Vector Variational-like Inequality Problems Research Articles

Strongly geodesic preinvexity and strongly invariant η-monotonicity on Riemannian manifolds and its application

This paper introduces the concepts of strongly geodesic preinvexity, strongly η-invexity of order m, and strongly invariant η-monotonicity of order m on Riemannian manifolds. Additionally, it discusses an important characterization of these functions under a condition, known as Condition C (The Condition C is defined in Remark 1 of this article), defined by Barani and Pouryayevali [J. Math. Anal. Appl. 328 (2007) 767–779]. The paper provides various non-trivial examples to support these definitions. Furthermore, it presents a significant characterization of strict η-minimizers (or η-minimizers) of order m for multi-objective optimization problems and a solution to the vector variational-like inequality problem.

Topological solutions of η-generalized vector variational-like inequality problems

Topological solutions of η-generalized vector variational-like inequality problems

A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in the proof. The admissibility of the function space topology and KKM-Theorem have played important role in proving the results.

Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

Existence results for generalized mixed vector variational-like inequality problems with exponential type invexities

In this paper, we study a new kind of existence of solution for set valued exponential type mixed vector variational-like inequality problem in Euclidean space and proposed αg-relaxed exponentially (γ,η)-monotone mapping. Moreover, we established an example in order to illustrate the main problem. We proved the existence results by KKM-technique with αg-relaxed exponentially (γ,η)-monotone mapping. Further, we give some consequences of the main result. The results presented in this paper unifies and extends some known results in this area.

New generalised mixed vector variational-like inequalities with semi-η-pseudomonotonicity

In this paper, we introduce the concept of semi-η-pseudomonotonicity for vector set-valued mappings. Using this concept, and utilising KKM technique and Nadler's Lemma, we establish some existence results for the new generalised mixed vector variational-like inequality problem in reflexive Banach spaces. Our results are the extension and improvement of some of the results given by Plubtieng and Thammathiwat (2014). Few examples are given to justify our new findings.

On vector variational-like inequalities and vector optimization problems with (G, α)-invexity

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.

Extended new generalized mixed vector variational-like inequalities in reflexive Banach spaces

In this paper, we introduce and study an extended new generalized mixed vector variational-like inequality (ENGMVVLI) problem in real reflexive Banach spaces. This problem is more general than many existing equilibrium problems and variational inequalities available in the literature. We prove the existence results of (ENGMVVLI) by utilizing the KKM lemma and Nadlar’s results with relaxed semi- α-monotone mappings. Also, by using the Brouwer’s fixed point theorem, we prove the existence of solutions for (ENGMVVLI) without using the monotonicity concept. Finally, we give an example to justify the validity of our results.

On composite vector variational-like inequalities and vector optimization problems

In this paper, we introduce a class of (weak) composite vector variational-like inequality problems and establish its relationship with composite vector optimization problem. We also prove the relation of a vector critical point in composite vector optimization problem with its weak efficient point, under the assumption of composite pseudo invexity. Using KKM Lemma, we derive result for existence of solutions of composite vector variational-like inequality problem. Furthermore, we define a gap function for the composite vector variational-like inequality problem and finally, as an application, we study a system of composite vector optimization problems and system of vector variational-like inequality problems, whose solutions imply the solution of Nash equilibrium problem. Examples are provided to illustrate the derived results.

Efficiency through variational-like inequalities with Lipschitz functions

In this work, first we introduce several notions of invexity and pseudoinvexity for a locally Lipschitz function by means of the generalized Jacobian. We study relationships between these concepts, in particular the implications between preinvexity and invexity. Next, we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite-dimensional (non necessarily Pareto) vector optimization problems with locally Lipschitz objective functions through solutions of vector variational-like inequality problems. These conditions are stated via the generalized Jacobian and under pseudoinvexity hypotheses, and they show that a vector optimization problem can be reformulated as a vector variational-like inequality problem. This work extends and improves several previous papers, where the objective function of the vector optimization problem is assumed to be differentiable, or being locally Lipschitz, the authors consider the componentwise subdifferential based on the Clarke’s generalized gradients of the components of the objective function. Throughout the paper some simple examples are given in order to illustrate the main concepts and results.

Exponential type vector variational-like inequalities and nonsmooth vector optimization problems

This paper is devoted to study a new class of exponential form of vector variational-like inequality problems comprised of locally Lipschitz functions having exponential type invexities. In the setting of Banach space, we investigate the relationship of nonsmooth exponential type vector variational-like inequality problems with vector optimization problems involving nonsmooth \((p,r)\)-invex functions. Also, we explore the conditions for solvability of the aforesaid nonsmooth exponential type vector variational-like inequality problems using Fan-KKM Theorem. Moreover, we provide examples to elucidate our results.

Exponential type vector variational-like inequalities and vector optimization problems with exponential type invexities

A new class of exponential form of vector variational-like inequality problems is introduced, and then the equivalence among (weakly) efficient solutions, vector critical points of vector optimization problem and the solutions of vector variational-like inequalities under the framework of (p,r)-invexity is established. To the best our knowledge, the presented results are new and highly application oriented to other results based on generalized invexities to the context of optimization problems in the literature.

Vector Variational-Like Inequalities with Generalized Semimonotone Mappings

We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using Kakutani-Fan-Glicksberg’s fixed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. The results presented in this paper generalize some known results for vector variational inequality in recent years.

GENERALIZED VECTOR MINTY'S LEMMA

In this paper, the author defines a new generalized <TEX>${\eta}$</TEX>, <TEX>${\delta}$</TEX>, <TEX>${\alpha}$</TEX>)-pseudomonotone mapping and considers the equivalence of Stampacchia-type vector variational-like inequality problems and Minty-type vector variational-like inequality problems for generalized (<TEX>${\eta}$</TEX>, <TEX>${\delta}$</TEX>, <TEX>${\alpha}$</TEX>)-pseudomonotone mappings in Banach spaces, called the generalized vector Minty's lemma.

Generalized vector variational-like inequalities and vector optimization

In this paper, we consider different kinds of generalized vector variational-like inequality problems and a vector optimization problem. We establish some relationships between the solutions of generalized Minty vector variational-like inequality problem and an efficient solution of a vector optimization problem. We define a perturbed generalized Stampacchia vector variational-like inequality problem and discuss its relation with generalized weak Minty vector variational-like inequality problem. We establish some existence results for solutions of our generalized vector variational-like inequality problems.

Vector variational-like inequality problem and vector optimization problem

In this paper, we consider (weak) vector variational-like inequality problems ((W)VLIP) and (weak) vector optimization problems ((W)VOP), and check their relationships through the existence of solutions in topological vector spaces.

Existence results for generalized nonlinear vector variational-like inequalities with set-valued mapping

In this paper, we introduce and study a class of generalized nonlinear vector variational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems and so on. Applying maximal element theorem, we prove the existence of its solutions in the setting of locally convex topological vector space.

Generalized Minty Vector Variational-Like Inequalities and Vector Optimization Problems

In this paper, we study the relationship among the generalized Minty vector variational-like inequality problem, generalized Stampacchia vector variational-like inequality problem and vector optimization problem for nondifferentiable and nonconvex functions. We also consider the weak formulations of the generalized Minty vector variational-like inequality problem and generalized Stampacchia vector variational-like inequality problem and give some relationships between the solutions of these problems and a weak efficient solution of the vector optimization problem.

A generalized mixed vector variational-like inequality problem

In this paper, we introduce relaxed η - α - P -monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η -complete semicontinuity and η -strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.

SOME CHARACTERIZATIONS FOR VECTOR VARIATIONAL-LIKE INEQUALITIES

In this paper, a class of $\eta$-PPM mappings $F$ where $F$ and $-F$ are both $\eta$-pseudomonotone are introduced, which are proper generalizations of the PPM mappings considered by Bianchi and Schaible. The solution sets of two kinds of vector variational-like inequality problems involving $\eta$-PPM mappings are characterized in Banach spaces. Furthermore, the solution sets of two classes of vector variational-like inequalities involving set-valued mappings are also characterized via the scalarization approach due to Konnov.