Mixed Variational Inequalities Research Articles

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

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.

Stochastic Bregman extragradient algorithm with line search for stochastic mixed variational inequalities

In this paper, we present a stochastic Bregman extragradient algorithm with line search for solving a class of stochastic mixed variational inequality, which not only does not require any information of the Lipschitz constant but allows to search a potentially larger step size per iteration. Compared with the existing algorithms, the proposed algorithm allows different step sizes in the prediction and correction steps, thus enhancing the algorithm's flexibility. Under the generalized monotonicity, we derive the almost sure convergence, the iteration complexity O ( 1 / ϵ ) and the oracle complexity O ( 1 / ϵ 2 ) for our algorithm. Furthermore, under the generalized strong monotonicity and the sample size increases at a geometric rate, the linear convergence rate of the proposed algorithm with respect to the Bregman distance between the iterations and solution is established. Numerical results demonstrate a favourable comparison of the proposed algorithm with existing ones.

Fixed-time neural networks with time-invariant and time-varying coefficients for mixed variational inequalities

This paper develops several fixed-time neural networks for solving mixed variational inequalities (MVIs). The proposed networks are highly efficient and with fixed-time convergence. First, based on the conventional forward-backward-forward neural network (FNN) and sliding mode control technique, a time-invariant fixed-time FNN (FxTFNN) is designed. Next, the Euclidean norm of FNN is introduced into FxTFNN to design the modified FxTFNN (MFxTFNN). It is shown that the proposed FxTFNN and MFxTFNN have fixed-time convergence properties and their settling-time functions are independent of the initial values. The proposed FxTFNN and MFxTFNN can be used to solve the Lasso problem and apply sparse signal reconstruction and image reconstruction. In addition, by introducing time-varying coefficients based on FxTFNN and MFxTFNN, time-varying FxTFNN (TFxTFNN) and time-varying MFxTFNN (TMFxTFNN) are developed. Finally, experimental results of numerical example, signal reconstruction, and image reconstruction are used to verify the effectiveness and superiority of the proposed neural networks.

Modified forward-backward algorithm for stochastic mixed variational inequality problems

Modified forward-backward algorithm for stochastic mixed variational inequality problems

Variance reduced forward-reflected-backward algorithm for solving nonconvex finite-sum mixed variational inequalities

Variance reduced forward-reflected-backward algorithm for solving nonconvex finite-sum mixed variational inequalities

Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities

Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities

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.

Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

Variable sample-size operator extrapolation algorithm for stochastic mixed variational inequalities

In this paper, we present a variable sample-size operator extrapolation algorithm for solving a class of stochastic mixed variational inequalities. One distinctive feature of our algorithm is that it updates a single search sequence by solving a prox-mapping subproblem and computing an evaluation of the expected mapping at each iteration and hence it may significantly reduce computation load. In particular, the iteration sequence generated by our algorithm always belongs to the feasible region. We show that, under some moderate conditions, the proposed algorithm can achieve O(1/T) ergodic convergence rate in terms of the expected restricted gap function, where T denotes the number of iterations. We derive some results related to the convergence rate of the Bregman distance between iterates and solutions, the iteration complexity, and the oracle complexity for the proposed algorithm when the sample size increases at a geometric rate. We also investigate the sublinear convergence rate in terms of the residual function under the generalized monotonicity condition. Numerical experiments on stochastic network game, stochastic sparse traffic assignment problems and sparse classification problem indicate that the proposed algorithm is promising compared with some existing algorithms.

On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities

Abstract A new class of stochastic differential equations (SDEs) is introduced in this article, which is driven by the generalized stochastic mixed variational inequality (GS-MVI). First, the property of the solution sets of the GS-MVI is proved by Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and Aumann’s measurable selection theorem. Next, we obtain the Carathéodory property of the solution set, with which the discussed SDEs can be transformed to stochastic differential inclusions (SDIs). The solution set of the proposed SDEs is proved to be nonempty through the existence of the solutions of the corresponding SDIs by the tools of fixed point theorem.

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.

A Proximal Neurodynamic Network With Fixed-Time Convergence for Equilibrium Problems and Its Applications.

This article proposes a novel fixed-time converging proximal neurodynamic network (FXPNN) via a proximal operator to deal with equilibrium problems (EPs). A distinctive feature of the proposed FXPNN is its better transient performance in comparison to most existing proximal neurodynamic networks. It is shown that the FXPNN converges to the solution of the corresponding EP in fixed-time under some mild conditions. It is also shown that the settling time of the FXPNN is independent of initial conditions and the fixed-time interval can be prescribed, unlike existing results with asymptotical or exponential convergence. Moreover, the proposed FXPNN is applied to solve composition optimization problems (COPs), l₁-regularized least-squares problems, mixed variational inequalities (MVIs), and variational inequalities (VIs). It is further shown, in the case of solving COPs, that the fixed-time convergence can be established via the Polyak-Lojasiewicz condition, which is a relaxation of the more demanding convexity condition. Finally, numerical examples are presented to validate the effectiveness and advantages of the proposed neurodynamic network.

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.

New Iterative Scheme Involving Self-Adaptive Method for Solving Mixed Variational Inequalities

Variational inequalities (VI) problems have been generalized and expanded in various ways. The VI principle has become a remarkable study area combining pure and applied research. The study of variational inequality in mathematics is significantly aided by providing an important framework by fixed-point theory. The concept of fixed-point theory can be considered an inherent component of the VI. We consider a mixed variational inequality (MVI) a useful generalization of a classical variational inequality. The projection method is not applicable to solve MVI due to the involvement of the nonlinear term ϕ. MVI is equivalent to fixed-point problems and the resolvent equation techniques. This technique is commonly used in the research on the existence of a solution to the MVI. This paper uses a new self-adaptive method using step size to modify the fixed-point formulation for solving the MVI. We will also provide the convergence of the proposed scheme. Our output could be seen as a significant refinement of the previously known results for MVI. A numerical example is also provided for the implementation of the generated algorithm.

Two-step inertial forward-reflected-backward splitting based algorithm for nonconvex mixed variational inequalities

This paper presents a modification of a recently studied Malitsky–Tam forward-reflected-backward splitting method with two-step inertial extrapolation to solve nonconvex mixed variational inequalities. We give global convergence results and nonasymptotic O(1/k) convergence rate of our proposed method under some appropriate conditions and present some mathematical justifications.

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.