Variational Inequalities Research Articles

Inertial Bilevel Variational Monotone Inclusion Problem in Banach Spaces

In this paper, we study accelerated Halpern-type iterative method for finding zero solution of sum of two monotone operators which solves variational inequality problem of inverse strongly monotone mapping in 2-uniformly convex and uniformly smooth real Banach spaces. The strong convergence of our proposed method is establish under some standard conditions imposed on parameters. Our theorems generalize many recently announced results in the literature.

Additive Schwarz Methods for Fourth-Order Variational Inequalities

Additive Schwarz Methods for Fourth-Order Variational Inequalities

A New Extragradient-Viscosity Method for a Variational Inequality, an Equilibrium Problem, and a Fixed Point Problem

In this research article, we introduce a novel iterative approach that builds upon a two-step extragradient-viscosity method. This method aims to find a common element among the solution set of a variational inequality, an equilibrium problem, and the set of common fixed points from a countable family of demicontractive mappings in a Hilbert space. We offer a robust convergence theorem for the proposed iterative scheme, considering certain well-conditioned parameters. Our findings represent an improvement over similar results already available in the existing literature. Furthermore, we demonstrate the applicability of our main result to W-mappings. Lastly, we present two numerical examples to exhibit the consistency and accuracy of our devised scheme.

New strong convergence analysis for variational inequalities and fixed point problems

In this paper, we obtain strong convergence results for solving variational inequality and fixed point problems using a combination of the Forward-Backward-Forward method and the Krasnoselkii–Mann iteration method with an inertial extrapolation step without assuming on-line rule of the inertial parameters and the iterates. Our results present a new way of choosing inertial parameters for strongly convergent algorithms to solve variational inequality and fixed point problems different from what is obtainable in the literature whereby on-line rule is assumed. We perform numerical tests to validate our theoretical analysis.

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

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

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

A gentle introduction to gradient-based optimization and variational inequalities for machine learning

Abstract The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradient-based methods remain essential—given the high dimensionality and large scale of machine-learning problems—but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variational inequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition.

Static and dynamic aspects of the principle of virtual work

AbstractThe paper presents a detailed analysis of the principle of virtual work in its statics and dynamics aspects. Special attention is paid to not exact formulation and interpretation of mechanical system motion equations based upon this principle, what is presented quite often in text books. The general form of the principle of virtual work and variational formulations of the problems in statics are presented and analyzed. Additionally, the dynamic implications of the principle of virtual work and dynamics problem formulation of a system subjected to scleronomic constraints are detailed. The basis for the formulation is the dynamic implication of the principle of virtual work that specifies the relation between the constraint reactions and the body accelerations. This relation takes the form of the variation inequality. As the result, the Gauss principle which determines the body acceleration vector is presented. Furthermore, a description of impact formulation, based upon the Carnot model, caused by the constraints is provided. The theoretical development is illustrated with examples of applications of the principle of virtual work.

A modified variational approach to noisy cell signaling.

Signaling in cells is full of noise and, hence, described with stochastic biochemical models. Thus, an efficient computation algorithm for these fluctuating reactions is much needed. Apart from the very popular Monte Carlo simulation, methods based on probability distributions are frequently desired due to their analytical tractability and possible numerical advantages in diverse circumstances, among which the variational approach is the most notable. In this paper, new basis functions are proposed to better depict possibly complex distribution profiles, and an extra regularization scheme is supplied to the variational equation to remove occasional degeneracy-induced singularities during the evolution. The new extension is applied to four typical biochemical reaction models and restores the Gillespie results accurately but with greatly reduced simulation time. This modified variational approach is expected to work in a wide range of cell signaling networks.

An electro-elastic contact problem with an internal state variable

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive foundation. General models for elastic materials with piezoelectirec effect, called electro-elastic materials. This paper is devoted to the study of the model involving a frictionless contact between an electro-elastic body characterized by long memory and a conductive adhesive foundation. The process is mechanically quasistatic and electrically static. We model the material with a general nonlinear electro-elastic constitutive law with internal state variable and the contact with normal compliance and unilateral constraint, where the adhesion is taken into account and a regularized electrical conductivity condition. The main feature of these models consists of the fact that their variational formulation is given by a history-dependent quasivariational inequality for the displacement field and a linear variational equation for the electric potential. A weak formulation for the model is given in the form of a coupled system for the displacement, the stress, the electric displacement, the electric potential, the bonding and the variable state internal fields. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and Banach fixed point arguments.

Expected Residual Minimization Method for a Class of Stochastic Tensor Variational Inequalities

Expected Residual Minimization Method for a Class of Stochastic Tensor Variational Inequalities

New Generalized Derivatives for Solving Variational Inequalities Using the Nonsmooth Newton Methods

AbstractVariational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python.

Dynamic spatial price equilibrium, dynamic user equilibrium, and freight transportation in continuous time: A differential variational inequality perspective

In this paper we provide a statement of dynamic spatial price equilibrium (DSPE) in continuous time as a basis for modeling freight flows in a network economy. The model presented describes a spatial price equilibrium due to its reliance on the notion that freight movements occur in response to differences between the local and distant prices of goods for which there is excess demand; moreover, local and distant delivered prices are equated at equilibrium. We propose and analyze a differential variational inequality (DVI) associated with dynamic spatial price equilibrium to study the Nash-like aggregate game at the heart of DSPE using the calculus of variations and optimal control theory. Our formulation explicitly considers inventory and the time lag between shipping and demand fulfillment. We stress that such a time lag cannot be readily accommodated in a discrete-time formulation. We provide an in-depth analysis of the DVI's necessary conditions that reveals the dynamic user equilibrium nature of freight flows obtained from the DVI, alongside the role played by freight transport in maintaining equilibrium commodity prices and the delivered-price-equals-local-price property of spatial price equilibrium. By intent, our contribution is wholly theoretical in nature, focusing on a mathematical statement of the defining equations and inequalities for dynamic spatial price equilibrium (DSPE), while also showing there is an associated differential variational inequality (DVI), any solution of which is a DSPE. The model of spatial price equilibrium we present integrates the theory of spatial price equilibrium in a dynamic setting with the path delay operator notion used in the theory of dynamic user equilibrium. It should be noted that the path delay operator used herein is based on LWR theory and fully vetted in the published dynamic user equilibrium literature. This integration is new and constitutes a significant addition to the spatial price equilibrium and freight network equilibrium modeling literatures. Among other things, it points the way for researchers interested in dynamic traffic assignment to become involved in dynamic freight modeling using the technical knowledge they already possess. In particular, it suggests that algorithms developed for dynamic user equilibrium may be adapted to the study of urban freight modelled as a dynamic spatial price equilibrium. As such, our work provides direction for future DSPE algorithmic research and application. However, no computational experiments are reported herein; instead, the computing of dynamic spatial price equilibria is the subject of a separate manuscript.

On quasi-inertial double extragradient-type iterations for variational inequalities and Bregman relatively asymptotically nonexpansive mapping

On quasi-inertial double extragradient-type iterations for variational inequalities and Bregman relatively asymptotically nonexpansive mapping

Solutions for a New Fractional Differential Dynamical System and Yosida Quasi-Inverse Variational Inequality in Hilbert Space

In this article, first we introduce and study a Yosida Quasi-inverse variational inequality problem (in short, YQIVI) in Hilbert space and then developed a new fractional differential dynamical system for the YQIVI. We prove the existence and uniqueness of solution for the suggested dynamical system. Further, using the Lyapunov function we also prove the asymptotic stability of the new dynamical system at the equilibrium point. Furthermore, using Rothe's time discretization method we investigate existence and uniqueness of solution of the proposed dynamical system. Finally, we provide a numerical example to demonstrate the credibility and efficacy of the dynamical system in solving the YQIVI.

A Projection-Type Implicit Algorithm for Finding a Common Solution for Fixed Point Problems and Variational Inequality Problems

This paper deals with the problem of finding a common solution for a fixed point problem for strictly pseudocontractive mappings and for a certain variational inequality problem. We propose a projection-type implicit averaged algorithm and establish the strong convergence of the sequences generated by this method to the common solution for the fixed point problem and the variational inequality problem. In order to illustrate the feasibility of the hypotheses and the superiority of our theoretical results over the existing literature, an example is also presented.

Variational Analysis of a Dynamic Contact Problem with Wear and Damage Involving Viscoelastic Materials with Long-Term Memory

We investigate a mathematical problem involving dynamic interaction between a viscoelastic body with long-term memory loss and an obstruction. The contact is frictional and bilateral, with a moving rigid base, resulting in wear of the contacting surface. The problem is expressed as a coupled system, with a hyperbolic quasi-variational inequality for displacement and a parabolic variational inequality for damage. We define a variational formulation for the model and demonstrate the existence of a single weak solution to the problem. The material behaviour is explained using a viscoelastic constitutive law that includes long-term memory and damage. Elastic deformations induce material deterioration, which is depicted by a parabolic inclusion. The proof is based on classical existence.

Modelling an emission trading scheme based on a dual-market mechanism in multimode networks with customised bus services

ABSTRACT The transport-related emissions are one of the main contributors to air pollution. In this paper, we construct an efficient and stable emission trading scheme (ETS) to reduce emissions. We build a dual-market mechanism for carbon emission permit (CEP) in the transportation system to control emissions at source, where CEPs are issued in a paid way in the primary market and traded freely in the secondary market. At the same time, we include customised bus (CB) and describe the competitive relationship between CB and private car (PC) in a multimode network. The market trading process and travel choice behaviours are formulated as a variational inequality problem (VIP). The VIP is solved using a relaxation algorithm, and the multi-round iteration strategy is used for updating permit volumes. Finally, we conduct numerical experiments based on Braess and Sioux Falls networks to demonstrate the merits of the ETS based on a dual-market mechanism.

A Comparative Study for Evaluating the Groundwater Inflow and Drainage Effect of Jinzhai Pumped Storage Power Station, China

Various hydrogeological problems like groundwater inflow, water table drawdown, and water pressure redistribution may be encountered in the construction of hydraulic projects. How to accurately predict the occurrence of groundwater inflow and assess the drainage effect during construction are still challenging problems for engineering designers. Taking the Jinzhai pumped storage power station (JPSPS) of China as an example, this paper aims to use different methods to calculate the water inflow rates of an underground powerhouse and evaluate the drainage effect caused by tunnel inflow during construction. The methods consist of the analytical formulas, the site groundwater rating (SGR) method, and the Signorini type variational inequality formulation. The results show that the analytical methods considering stable water table may overestimate the water inflow rates of caverns in drained conditions, whereas the SGR method with available hydro-geological parameters obtains a qualitative hazard assessment in the preliminary phase. The numerical solutions provide more precise and reliable values of groundwater inflow considering complex geological structures and seepage control measures. Moreover, the drainage effects, including a seepage-free surface, pore water pressure redistribution, and hydraulic gradient, have been accurately evaluated using various numerical synthetic cases. Specifically, the faults intersecting on underground caverns and drainage structures significantly change the groundwater flow regime around caverns. This comparative study can not only exactly identify the capabilities of the methods for cavern inflow in drained conditions, but also can comprehensively evaluate the drainage effect during cavern construction.

Primal-Dual Extrapolation Methods for Monotone Inclusions Under Local Lipschitz Continuity

In this paper, we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone, whereas the other is locally Lipschitz continuous. We propose primal-dual (PD) extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of [Formula: see text] and [Formula: see text], measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an ε-residual solution of strongly and nonstrongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity [Formula: see text]. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an ε-KKT or ε-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under local Lipschitz continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods. Funding: This work was partially supported by the National Science Foundation [Grant IIS-2211491], the Office of Naval Research [Grant N00014-24-1-2702], and the Air Force Office of Scientific Research [Grant FA9550-24-1-0343].