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.

Some Classes of Exponentially General Variational-like Inequalities

Some Classes of Exponentially General Variational-like Inequalities

Solvability of the System of Extended Nonlinear Mixed Variational-Like Inequalities and Proximal Dynamical System

In this article, our goal is to study the system of extended nonlinear mixed variational-like inequalities (in short, SENMVLI) with a nonconvex functional in the setting of real Hilbert spaces and discuss the existence of solution of our considered problem. We propose a three-step iterative algorithm to calculate the approximate solutions of SENMVLI and investigate the convergence analysis as well as stability analysis of the proposed algorithm. Furthermore, we also study the proximal dynamical system for SENMVLI and prove that the trajectory of the solution of the extended proximal dynamical system converges globally exponentially to a unique solution of SENMVLI. Our suggested iterative algorithm and results have become the significant improvement, enhancement, and generalization of many previously known results in the literature.

Generalized Set-valued Nonlinear Variational-like Inequalities and Fixed Point Problems: Existence and Approximation Solvability Results

Generalized Set-valued Nonlinear Variational-like Inequalities and Fixed Point Problems: Existence and Approximation Solvability Results

New classes of higher order variational-like inequalities

New classes of higher order variational-like inequalities

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

Perturbed Mixed Variational-like Inequalities and Auxiliary Principle Pertaining to a Fuzzy Environment

Convex and non-convex fuzzy mappings are well known to be important in the research of fuzzy optimization. Symmetry and the idea of convexity are closely related. Therefore, the concept of symmetry and convexity is important in the discussion of inequalities because of how its definition behaves. This study aims to consider new class of generalized fuzzy variational-like inequality for fuzzy mapping which is known as perturbed fuzzy mixed variational-like inequality. We also introduce strongly fuzzy mixed variational inequality, as a particular case of perturbed fuzzy mixed variational-like inequality which is also a new one. Furthermore, by using the generalized auxiliary principle technique and some new analytic techniques, some existence results and efficient numerical techniques of perturbed fuzzy mixed variational-like inequality are established. As exceptional cases, some known and new results are obtained. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

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.

System of generalized nonlinear variational-like inequalities and nearly asymptotically nonexpansive mappings: graph convergence and fixed point problems

System of generalized nonlinear variational-like inequalities and nearly asymptotically nonexpansive mappings: graph convergence and fixed point problems

Existence and Approximation of Solution for Vector Mixed Variational-Like Inequality Under Densely (η, f)-C-Pseudomonotone Mappings

In this paper we study vector mixed variational inequality problem under (η, C)-pseudomonotonicity and densely (η, f)-C-pseudo-monotonocity in reflexive Banach space. The existence and uniqueness of solutions have been established with the help of KKM technique and further we have proposed iterative algorithm to find the approximate solution of vector mixed variational inequality by defining an auxiliary problem.

A proximal iterative algorithm for system of generalized nonlinear variational-like inequalities and fixed point problems

ABSTRACT In this paper, we are concerned with the investigation problem of finding a common element of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational-like inequalities. For this end, based on P-η-proximal mapping, an iterative algorithm is constructed and under some suitable conditions, the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common element of the above two sets are proved.

Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-η-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion M-η-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every M-η-proximal mapping is actually P-η-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out.

Some Iterative Schemes for Solving Mixed Equilibrium Variational-like Inequalities

Some new types of equilibrium variational-like inequalities are considered, which is called the bifunction mixed equilibrium variational-like inequalities. The auxiliary principle technique is used to construct some iterative schemes to solve these new equilibrium variational-like inequalities. Convergence of the suggested schemes is discussed under relaxed conditions. Several special cases are discussed as applications of the main results. The ideas and techniques may be starting point for future research.

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

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

Second order evolutionary partial differential variational-like inequalities

<abstract><p>This work is dedicated to the study of a second order evolutionary partial differential variational-like inequality in Banach space. We obtain the mild solution of our problem by applying the concept of strongly continuous cosine family of bounded linear operators, fixed point theorem for condensing multi-valued operators and measure of non-compactness. It is proved that the solution set of mixed variational-like inequalities is non-empty, bounded, closed and convex.</p></abstract>

Some new classes of preinvex functions and directional variational-like inequalities

In this paper, we introduce and study some new classes of invex sets and preinvex functions with respect to an arbitrary function k and the bifunction ?(.,.); which are called the generalized preinvex functions. These functions are nonconvex functions and include the preinvex function, convex functions and k-convex as special cases. We study some properties of generalized preinvex functions. It is shown that the minimum of generalized preinvex functions on the generalized invex sets can be characterized by a class of variational inequalities, which is called the directional variational-like inequalities. Using the auxiliary technique, several new inertial type methods for solving the directional variational-like inequalities are proposed and analyzed . Convergence analysis of the proposed methods is considered under suitable conditions. Some open problems are also suggested for future research.

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

Preinvexity of $ n $-dimensional fuzzy number-valued functions: characterization, variational inequality and optimization problems

<abstract><p>In this paper, the preinvexity of <italic>n</italic>-dimensional fuzzy number-valued functions are defined and discussed by means of the partial order relation in <italic>n</italic>-dimensional fuzzy number space which including preinvexity, weak preinvexity, strict preinvexity, weakly strict preinvexity, prequasiinvexity, weak prequasiinvexity, strict prequasiinvexity, weakly strict prequasiinvexity, and so on. In addition, their interrelations of the preinvexity of <italic>n</italic>-dimensional fuzzy number-valued functions are discussed, and some counterexamples are given. Furthermore, the two-parameter optimization problem, $ n $-dimensional fuzzy variational-like inequality and optimality conditions related to $ n $-dimensional preinvex fuzzy number-valued functions are discussed.</p></abstract>

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>