Mixed Variational Inequalities 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.

Stable Proximal Dynamical System for Mixed Variational Inequalities in Hilbert Spaces

In this paper we focus on solving mixed variational inequalities by proximal fixed-time dynamical systems, in which the solution of

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

A fixed-time stable forward–backward dynamical system for solving generalized monotone inclusions

AbstractWe propose a forward–backward splitting dynamical system for solving inclusion problems of the form $$0\in A(x)+B(x)$$ 0 ∈ A ( x ) + B ( x ) in Hilbert spaces, where A is a maximal operator and B is a single-valued operator. Involved operators are assumed to satisfy a generalized monotonicity condition, which is weaker than the standard monotone assumptions. Under mild conditions on parameters, we establish the fixed-time stability of the proposed dynamical system. In addition, we consider an explicit forward Euler discretization of the dynamical system leading to a new forward backward algorithm for which we present the convergence analysis. Applications to other optimization problems such as constrained optimization problems, mixed variational inequalities, and variational inequalities are presented and some numerical examples are given to illustrate the theoretical results.

Analysis of relaxed inertial method for non-convex mixed variational inequalities

Analysis of relaxed inertial method for non-convex mixed variational inequalities

Mixed Variational Inequalities and Nonconvex Analysis

In this expository paper, we provide an account of fundamental aspects of mixed variational inequalities with major emphasis on the computational properties, various generalizations, dynamical systems, nonexpansive mappings, sensitivity analysis and their applications. Mixed variational inequalities can be viewed as novel extensions and generalizations of variational principles. A wide class of unrelated problems, which arise in various branches of pure and applied sciences are being investigated in the unified framework of mixed variational inequalities. It is well known that variational inequalities are equivalent to the fixed point problems. This equivalent fixed point formulation has played not only a crucial part in studying the qualitative behavior of complicated problems, but also provide us numerical techniques for finding the approximate solution of these problems. Our main focus is to suggest some new iterative methods for solving mixed variational inequalities and related optimization problems using resolvent methods, resolvent equations, splitting methods, auxiliary principle technique, self-adaptive method and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis of the mixed variational inequalities is studied using the resolvent equations method. Iterative methods for solving some new classes of mixed variational inequalities are proposed and investigated. Our methods of discussing the results are simple ones as compared with other methods and techniques. Results proved in this paper can be viewed as significant and innovative refinement of the known results.

The mixed method with two Lagrange multiplier formulations for the Signorini problem

Introducing stress as a new unknown, we consider the Signorini problem in the mixed form. The well-posedness theory of the continuous and discrete mixed variational inequalities has been established by reformulating them into projection problems. Two Lagrange multiplier formulations are introduced utilizing different projections. The equivalency among the Lagrange multiplier formulations and the mixed variational inequality is demonstrated in continuous and discrete sense, respectively. For the discrete mixed variational inequality, we develop the error estimates. Based on two Lagrange multiplier formulations, we design two Active/Inactive set algorithms and investigate the convergence. Several numerical experiments are conducted to verify the theoretical convergence rates of the finite element discretization and the iteration algorithms.

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.

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.