Mixed Variational Inequality Problems Research Articles

Stochastic golden ratio algorithm to non-convex stochastic mixed variational inequality problem

Stochastic golden ratio algorithm to non-convex stochastic mixed variational inequality problem

A new proximal gradient method for solving mixed variational inequality problems with a novel explicit stepsize and applications

In this paper, we propose a new algorithm for solving monotone mixed variational inequality problems in real Hilbert spaces based on proximal gradient method. Our new algorithm uses a novel explicit stepsize which is proved to be increasing to a positive limitation. This property plays an important role in improving the speed of the algorithm. To the best of our knowledge, it is the first time such a kind of stepsize has been proposed for the proximal gradient method solving mixed variational inequality problems. We prove the weak convergence and strong convergence with R-linear rate of our new algorithm under standard assumptions. The reported numerical simulations for applications in sparse logistic regression and image deblurring reveal the significant efficacy performance of our proposed method compared to the recent ones.

Projection algorithms with adaptive step sizes for multiple output split mixed variational inequality problems

Projection algorithms with adaptive step sizes for multiple output split mixed variational inequality problems

Solutions for the Nonlinear Mixed Variational Inequality Problem in the System

Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, we suggest a stability result for the SNMVIPs by perturbing the duality mappings. Furthermore, we present an optimal control problem that is governed by the SNMVIPs and show that it can be solved.

On the intermixed method for mixed variational inequality problems: another look and some corrections

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.

A novel predefined-time neurodynamic approach for mixed variational inequality problems and applications

In this paper, we propose a novel neurodynamic approach with predefined-time stability that offers a solution to address mixed variational inequality problems. Our approach introduces an adjustable time parameter, thereby enhancing flexibility and applicability compared to conventional fixed-time stability methods. By satisfying certain conditions, the proposed approach is capable of converging to a unique solution within a predefined-time, which sets it apart from fixed-time stability and finite-time stability approaches. Furthermore, our approach can be extended to address a wide range of mathematical optimization problems, including variational inequalities, nonlinear complementarity problems, sparse signal recovery problems, and nash equilibria seeking problems in noncooperative games. We provide numerical simulations to validate the theoretical derivation and showcase the effectiveness and feasibility of our proposed method.

Modified forward-backward algorithm for stochastic mixed variational inequality problems

Modified forward-backward algorithm for stochastic mixed variational inequality problems

The Study of Coordinate-Wise Decomposition Descent Method for Optimization Problems

The aim of this paper is to consider a general non-stationary optimization problem whose objective function need not be smooth in general and only approximation sequences are known instead of exact values of the functions. We apply a two-step technique where approximate solutions of a sequence of a generalized mixed variational inequality problem (GMVIP) are inserted in the iterative method of a selective coordinate-wise decomposition descent method. Its convergence is achieved under coercivity-type assumptions.

The forward–backward splitting method for finding the minimum like-norm solution of the mixed variational inequality problem

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set, which is by itself an optimal set of another mixed variational inequality problem in a Hilbert space. Regularized forward–backward splitting method is applied to find the minimum like-norm solution of the mixed variational inequality problem under investigation.

Deterministic Bi-Criteria Model for Solving Stochastic Mixed Vector Variational Inequality Problems

In this paper, we consider stochastic mixed vector variational inequality problems. Firstly, we present an equivalent form for the stochastic mixed vector variational inequality problems. Secondly, we present a deterministic bi-criteria model for giving the reasonable resolution of the stochastic mixed vector variational inequality problems and further propose the approximation problem for solving the given deterministic model by employing the smoothing technique and the sample average approximation method. Thirdly, we obtain the convergence analysis for the proposed approximation problem while the sample space is compact. Finally, we propose a compact approximation method when the sample space is not a compact set and provide the corresponding convergence results.

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a mixed variational inequality problem (MVIP) within a fixed time, where the time of convergence is finite and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined, and converges to the unique solution of the associated MVIP within a fixed time. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumption of strong monotonicity is relaxed to that of strong pseudomonotonicity. Finally, it is shown that the solution obtained using the forward-Euler discretization of the proposed modified proximal dynamical system converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of time steps, independent of the initial conditions.

One-step Bregman projection methods for solving variational inequalities in reflexive Banach spaces

ABSTRACT We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature.

An intermixed method for solving the combination of mixed variational inequality problems and fixed-point problems

In this paper, we introduce an intermixed algorithm with viscosity technique for finding a common solution of the combination of mixed variational inequality problems and the fixed-point problem of a nonexpansive mapping in a real Hilbert space. Moreover, we propose the mathematical tools related to the combination of mixed variational inequality problems in the second section of this paper. Utilizing our mathematical tools, a strong convergence theorem is established for the proposed algorithm. Furthermore, we establish additional conclusions concerning the split-feasibility problem and the constrained convex-minimization problem utilizing our main result. Finally, we provide numerical experiments to illustrate the convergence behavior of our proposed algorithm.

A neural network for solving the generalized inverse mixed variational inequality problem in Hilbert Spaces

<abstract><p>In this paper, we study and analyze the generalized inverse mixed variational inequality. The existence and uniqueness of the solution of such problem are proposed. The neural network associated with the generalized inverse mixed variational inequality is presented, and moreover, the Wiener-Hopf equation which the solution of the equation is equivalent to the solution of the generalized inverse mixed variational inequality, is considered. The stability and existence of solution of such neural network are proved. Finally, we introduce some algorithms which are constructed by the concept of the neural network and display a numerical example for understanding our results.</p></abstract>

A mixed variational inequality method for solving Signorini problems

This paper presents a new iterative method for solving Signorini problems. Boundary integral equations are established by discretizing the boundary of the problem, and the system equations related only to the unknown variables on the boundary of the problem are derived. Then, the Signorini boundary conditions of the problem are expressed as equivalent variational inequalities. By considering a simultaneous system of equations and variational inequalities, the Signorini problem is reduced to a mixed variational inequality problem. Finally, a compatibility iteration algorithm for solving mixed variational inequalities is designed based on the projection-contraction algorithm. Several examples are tested to demonstrate the accuracy and effectiveness of the proposed algorithm.

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.

Two fast variance-reduced proximal gradient algorithms for SMVIPs-Stochastic Mixed Variational Inequality Problems with suitable applications to stochastic network games and traffic assignment problems

In this paper, we propose two proximal gradient algorithms with variance reduction for stochastic mixed variational inequality problems. One is a proximal extragradient algorithm and another is a proximal forward–backward–forward algorithm. Under the monotonicity assumption on the mapping F and other moderate conditions, we derive some asymptotic convergence properties and O(1/k) convergence rate in terms of the restricted gap function values for the proposed algorithms. Furthermore, under the bounded metric subregularity condition, we investigate the linear convergence rate and oracle complexity bounds for the proposed algorithms when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate of ⌈k+1⌉−s with s>0, the mean-squared distance of the iterates to the solution set decays at a corresponding polynomial rate, while the iterations and oracle complexities to obtain an ε-solution are O(1/ε1/s) and O(1/ε1+1/s) respectively. Finally, some numerical experiments on stochastic network games and traffic assignment problems indicate that the proposed algorithms are efficient.

Mixed Variational Inequality Problem Involving Generalized Yosida Approximation Operator in q‐Uniformly Smooth Banach Space

A mixed variational inequality problem involving generalized Yosida approximation operator is considered and studied in q‐uniformly smooth Banach space. We have shown that mixed variational inequality problem involving generalized Yosida approximation operator is equivalent to a fixed‐point equation. The fixed‐point formulation is applied to establish an algorithm to obtain the solution of mixed variational inequality problem involving generalized Yosida approximation operator. Convergence criteria are also discussed. In support of our main result, we provide an example using Matlab program together with a computation table and convergence graphs. To check the validity of mixed variational inequality problem involving generalized Yosida approximation operator and its fixed‐point formulation, we construct one more example.

Stability and Convergence Analysis for Set-Valued Extended Generalized Nonlinear Mixed Variational Inequality Problems and Generalized Resolvent Dynamical Systems

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.