Variational Inequalities Research Articles

Universal methods for variational inequalities: Deterministic and stochastic cases

In this paper, we propose universal proximal mirror methods to solve the variational inequality problem with Hölder-continuous operators in both deterministic and stochastic settings. The proposed methods automatically adapt not only to the oracle’s noise (in the stochastic setting of the problem) but also to the Hölder continuity of the operator without having prior knowledge of either the problem class or the nature of the operator information. We analyzed the proposed algorithms in both deterministic and stochastic settings and obtained estimates for the required number of iterations to achieve a given quality of a solution to the variational inequality. We showed that, without knowing the Hölder exponent and Hölder constant of the operators, the proposed algorithms have the least possible in the worst-case sense complexity for the considered class of variational inequalities. We also compared the resulting stochastic algorithm with other popular optimizers for the task of image classification.

Nonlinear stability of smooth multi-solitons for the Dullin-Gottwald-Holm equation

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δH1/δu=m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work.

Physics and geometry from a Lagrangian: Dirac spinors on a generalised frame bundle

We clarify the structure obtained in Hélein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle, starting from a structure-less differentiable 10-manifold [17]. The obtained structure is locally equivalent to a frame bundle thus we term it “generalised frame bundle”. In the same time, we enrich the model with a Dirac spinor coupled to the Einstein-Cartan spacetime. The obtained variational equations generalise the usual Einstein-Cartan-Dirac field equations in the sense that they are shown to imply the usual field equations when the generalised frame bundle is a standard frame bundle.

An Alternated Inertial Projection and Contraction Algorithm for Solving Quasimonotone Bilevel Variational Inequalities with Application to Optimal Control Problems

An Alternated Inertial Projection and Contraction Algorithm for Solving Quasimonotone Bilevel Variational Inequalities with Application to Optimal Control Problems

A new class of generalized Nash-population games via variational inequalities and fixed points

A new class of generalized Nash-population games via variational inequalities and fixed points

Boundary optimal control problems for parabolic variational inequalities of bilateral obstacle type

This paper is concerned with the optimality system for an optimal control problem governed by parabolic bilateral obstacle problems with Robin boundary conditions. The control-state dynamics is described by variational inequalities and the control on the boundary is considered. Our analysis relies on a regularization method of approximating the variational inequality by a series of semilinear PDEs. The Lipschitz continuity and directionally differentiability in the weak sense for the control-to-state mapping are proved. Then, the necessary optimality conditions for approximate optimal control problems are derived. Finally, the first-order optimality system including complementarity conditions for the original problem is established by a limiting procedure. The result is carefully compared with those obtained in existing literature.

Method with batching for stochastic finite-sum variational inequalities in non-Euclidean setting

Variational inequalities are a universal optimization paradigm that incorporate classical minimization and saddle point problems. Nowadays more and more tasks require to consider stochastic formulations of optimization problems. In this paper, we present an analysis of a method that gives optimal convergence estimates for monotone stochastic finite-sum variational inequalities. In contrast to the previous works, our method supports batching, does not lose the oracle complexity optimality and uses an arbitrary Bregman distance to take into account geometry of the problem. Paper provides experimental confirmation to algorithm’s effectiveness.

Correction: A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

Correction: A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

Galerkin Kantorovich Method for Solving the Terzaghi’s One-Dimensional Soil Consolidation Equation

This paper presents the Galerkin-Kantorovich variational method for solving the Terzaghi’s one-dimensional consolidation equation for two-way drainage conditions. The solution was considered as an infinite series of known coordinate (shape) functions and unknown function of time which we sought such that the resulting functional is minimized. The shape functions satisfied the hydraulic boundary conditions at the boundary of the consolidating soil. Galerkin-Kantorovich variational integral equation was thus formulated for the initial boundary value problem using residual minimization principles. The solution resulted in a system of first order ordinary differential equations in which was solved for Orthogonalization principles were used to obtain the integration constants in terms of initial pore water pressure, thus yielding the general solution. Solutions for constant initial excess pore water pressure were obtained and found to be the closed-form solution. The solutions were presented in terms of global (average) degrees of consolidation and tabulated. The results obtained were exact and identical with results previously found using separation of variables techniques.

A Hybrid Iterative Method for Elliptic Variational Inequalities of the Second Kind

A Hybrid Iterative Method for Elliptic Variational Inequalities of the Second Kind

An inertial extragradient method for solving strongly pseudomonotone equilibrium problems in Hilbert spaces

In this work, we propose an inertial extragradient method for solving strongly pseudomonotone equilibrium problems utilizing a novel self-adaptive stepsize approach. We establish the R-linear convergence rate of the proposed method without prior knowledge of the Lipschitz-type constants associated with the bifunction. We also discuss the application of the obtained results to variational inequality problems involving strongly pseudomonotone and Lipschitz continuous mapping. Numerical examples are presented to illustrate the efficiency of the proposed method.

Gauss-Seidel Type Iterative Algorithm for a Generalized System of Extended Non-linear Variational Inequalities

This study is focused on a new generalized system of extended non-linear variational inequalities (GSENVI, for short) involving 3k-distinct non-linear relaxed cocoercive operators. We give the equivalent formulation of GSENVI in a more convenient form by using auxiliary principal technique. Through the projection technique, we demonstrate that the non-linear projection equations are analogous to equivalent form of GSENVI. By the use of alternative fixed point formulations, we proposed the k-steps Gauss-Seidel type iterative algorithms to obtain an approximate solution of the GSENVI. Further, we discuss the convergence of proposed k-step Gauss Seidel type iterative algorithms. Several special cases of GSENVI are discussed for the reliability of our findings.

Relaxed projection methods for solving variational inequality problems

Relaxed projection methods for solving variational inequality problems

A Ramsey-Type Equilibrium Model with Spatially Dispersed Agents

AbstractWe present a spatial and time-continuous Ramsey-type equilibrium model for households and firms that interact on a spatial domain to model labor mobility in the presence of commuting costs. After discretization in space and time, we obtain a mixed complementarity problem that represents the spatial equilibrium model. We prove existence of equilibria using the theory of finite-dimensional variational inequalities and derive a tailored diagonalization method to solve the resulting large-scale instances. Finally, we present a case study that highlights the influence of commuting costs and show that the model allows to analyze transitory effects of industrial agglomeration that emerge and vanish over time as in the real economy.

Study of a boundary value problem governed by the general elasticity system with a new boundary conditions in a thin domain

Abstract The aim of this work is the study of a nonlinear boundary value problem which theoretically generalizes the Lamé system with disturbance in a thin 3D domain with friction and a generalized boundary condition. For the resolution of the considered problem and after the variational formulation, we construct an operator from the variational problem. Then we prove that this operator has certain properties which allows us to apply the theorem of existence and uniqueness of the solution of variational inequalities of the 2nd kind. Finally, using a change of scale, we transport the variational problem to an equivalent problem defined on a domain independent of the parameter ζ {{\zeta}} and subsequently we obtain the limit problem and the generalized weak equations of the initial problem.

Inertial Halpern-type methods for variational inequality with application to medical image recovery

In this paper, we propose inertial Halpern-type algorithms involving a quasi-monotone operator for approximating solutions of variational inequality problems which are fixed points of quasi-nonexpansive mappings in reflexive Banach spaces. We use Bregman distance functions to enhance the efficiency of our algorithms and obtain strong convergence results, even in cases where the Lipschitz constant of the operator involved is unknown a priori. Furthermore, we illustrate the practical applicability of our methods through numerical experiments. Notably, our algorithms excel when compared to recent techniques in the literature. Of particular significance is their successful application in restoring computed tomography medical images that have been affected by motion blur and random noise. Our algorithms consistently outperform established state-of-the-art methods in all conducted experiments, showcasing their competitiveness and potential to advance variational inequality problem-solving, especially in the field of medical image recovery.

The Role of Cultural Variation and Economic Inequality in the Prevalence of Social Entrepreneurship

The Role of Cultural Variation and Economic Inequality in the Prevalence of Social Entrepreneurship

Error‐transfer‐theory based adaptive filtering algorithm for external supported tracking

AbstractAn improved filtering algorithm, called the EttAF adaptive filtering, is proposed in this paper. Compared with the available algorithm, the equivalent observation error in the fusion centre from the external sensor is considered, which makes the observation noise matrix of the filter dynamically adjust according to the relative relationship between the fusion centre and the external sensor. The strong manoeuvring target is modelled based on the CS‐jerk model, and the error adaptive variation equation is derived for the external data. Simulation results show that compared to traditional algorithm the EttAF algorithm can effectively improve the accuracy of position and velocity by 10.8% and 22.3% respectively.

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.

A stochastic ridesharing user equilibrium model with origin-destination-based ride-matching strategy

Ridesharing, as an emerging mode of modern urban transportation, has gained widespread popularity due to its ability to improve convenience and reduce the expenses in daily travel. To capture its impact on the network-wide traffic flow pattern, this paper develops a new variational inequality (VI) formulation for modeling the stochastic ridesharing user equilibrium (SRUE) problem considering an origin–destination (O-D) -based ride-matching strategy. The proposed SRUE problem is formulated based on a hierarchical logit choice model to depict the multi-stage decision-making (i.e., role choice, ride-matching, and route choice) of the ridesharing travelers. The SRUE model also reflects a two-sided market equilibrium considering the interaction effect between the ridesharing market and the road traffic equilibrium market. The logit choice model captures the ride-matching problem between the drivers and the passengers traveling on the same/different O-D pairs. Moreover, the lower- and upper-bound constraints for a ridesharing driver picking up passengers are established at the O-D level. For solving the SRUE model with O-D-based ride-matching constraints, the Lagrangian dual method (LDM) is adopted by incorporating the Barzilai-Borwein (BB) step size to update the dual variables, which accelerates the convergence of the LDM. Three numerical examples are conducted to illustrate the features of the proposed SRUE model and demonstrate the performance of the solution algorithm.