Vector Variational-like Inequalities Research Articles

On vector variational E-inequalities and differentiable vector optimization problem

AbstractIn this paper, in order to characterize optimality of a class of nonconvex differentiable multiobjective programming problems, we introduce two new types of vector variational-like inequalities, namely, a weak variational E-inequality and a vector variational E-inequality. Namely, under (strictly) E-convexity, we prove relationships between the aforesaid vector variational-like inequalities and differentiable vector optimization problems. Further, under (strictly) pseudo-E-convexity, we are in position to identify vector critical E-points, weak E-Pareto (E-Pareto) solutions of differentiable vector optimization problems and solutions of these introduced vector variational-like inequalities.

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.

On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

On minty variational principle for quasidifferentiable vector optimization problems

This paper deals with quasidifferentiable vector optimization problems involving invex functions wrt convex compact sets. We present vector variational-like inequalities of Minty type and of Stampacchia type in terms of quasidifferentials denoted by (QMVVLI) and (QSVVLI), respectively. By utilizing these variational inequalities, we infer vital and adequate optimality conditions for an efficient solution of the quasidifferentiable vector optimization problem involving invex functions wrt convex compact sets. We also establish various results for the solutions of the corresponding weak versions of the vector variational-like inequalities in terms of quasidifferentials.

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

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

A study on vector variational-like inequalities using convexificators and application to its bi-level form

<p style='text-indent:20px;'>This paper deals with the weak versions of the vector variational-like inequalities, namely Stampacchia and Minty type under invexity in the framework of convexificators. The connection between both the problems along with the link to vector optimization problem are analyzed. An application to nonconvex mathematical programming has also been presented. Further, the bi-level version of these problems is formulated and a procedure to obtain the solution involving the auxiliary principle technique is described in detail. We have shown that the iterative algorithm with the help of which we get the approximate solution converges strongly to the exact solution of the problem.</p>

Generalized regularized gap functions and error bounds for generalized vector variational-like inequalities

Generalized regularized gap functions and error bounds for generalized vector variational-like inequalities

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.

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.

Vector Optimization Problems and Generalized Vector Variational-Like Inequalities

In this paper, some properties of pseudoinvex functions, defined by means of limiting subdifferential, are discussed. Furthermore, the Minty vector variational-like inequality, the Stampacchia vector variational-like inequality, and the weak formulations of these two inequalities defined by means of limiting subdifferential are studied. Moreover, some relationships between the vector variational-like inequalities and vector optimization problems are established.

On Generalized Minty and Stampacchia Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problem Involving Higher Order Strong Invexity

On Generalized Minty and Stampacchia Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problem Involving Higher Order Strong Invexity

Stability analysis of a class of non-newtonian fluids based on generalized vector variational-like inequalities on riemannian manifold

Fluid mechanics is a branch of mechanics. It is the science of studying fluid phenomena and related mechanical behaviors. So far, the mutual soaking and fusion between fluid mechanics and other disciplines have formed many branches, many physicists and Mathematicians are working on this aspect of research. The classical Newtonian fluid mechanics believe that in parallel flow, the shear force is proportional to the shear rate, and the proportional coefficient is called the viscosity coefficient. In recent years, with the increasing importance of non-Newtonian fluids, people have found that research on non-Newtonian flow is necessary. The variational inequalities based on Riemannian manifolds were first proposed by S.Z.Németh. S.Z.Németh studied the existence of solutions to variational inequalities on Hadamard manifolds. For the equivalence of non-differentiable vector optimization problems on Riemannian manifolds and generalized weak vector variational-like inequalities, we give some new conclusions.

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.

On vector variational-like inequalities involving right upper-Dini-derivative functions

In this paper, we introduce (weak) Stampacchia and Minty arcwise connected vector variational-like inequalities in the term of right upper-Dini-derivative and establish not only the relations of introduced inequalities with vector optimization problems but also the existence results, by using KKM-Fan theorem and Brouwer fixed point theorem. Examples are provided to illustrate the derived results.

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.

New generalised mixed vector variational-like inequalities with semi-<i>η</i>-pseudomonotonicity

New generalised mixed vector variational-like inequalities with semi-<i>η</i>-pseudomonotonicity

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.

Vector critical points and generalized quasi-efficient solutions in nonsmooth multi-objective programming

In this work, several extended approximately invex vector-valued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higher-order (weak) quasi-efficiency with respect to a function are proposed for a multi-objective programming. Under the introduced generalization of higher-order approximate invexities assumptions, we prove that the solutions of generalized vector variational-like inequalities in terms of the generalized Jacobian are the generalized quasi-efficient solutions of nonsmooth multi-objective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function.

A class of generalized invex functions and vector variational-like inequalities

In this paper, a class of generalized invex functions, called (alpha,rho,eta)-invex functions, is introduced, and some examples are presented to illustrate their existence. Then we consider the relationships of solutions between two types of vector variational-like inequalities and multi-objective programming problem. Finally, the existence results for the discussed variational-like inequalities are proposed by using the KKM-Fan theorem.