Linear Variational Inequalities Research Articles

A Non- Linear Hyperbolic Variational Inequality

A Non- Linear Hyperbolic Variational Inequality

Quantitative stability for stochastic tensor variational inequalities

ABSTRACT The goal of this article is to investigate a class of stochastic tensor variational inequalities (denoted by STVI), which is a generalization of the stochastic linear variational inequalities (denoted by SLVI) and a subclass of stochastic variational inequalities (denoted by SVI). The quantitative stability of the expected residual minimization (ERM) problem for the STVI is analysed. Firstly, the existence of solutions of the ERM problem and its perturbed problem are considered. Then, the quantitative stability of the ERM problem is derived under suitable probability metrics. Finally, under some suitable assumptions, the rates of convergence of the sample average approximation (SAA) method for the optimal solution sets of the perturbed problems are discussed.

Existence of Optimal Sets for Linear Variational Equations and Inequalities

Existence of Optimal Sets for Linear Variational Equations and Inequalities

Robust fuzzy adaptive yaw moment control of humanoid robot with unknown backlash nonlinearity

To achieve an excellent dynamic balance during humanoid robot walks, it is essential to counteract the undesired yaw moment and deal with the nonlinearities existing in the system dynamics. In this paper, the problems of the dynamic balance optimization and yaw moment control for humanoid robot are investigated. First we formulate a constrained dynamic model of compliant joints for robots. Then, a yaw moment control approach based on quadratic objective function is proposed for computing optimal ankle torque to equilibrate the undesired yaw moment. In order to deal with the complicated optimization problem with inequality and equality constraints, a Quadratic Programming (QP) solver based on Linear Variational Inequality (LVI) is adopted. In comparison with the yaw moment control approaches studied previously, the main contributions of this paper are summarized as follows. 1) our newly proposed control scheme is more effective in counteracting the undesired yaw moment for the case of handling object-carrying tasks during walking. 2) a novel adaptive backlash inverse is included in the proposed scheme to cancel the unknown actuator backlash nonlinearity without a prior knowledge of the backlash nonlinearity. By the strict Lyapunov argument, the asymptotic convergence of the system tracking error to arbitrarily small neighborhood of the origin is proved. The simulation examples involving humanoid robot walking are constructed to validate the proposed algorithm.

Stable model reduction for linear variational inequalities with parameter-dependent constraints

We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-disks with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

A three-step stabilized algorithm for the Navier-Stokes type variational inequality

In the light of two-grid finite element discretization, this paper is concerned with a three-step stabilized algorithm for the Navier-Stokes equations with nonlinear slip boundary conditions of friction type. The proposed algorithm proceeds three steps: the first step solves a stabilized nonlinear variational inequality problem on a coarse grid, the second step solves a stabilized linear variational inequality problem based on Newton iteration on a fine grid, and the third step solves the same stabilized linear problem with only different right-hand sides on a fine grid. We analyze the stability of the presented algorithm, and derive error estimate of the approximate solutions in L2 norms of the velocity gradient and pressure. Finally, we demonstrate the effectiveness of the present algorithm by two numerical tests.

Projection and contraction method for updating simultaneously mass and stiffness matrices

This paper deals with the problem of updating simultaneously mass and stiffness matrices. The desired matrix properties, including satisfaction of the characteristic equation, symmetry, positive semidefiniteness, and boundedness, are imposed as side constraints to form the matrix pencil minimization problem. This problem is related to finite element model updating and damage detection of a structure. Using partial Lagrangian multipliers, we convert the matrix pencil minimization problem into a monotone matrix linear variational inequality, and develop the projection and contraction method. Based on the special structure of the matrix linear variational inequality, we derive an efficient implementation of the projection and contraction method for large-scale problems. Several numerical tests are presented to illustrate the performance and application of the proposed method.

A Support Vector Neural Network for P300 EEG Signal Classification

Brain computer interface (BCI) P300 speller can help severely disabled patients communicate and control with external machines or robots, so that the classification methods of P300 Electroencephalogram (EEG) signal play an important role in the development of BCI system and technology. In this paper, a novel support vector neural network (SVNN) is proposed and developed to obtain more accurate and effective EEG classification results. It is the first time to combine linear variational inequality (LVI) based primal-dual neural network with convex quadratic programming problem based on support vector machine to solve the classification problem. It has been proved that the support vector neural network globally converges to the optimal solution of convex optimization problem and iterates the parameters in the form of matrix, which means that the method has global convergence and parallelism. The proposed SVNN method is used to solve the classification problem of P300 EEG signals. Experimental results on data set IIb from BCI competition II and data set II from BCI competition III show that the accuracy of the proposed SVNN method is 100% and 98%, respectively. Compared with most of the state-of-theart algorithms, SVNN has the highest recognition accuracy and information transfer rate.

Numerical simulation for European and American option of risks in climate change of Three Gorges Reservoir Area

Abstract With the climate change processes over times, all professions and trades in Three Gorges Reservoir Area will be influenced. One of the biggest challenges is the risk of rising sea level. In this situation, a large number of uncertainties for climate changes will be faced in Three Gorges Reservoir Area. Therefore, it is of importance to investigate the complexity of decision making on investing in the long term rising sea level risk related projects in Three Gorges Reservoir Area. This paper investigates the sea level and the temperature as the underlying assets in Three Gorges Reservoir Area. A real option model is constructed to evaluate potential sea level rising risk. We formulate European and American real option models into a linear parabolic variational inequalities and propose a power penalty approach to solve it. Then we obtain a nonlinear parabolic equation. It shows that the nonlinear parabolic equation is unique and solvable. Also, the solutions of the nonlinear parabolic equation converge to the solutions of the parabolic variational inequalities at the rate of order O(λ −k/2). Since the analytic solution of nonlinear parabolic equation is difficult to obtain, a fitted finite volume method is developed to solve it in case of European and American options, and the convergence of the nonlinear parabolic equation is obtained. An empirical analysis is presented to illustrate our theoretical results.

Asymptotic Behavior of a Bingham Flow in Thin Domains with Rough Boundary

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter \(\epsilon \) denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when \(\epsilon \) tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law.

Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities

<p style='text-indent:20px;'>This paper focuses on the quantitative stability analysis of the expected residual minimization (ERM) formulation for a class of stochastic linear variational inequalities. Firstly, the existence of solutions of the ERM formulation and its perturbed problem is discussed. Then, the quantitative stability of the ERM formulation is derived under suitable probability metrics. Finally, the sample average approximation (SAA) problem of the ERM formulation is studied, and the rates of convergence of optimal solution sets are obtained under different assumptions.</p>

Quantitative stability of two-stage stochastic linear variational inequality problems with fixed recourse

ABSTRACT This paper focus on the quantitative stability of a class of two-stage stochastic linear variational inequality problems whose second stage problems are stochastic linear complementarity problems with fixed recourse matrix. Firstly, we discuss the existence of solutions to this two-stage stochastic problems and its perturbed problems. Then, by using the corresponding residual function, we derive the quantitative stability of this two-stage stochastic problem under Fortet-Mourier metric. Finally, we study the sample average approximation problem, and obtain the convergence of optimal solution sets under moderate assumptions.

Quantitative analysis for a class of two-stage stochastic linear variational inequality problems

This paper considers a class of two-stage stochastic linear variational inequality problems whose first stage problems are stochastic linear box-constrained variational inequality problems and second stage problems are stochastic linear complementary problems having a unique solution. We first give conditions for the existence of solutions to both the original problem and its perturbed problems. Next we derive quantitative stability assertions of this two-stage stochastic problem under total variation metrics via the corresponding residual function. Moreover, we study the discrete approximation problem. The convergence and the exponential rate of convergence of optimal solution sets are obtained under moderate assumptions respectively. Finally, through solving a non-cooperative game in which each player’s problem is a parameterized two-stage stochastic program, we numerically illustrate our theoretical results.

Two Hybrid End-Effector Posture-Maintaining and Obstacle-Limits Avoidance Schemes for Redundant Robot Manipulators

To fulfill path tracking tasks with the end-effector posture controlled in a complex environment, maintaining the robot manipulator end-effector posture and avoiding obstacles are two important issues needed to be considered. In this paper, two hybrid end-effector posture-maintaining and obstacle-limits avoidance (hybrid PM-OLA) schemes are proposed and investigated for motion planning of redundant robot manipulators, which are based on the quadratic programming (QP) framework. The end-effector posture-maintaining, obstacle-avoidance, and the joint-angular-limits are formulated as an equality constraint, inequality constraint, and bound constraint into the QP problem. With these hybrid PM-OLA schemes, the robot manipulator can avoid the obstacle and joint physical limits when executing end-effector tasks. The hybrid PM-OLA schemes are finally transformed into linear variational inequalities and solved by a recurrent neural network. Computer simulations and physical experiments substantiate the effectiveness, accuracy, safety, and the practicability of the proposed hybrid PM-OLA schemes. Comparisons with other schemes show that the proposed hybrid PM-OLA schemes are more suitable for applications.

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.

A novel neurodynamic reaction-diffusion model for solving linear variational inequality problems and its application

In this paper, we present a new delayed projection neural network with reaction-diffusion terms for solving linear variational inequality problems. The proposed neural network possesses a simple one-layer structure. By employing the differential inequality technique and constructing a new Lyapunov–Krasovskii functional, we derive some novel sufficient conditions ensuring the globally exponential stability. These conditions are dependent on diffusions and the monotonicity assumption is unnecessary. Furthermore, the considered neural network can solve quadratic programming problems. Finally, several applicable examples are provided to illustrate the satisfactory performance of the proposed neural network.

A class of two-stage distributionally robust games

An \begin{document}$ N $\end{document} -person noncooperative game under uncertainty is analyzed, in which each player solves a two-stage distributionally robust optimization problem that depends on a random vector as well as on other players' decisions. Particularly, a special case is considered, where the players' optimization problems are linear at both stages, and it is shown that the Nash equilibrium of this game can be obtained by solving a conic linear variational inequality problem.

Asymptotic analysis of a Bingham fluid in a thin T-like shaped structure

We study the steady incompressible flow of a Bingham fluid in a thin T-like shaped domain, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. This phenomenon is described by non linear variational inequalities. By letting the parameter describing the thickness of the thin domain tend to zero, we derive two uncoupled problems corresponding to the two branches of the T-like shaped structure. We then analyze and give a physical justification of the limit problem.

Robust solutions to box-constrained stochastic linear variational inequality problem

We present a new method for solving the box-constrained stochastic linear variational inequality problem with three special types of uncertainty sets. Most previous methods, such as the expected value and expected residual minimization, need the probability distribution information of the stochastic variables. In contrast, we give the robust reformulation and reformulate the problem as a quadratically constrained quadratic program or convex program with a conic quadratic inequality quadratic program, which is tractable in optimization theory.

A variational inequality approach for constrained multifacility Weber problem under gauge

The classical multifacility Weber problem (MFWP) is one of the most important models in facility location. This paper considers more general and practical case of MFWP called constrained multifacility Weber problem (CMFWP), in which the gauge is used to measure distances and locational constraints are imposed to facilities. In particular, we develop a variational inequality approach for solving it. The CMFWP is reformulated into a linear variational inequality, whose special structures lead to new projection-type methods. Global convergence of the projection-type methods is proved under mild assumptions. Some preliminary numerical results are reported which verify the effectiveness of proposed methods.