variational problems to identify with Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

scholarly journals On the Generic Uniqueness of Pareto-Efficient Solutions of Vector Optimization Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Dejin Zhang ◽  
Shuwen Xiang ◽  
Yanlong Yang ◽  
Xicai Deng
Keyword(s):  
Vector Optimization ◽  
Efficient Solution ◽  
Optimization Problems ◽  
Baire Category ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Weakly Efficient Solution ◽  
Weakly Efficient Solutions ◽  
Pareto Efficient ◽  
Generic Uniqueness

In this paper, the generic uniqueness of Pareto weakly efficient solutions, especially Pareto-efficient solutions, of vector optimization problems is studied by using the nonlinear and linear scalarization methods, and some further results on the generic uniqueness are proved. These results present that, for most of the vector optimization problems in the sense of the Baire category, the Pareto weakly efficient solution, especially the Pareto-efficient solution, is unique. Furthermore, based on these results, the generic Tykhonov well-posedness of vector optimization problems is given.

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