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.

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 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 Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xiaohong Hu ◽  
Zhimiao Fang ◽  
Yunxuan Xiong
Keyword(s):  
Saddle Point ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Well Posedness ◽  
Scalar Problem ◽  
New Type ◽  
Strict Efficiency ◽  
Multiplier Theorems

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimization problem involving nearly convexlike set-valued maps. Two scalarization theorems and two Lagrange multiplier theorems for strict efficiency in vector optimization involving nearly convexlike set-valued maps are established. A dual is proposed and duality results are obtained in terms of strictly efficient solutions. A new type of saddle point, called strict saddle point, of an appropriate set-valued Lagrange map is introduced and is used to characterize strict efficiency.

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 Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

2019 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Maria B. Donato ◽  
Rafaela Osuna-Gómez ◽  
Monica Milasi
Keyword(s):  
Optimality Conditions ◽  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Hadamard Manifolds ◽  
Vector Equilibrium Problems ◽  
Topological Vector Spaces ◽  
Weakly Efficient Solutions ◽  
Vector Equilibrium

The aim of this paper is to obtain Karush-Kuhn-Tucker optimality conditions for weakly efficient solutions to 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 solutions to the constrained vector optimization problem are presented. As well as some examples. The results presented in this paper generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces and real Banach spaces to Hadamard manifolds, respectively.

scholarly journals On generalized approximate convex functions and variational inequalities  

2020 ◽  
Author(s):  
Bhuwan Chandra Joshi
Keyword(s):  
Variational Inequalities ◽  
Vector Optimization ◽  
Convex Functions ◽  
Optimization Problem ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Locally Lipschitz ◽  
Approximate Vector Variational Inequalities ◽  
Approximate Efficient Solutions ◽  
Clarke Subdifferentials

In this paper, we consider a vector optimization problem involving locally Lipschitz generalized approximately convex functions and provide several concepts of approximate efficient solutions. We formulate approximate vector variational inequalities of Minty and Stampacchia type under the framework of Clarke subdifferentials and use these inequalities as a tool to characterize an approximate efficient solution of the vector optimization problem.

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.

Optimality and duality for vector optimization problem with non-convex feasible set

OPSEARCH ◽  
2019 ◽  
Vol 57 (1) ◽  
pp. 1-12 ◽  
Author(s):  
S. K. Suneja ◽  
Sunila Sharma ◽  
Priyanka Yadav
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Feasible Set ◽  
Optimality And Duality

scholarly journals Vector variational inequalities and their relations with vector optimization

2013 ◽  
Vol 4 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Surjeet Kaur Suneja ◽  
Bhawna Kohli
Keyword(s):  
Variational Inequalities ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Clarke Subdifferential ◽  
Vector Variational Inequalities ◽  
Regularity Assumption ◽  
Weak Minimum ◽  
Minimum Solution ◽  
Clarke Subdifferentials

In this paper, K- quasiconvex, K- pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where   is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) -pseudomonotonicity, (strict) K- naturally quasimonotonicity and K- quasimonotonicity of Clarke subdifferential maps have also been defined. Further, we introduce Minty weak (MVVIP) and Stampacchia weak (SVVIP) vector variational inequalities over arbitrary cones. Under regularity assumption, we have proved that a weak minimum solution of vector optimization problem (VOP) is a solution of (SVVIP) and under the condition of K- pseudoconvexity we have obtained the converse for MVVIP (SVVIP). In the end we study the interrelations between these with the help of strict K-naturally quasimonotonicity of Clarke subdifferential map.

Derivatives of Hadamard type in vector optimization

2018 ◽  
Vol 68 (2) ◽  
pp. 421-430
Author(s):  
Karel Pastor
Keyword(s):  
Nonlinear Analysis ◽  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimality Condition ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Dini Derivatives ◽  
Scalar Optimality ◽  
Derivatives Of

Abstract In our paper we will continue the comparison which was started by Vsevolod I. Ivanov [Nonlinear Analysis 125 (2015), 270–289], where he compared scalar optimality conditions stated in terms of Hadamard derivatives for arbitrary functions and those which was stated for ℓ-stable functions in terms of Dini derivatives. We will study the vector optimization problem and we show that also in this case the optimality condition stated in terms of Hadamard derivatives is more advantageous.

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