Variational Inequalities In Banach Spaces Research Articles

Three novel inertial subgradient extragradient methods for quasi-monotone variational inequalities in Banach spaces

Three novel inertial subgradient extragradient methods for quasi-monotone variational inequalities in Banach spaces

New algorithms for solving pseudo-monotone variational inequalities in Banach spaces

In this paper, we introduce new algorithms for finding a solution of a variational inequality problem involving pseudo-monotone operator which is also a fixed point of a Bregman relatively inexpensive mapping in $p$-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. We prove weak and strong convergence theorems for proposed algorithms. Finally, we give some numerical experiments for supporting our main results.

An upper bound for a condition number theorem of variational inequalities

Nonlinear variational inequalities in Banach spaces are considered. A notion of (absolute) condition number with respect to the right-hand side is introduced. A distance among variational inequalities is defined. We prove that the distance to suitably restricted ill-conditioned variational inequalities is bounded from above by a multiple of the reciprocal of the condition number. By using an analogous lower bound of the companion paper [14], we obtain a full condition number theorem for variational inequalities. The particular case of convex optimization problems is also considered. Known results dealing with optimization problems are thereby generalized.

Operator extrapolation method for variational inequalities in Banach spaces

Operator extrapolation method for variational inequalities in Banach spaces

On the Approximation of Some Evolutionary Inclusions in Banach Spaces

This paper deals with an original application of method of finite difference approximations for evolutionary variational inequalities in Banach spaces. The main result of the study is the suggestion of a procedure and the demonstration of a theorem that, with a different approach, respect to the well known methods present in literature, proves some properties of the solutions of evolutionary variational inequalities.

Strong convergent inertial Tseng’s extragradient method for solving non-Lipschitz quasimonotone variational inequalities in Banach spaces

Strong convergent inertial Tseng’s extragradient method for solving non-Lipschitz quasimonotone variational inequalities in Banach spaces

Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces*

Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces*

On the Hadamard Well-Posedness of Generalized Mixed Variational Inequalities in Banach Spaces

We introduce a new concept of Hadamard well-posedness of a generalized mixed variational inequality in a Banach space. The relations between the Levitin–Polyak well-posedness and Hadamard well-posedness for a generalized mixed variational inequality are studied. The characterizations of Hadamard well-posedness for a generalized mixed variational inequality are established.

Modified projected subgradient method for solving pseudomonotone equilibrium and fixed point problems in Banach spaces

Motivated by the work of D.V. Hieu and J.-J. Strodiot [Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl., (2018), 20:131], we introduce a new projected subgradient method for solving pseudomonotone equilibrium and fixed point problem in Banach spaces. The main iterative steps in the proposed method use a projection method and do not require any Lipschitz-like condition on the equilibrium bifunction. A strong convergence result is proved under mild conditions and we applied our algorithm to solving pseudomonotone variational inequalities in Banach spaces. Also, we provide some numerical examples to illustrate the performance of the proposed method and compare it with other methods in the literature.

Scalarization and Optimality Conditions for the Approximate Solutions to Vector Variational Inequalities in Banach Spaces

In this paper, we present the notion of approximate solutions for vector variational inequalities in Banach spaces, which extends the existing approximate solutions. It is shown that, under the cone subconvexlikeness, our new approximate solutions can be characterized by the approximate solutions of linear scalar problem by means of the convex separation theorem. For the nonconvex cases, based on the nonconvex separation theorem with a special scalar functional introduced by Hiriart–Urruty, we get the nonlinear scalarization results for the approximate solutions. In order to establish optimality conditions without convexity assumption, we calculate the subdifferential of the Hiriart–Urruty nonlinear scalar functional. And based on the scalar characterizations, optimality conditions are obtained by using Ekeland variational principle and Fermat rule for scalar optimization problems. Finally, we consider the special case of vector inequality problems in finite dimensional Euclidean space, and establish some relations between the approximate solutions of vector inequality problems and nonsmooth vector optimization problems.

Single projection algorithm for variational inequalities in Banach spaces with application to contact problem

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern’s iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.

A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces

In this paper, we introduce an algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in Banach space. We modify the subgradient extragradient methods with a new and simple iterative step size, the strong convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, a numerical experiment is presented to show the efficiency and advantage of the proposed algorithm. Our results generalize some of the work in Hilbert spaces to Banach spaces.

Stability analysis of partial differential variational inequalities in Banach spaces

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

A Modified Thakur Three-Step Iterative Algorithm to Garcia-Falset Mappings and Variational Inequalities

In this paper, we suggest and analyze a modified Thakur three-step iterative algorithm to approximate a common element of the set of common fixed points of Garcia-Falset mappings and the set of solutions of some variational inequalities in Banach spaces. We also establish strong convergence theorems for a common solution of the above-said problems by the proposed iterative algorithm without the compactness assumption. The methods in this paper are novel and different from those given in many other papers. And the results are the extension and improvement of the recent results announced by many others.

On the existence and approximation of solutions for bilevel vector variational inequalities in Banach spaces

On the existence and approximation of solutions for bilevel vector variational inequalities in Banach spaces

New Algorithms for a Class of Accretive Variational Inequalities in Banach Spaces

In this paper, for finding a common zero of a finite family of m-accretive mappings in uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm, we propose an implicit iteration algorithm and an explicit one, based on a convex combination of the steepest-descent method and a composition of resolvents. We also show that our main algorithm contains some iterative ones in literature as special cases. Finally, we give numerical examples for illustration.

An existence principle for variational inequalities in Banach spaces

The present article deals with existence and uniqueness results for a variational inequality posed on a real smooth Banach space. This allows us to generalize the known results of this type given in Hilbert spaces.

Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem.

Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.