System Of 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.

Multi-step inertial algorithms for equilibrium, fixed point, general systems of variational inequalities and split feasibility problems

Multi-step inertial algorithms for equilibrium, fixed point, general systems of variational inequalities and split feasibility problems

Approximating Solution of System of Variational Inequality Constrained Split Common Fixed Point Problem in Hilbert Spaces

Approximating Solution of System of Variational Inequality Constrained Split Common Fixed Point Problem in Hilbert Spaces

Inertial‐like Bregman projection method for solving systems of variational inequalities

In this paper, we propose a self‐adaptive inertial‐like algorithm with Bregman distance for approximating a common solution of systems of variational inequalities for a class of monotone and Lipschitz continuous mappings in real reflexive Banach spaces. Our algorithm is constructed without using hybrid projection method and shrinking projection method, and its strong convergence is proved without the prior information of the Lipschitz constant of the mapping. Finally, we provide some numerical experiments to illustrate the performance of the newly proposed method including a comparison with related works in solving signal restoration problems.

VARIATIONAL ANALYSIS OF A VISCOELASTIC FRICTIONAL CONTACT WITH LONG-TERM MEMORY BODY WITH THERMAL EFFECTS

In this article we study a mathematical model which describes the quasi-static process of contact between a piezoelectric body with long-term memory and an obstacle. The contact is modeled with a normal conformity condition and a version of Coulom's law. The evolution of temperature is described by a first kind evolution equation. The problem is formulated as a system of scalable elliptical variational inequalities for displacement, and a variational equality for electrical stress. We prove the existence of a unique weak solution to the problem. The proof is based on arguments from time-dependent variational inequalities, differential equations and fixed point.

Subgradient-like extragradient algorithms for systems of variational inequalities with constraints

In this paper, we introduce a modified viscosity subgradient-like extragradient implicit rule with line-search process for finding a solution of a general system of variational inequalities (GSVI) with a variational inequality (VIP) and a fixed-point (FPP) constraints in Hilbert spaces. The suggested algorithms are based on the subgradient extragradient method with line-search process, hybrid Mann implicit iteration method, and composite viscosity approximation method. Under suitable restrictions, we demonstrate the strong convergence of the suggested algorithm to a solution of the GSVI with the VIP and FPP constraints, which is a unique solution of a certain hierarchical variational inequality.

A study of Wiener-Hopf dynamical systems for variational inequalities in the setting of fractional calculus

<abstract><p>In this paper, we consider a new fractional dynamical system for variational inequalities using the Wiener Hopf equations technique. We show that the fractional Wiener-Hopf dynamical system is exponentially stable and converges to its unique equilibrium point under some suitable conditions. We also discuss some special cases, which can be obtained from our main results.</p></abstract>

On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems

Abstract In a real Banach space, let the VI indicate a variational inclusion for two accretive operators and let the CFPP denote a common fixed point problem of countably many nonexpansive mappings. In this article, we introduce a generalized extragradient implicit method for solving a general system of variational inequalities (GSVI) with the VI and CFPP constraints. Strong convergence of the suggested method to a solution of the GSVI with the VI and CFPP constraints under some suitable assumptions is established.

Coupled systems of nonlinear variational inequalities and applications

In this paper we investigate the existence of solutions for a system consisting of two inequalities of variational type. Each inequality is formulated in terms of a nonlinear functional χ and ψ, respectively and a coupling functional B. We consider two sets of assumptions (Hχi), (Hψj) and (HBk), i,j,k∈{1,2} and we show that, if the constraints sets are bounded, then a solution exists regardless if we assumed the first or the second hypothesis on χ, ψ or B, thus obtaining eight possibilities. When the constraint sets are unbounded a coercivity condition is needed to ensure the existence of solutions. We provide two such conditions. An application, arising from Contact Mechanics, in the form of a partial differential inclusion driven by the Φ-Laplace operator is presented in the last section.

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.

A free boundary problem arising from a multi-state regime-switching stock trading model

In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock whose price is driven by unobservable market status and noise processes. The free boundary problem is a variational inequality system of three functions with a degenerate operator. We prove that all the four switching free boundaries are no-overlapping, monotonic and C∞-smooth by the approximation method. We also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.

A Valuable Remark on Lipschitz in the First Variable Definition and System of Nonlinear Variational Inequalities

The goal of this paper is to present a critique on the incorrect use of the Lipschitz definition concerning the first variable and/or the second variable in the literature on the system of variational inequalities by many authors. The possible impact of this paper is rather important, it questions the results of different authors, particularly when taking into account that some of these papers are published in quite good mathematical journals. As a result, not only that the proofs are wrong, but also the credibility of the theorems themselves is compromised. In addition, this paper illustrate, using a counterexample, that there is an error in setting up first variable definition and the results obtained in listed references do not hold up in H × H.

Nonzero-sum impulse games with regime switching

This paper is concerned with a two-player nonzero-sum stochastic differential game, where both players use impulse controls and the state process evolves as a regime switching diffusion. Based on a system of variational inequalities, a verification theorem as a sufficient criterion for optimality is established. Nash equilibrium strategies for the two players, indicating when and how it is optimal to intervene, are given in terms of the obstacle parts of the variational inequalities.

Strong convergence theorem for a general system of variational inequalities, equilibrium problems, and fixed point problems

Strong convergence theorem for a general system of variational inequalities, equilibrium problems, and fixed point problems

On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set.

New Parallel Fixed Point Algorithms and Their Application to a System of Variational Inequalities

In this study, considering the advantages of parallel fixed point algorithms arising from their symmetrical behavior, new types of parallel algorithms have been defined. Strong convergence of these algorithms for certain mappings with altering points has been analyzed, and it has been observed that their convergence behavior is better than existing algorithms with non-simple samples. In addition, the concept of data dependency for these algorithms has been examined for the first time in this study. Finally, it has been proven that the solution of a variational inequality system can be obtained using newly defined parallel algorithms under suitable conditions.

Convergence theorems for solving a system of pseudomonotone variational inequalities using Bregman distance in Banach spaces

Convergence theorems for solving a system of pseudomonotone variational inequalities using Bregman distance in Banach spaces

Novel Algorithms for Solving a System of Absolute Value Variational Inequalities

The goal of this paper is to study a new system of a class of variational inequalities termed as absolute value variational inequalities. Absolute value variational inequalities present a rational, pragmatic, and novel framework for investigating a wide range of equilibrium problems that arise in a variety of disciplines. We first develop a system of absolute value auxiliary variational inequalities to calculate the approximate solution of the system of absolute variational inequalities, and then by employing the projection technique, we prove the existence of solutions of the system of absolute value auxiliary variational inequalities. By utilizing an auxiliary principle and the existence result, we propose several new iterative algorithms for solving the system of absolute value auxiliary variational inequalities in the frame of four different operators. Furthermore, the convergence of the proposed algorithms is investigated in a thorough manner. The efficiency and supremacy of the proposed schemes is exhibited through some special cases of the system of absolute value variational inequalities and an illustrative example. The results presented in this paper are more general and rehash a number of some previously published findings in this field.

Hybrid optimal impulse control

This paper is concerned with an optimal impulse control problem under a hybrid diffusion (or, regime switching) model, where the state of the system consists of a number of diffusions coupled by a continuous-time finite-state Markov chain. The objective is to minimize the expected discounted cost from exerting the impulse control in the infinite horizon. Based on the dynamic programming principle (DPP), the value function of the hybrid optimal impulse control problem is shown to be the unique viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation, which is in the form of a coupled system of variational inequalities. Moreover, a verification theorem as the sufficient condition for optimality of a solution is also established. The optimal impulse control, indicates when and how it is optimal to intervene, is described by the obstacle part of the HJB equation. Finally, the general theoretical results are applied to an optimal cash management problem. The value function in closed-form and an explicit optimal policy are obtained, which exhibit clearly the effect of regime switching on the agent’s decision.

Challenges in Optimization with Complex PDE-Systems

The workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed.