System Of Variational Inequalities Research Articles

Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.

A fully implicit constraint-preserving simulator for the black oil model of petroleum reservoirs

Due to the rapid advancement of supercomputing resource, there is a growing interest in developing parallel algorithms for the large-scale reservoir simulation. In this paper, we present a parallel and fully implicit simulator for the black oil model based on the variational inequality (VI) framework, which can be used to enforce important mathematical and physical properties to obtain accurate constraint-preserving solutions. In other words, this framework ensures the predicted solution to stay within the physical range. In the proposed approach, the black oil model is reformulated as a variational inequality system that naturally satisfies the basic boundedness requirement of the solution, and then a fully implicit finite volume method is applied to discretize the model equations. In addition to that, a number of nonlinear and linear fast solver technologies, including a variant of inexact Newton methods and the domain decomposition based preconditioners, are employed to guarantee the robustness and parallel scalability of the simulator. A particular emphasis of the proposed framework is placed on the parallel and algorithmic performance of the variational inequality approach across large-scale and heterogeneous problems. Several numerical results pertaining to the problems in one, two and three dimensions are presented to illustrate the efficiency, robustness, and the overall performance of the fully implicit constraint-preserving simulator.

Strong Convergence Theorems for Variational Inequalities and Common Fixed-Point Problems Using Relaxed Mann Implicit Iteration Methods

Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a countable family of uniformly Lipschitzian pseudocontractive mappings. More convergence theorems are proved under some suitable weak condition in both 2-uniformly smooth and uniformly convex Banach spaces.

Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.

Extremal solutions in systems of variational inequalities with multivalued mappings

ABSTRACT In this paper, we use a sub-supersolution method to study systems of variational inequalities of the form: , where and are multivalued mappings with possibly non-power growths and is a closed, convex set. We introduce a concept of mixed extremal solutions in the set-theoretic sense and prove the existence of such solutions between sub- and supersolutions. We also show the existence of least and greatest solutions of the above system between sub- and supersolutions if the lower order terms have certain increasing properties.

Systems of Variational Inequalities with Nonlinear Operators

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces.

Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems

This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI), and introduces some Mann-type implicit iteration methods for solving it. Norm convergence of the proposed methods of the iteration methods is guaranteed under some suitable assumptions.

Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems

In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions.

Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces.

Systems of variational inequalities with hierarchical variational inequality constraints for Lipschitzian pseudocontractions

Systems of variational inequalities with hierarchical variational inequality constraints for Lipschitzian pseudocontractions

Systems of variational inequalities with hierarchical variational inequality constraints for asymptotically nonexpansive and pseudocontractive mappings

The purpose of this paper is to introduce and analyze a hybrid extragradient-like implicit rule for finding a common solution of a general system of variational inequalities (GSVI) and a common fixed point problem (CFPP) of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Here the hybrid extragradient-like implicit rule is based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the CFPP, which solves a certain variational inequality on their common solution set. As an application, we give an algorithm to solve common fixed point problems of nonexpansive and pseudocontractive mappings, variational inequality problems and generalized mixed equilibrium problems in Hilbert spaces.

Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

Iterations for systems of variational inequalities, fixed points of pseudocontractions and zeros of accretive operators

In this paper, we introduce implicit composite three-step Mann iterations for finding a common solution of a general system of variational inequalities, a fixed point problem of a countable family of pseudocontractive mappings and a zero problem of an accretive operator in Banach spaces. Strong convergence of the suggested iterations are given.

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.

General iterative methods for systems of variational inequalities with the constraints of generalized mixed equilibria and fixed point problem of pseudocontractions

In this paper, we introduce two general iterative methods (one implicit method and one explicit method) for finding a solution of a general system of variational inequalities (GSVI) with the constraints of finitely many generalized mixed equilibrium problems and a fixed point problem of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed implicit and explicit iterative methods to a solution of the GSVI with the above constraints, which is the unique solution of a certain variational inequality. The results presented in this paper improve, extend, and develop the corresponding results in the earlier and recent literature.

Steady flows of an Oldroyd fluid with threshold slip

We consider a mathematical model that describes 3D steady flows of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under mixed boundary conditions, including a threshold-slip boundary condition. Using the concept of weak solutions, we reduce the original slip problem to a coupled system of variational inequalities and equations for the velocity field and stresses. For arbitrary large data (forcing and boundary data) and suitable material constants, we prove the existence of weak solutions and establish some of their properties.

Driver-Rider Cost-Sharing Strategies and Equilibria in a Ridesharing Program

The rapid development of smartphone technology has led to the increased popularity of dynamic ridesharing apps used to organize ad hoc ridesharing trips between strangers on short notice. To support such real-time on-demand services, cost-sharing between drivers and riders is commonly centrally determined by ridesharing apps according to prescribed rules. To highlight the impacts of appropriate cost-sharing strategies on the success of ridesharing programs, this paper models the mode choices of a group of heterogeneous travelers with continuously distributed values of time in a single-corridor network, considering the complex interactions between travelers’ mode choices and the attractiveness of ridesharing in terms of rider/driver waiting/detouring times and matching probabilities. The equilibrium state under any given cost-sharing strategy is described by a system of variational inequalities based on which the existence of equilibria is established. With the proposed modeling framework, various cost-sharing strategies are examined to avoid mode shifts among transit users to autos and/or reduce vehicular traffic in the short run; the necessary conditions for cost-sharing strategies to sustain participation and/or reduce vehicle usage are explicitly provided. It is shown that when driving alone is faster but more expensive than public transit, no cost-sharing strategy exists to sustain an active ridesharing platform without inducing transit users to join the ridesharing program. Moreover, the existence of cost-sharing strategies capable of reducing vehicular traffic on the road is not always guaranteed, depending on the costs of driving alone and taking public transit in the considered corridor, fuel prices, and travelers’ prioritization of safety and privacy. Furthermore, it is found that the initial state with no ridesharing participants is an equilibrium under any cost-sharing strategy if the additional cost incurred by a traveler through participating in a ridesharing program is nonnegative. This explains the difficulty of initiating a ridesharing program and implies the initial necessity of subsidizing all intended riders and/or drivers to encourage participation. The online appendix is available at https://doi.org/10.1287/trsc.2017.0801 .

Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces

Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces

Optimal switching under a hybrid diffusion model and applications to stock trading

This paper is concerned with the optimal switching problem under a hybrid diffusion (or, regime switching) model in an infinite horizon. The state of the system consists of a number of diffusions coupled by a finite-state continuous-time Markov chain. Based on the dynamic programming principle, the value function of our optimal switching problem is proved to be the unique viscosity solution to the associated system of variational inequalities. The optimal switching strategy, indicating when and where it is optimal to switch, is given in terms of the switching and continuation regions. In many applications, the underlying Markov chain has a large state space and exhibits two-time-scale structure. In this case, a singular perturbation approach is employed to reduce the computational complexity involved. It is shown that as the time-scale parameter ε goes to zero, the value function of the original problem converges to that of a limit problem. The limit problem is much easier to solve, and its optimal switching solution leads to an approximate solution to the original problem. Finally, as an application of our theoretical results, an example concerning the stock trading problem in a regime switching market is provided. It is emphasized that, this paper is the first time to introduce the optimal switching as a general framework to study the stock trading problem, in view of their inherent connection. Both optimal trading rule and convergence result are numerically demonstrated in this example.

A new general system of variational inequalities for convergence theorem and application

In this present paper, we propose a modified form of generalized system of variational inequalities and introduce an iterative scheme for finding a common element of the set of fixed points of nonexpansive mapping and the solution set of the proposed problem in the framework of real Hilbert spaces. We prove a strong convergence theorem of the proposed iterative scheme. Applying our main result, we prove strong convergence theorems of the standard constrained convex optimization problem and the split feasibility problem. In support of our main result, a numerical example is also presented.