A predictor-corrector interior-point algorithmfor monotone variational inequality problems

Mehrotra's recent suggestion of a predictor corrector variant of primal dual interior point method for linear programming is currently the interior point method of choice for linear programming. In this work the authors give a predictor corrector interior point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level 1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.

RELAXED INERTIAL METHODS FOR SOLVING SPLIT VARIATIONAL INEQUALITY PROBLEMS WITHOUT PRODUCT SPACE FORMULATION

Many methods have been proposed in the literature for solving the split variational inequality problem.Most of these methods either require that this problem is transformed into an equivalent variational inequality problem in a product space,or that the underlying operators are co-coercive.However,it has been discovered that such product space transformation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting nature of the split variational inequality problem.On the other hand,the co-coercive assumption of the underlying operators would preclude the potential applications of these methods.To avoid these setbacks,we propose two new relaxed inertial methods for solving the split variational inequality problem without any product space transformation,and for which the underlying operators are freed from the restrictive co-coercive assumption.The methods proposed,involve projections onto half-spaces only,and originate from an explicit discretization of a dynamical system,which combines both the inertial and relaxation techniques in order to achieve high convergence speed.Moreover,the sequence generated by these methods is shown to converge strongly to a minimum-norm solution of the problem in real Hilbert spaces.Furthermore,numerical implementations and comparisons are given to support our theoretical findings.

A Smoothing Newton Method for the Box Constrained Variational Inequality Problems

The box constrained variational inequality problem can be reformulated as a nonsmooth equation by using median operator.In this paper,we present a smoothing Newton method for solving the box constrained variational inequality problem based on a new smoothing approximation function.The proposed algorithm is proved to be well defined and convergent globally under weaker conditions.

A HYBRID METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEMS

By using Fukushima's differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993.In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix Δ F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty,polyhedral,closed and convex is proposed.Armijo type line search and trust region strategies as well as Fukushima's differentiable merit function are incorporated into the method.It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al.,the method reduces to the basic Newton method and hence the rate of convergence is quadratic.Computational experiences show the efficiency of the proposed method.

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES

In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping.A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions.Finally,we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.

New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.

UNCONSTRAINED METHODS F0R GENERALIZED NONLINEAR COMPLEMENTARITY AND VARIATIONAL INEQUALITY PROBLEMS

In this papert we construct unconstrained methods for the generalized nonlinearcomplementarity problem and variational inequalities. Properties of the correspon-dent unconstrained optimization problem are studied. We apply these methods tothe subproblems in trust region method, and study their interrelationships. Nu-merical results are also presented.

A TRUST REGION-TYPE METHOD FOR SOLVINGMONOTONE VARIATIONAL INEQUALITY

Presents a study which proposed to introduce a trust region-type modification of Newton method for the monotone inequality problem using merit function. Concepts of monotone mapping; Proof of convergence of algorithm variational inequality trust region; Results.

Nested Alternating Direction Method of Multipliers to Low-Rank and Sparse-Column Matrices Recovery

The task of dividing corrupted-data into their respective subspaces can be well illustrated,both theoretically and numerically,by recovering low-rank and sparse-column components of a given matrix.Generally,it can be characterized as a matrix and a 2,1-norm involved convex minimization problem.However,solving the resulting problem is full of challenges due to the non-smoothness of the objective function.One of the earliest solvers is an 3-block alternating direction method of multipliers(ADMM)which updates each variable in a Gauss-Seidel manner.In this paper,we present three variants of ADMM for the 3-block separable minimization problem.More preciously,whenever one variable is derived,the resulting problems can be regarded as a convex minimization with 2 blocks,and can be solved immediately using the standard ADMM.If the inner iteration loops only once,the iterative scheme reduces to the ADMM with updates in a Gauss-Seidel manner.If the solution from the inner iteration is assumed to be exact,the convergence can be deduced easily in the literature.The performance comparisons with a couple of recently designed solvers illustrate that the proposed methods are effective and competitive.

IMPROVED GRADIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {An}n∈N be a family of monotone and Lipschitz continuos mappings of C into E＊. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [i0] for solving the variational inequality problem for {AN} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.

New cooperative projection neural network for nonlinearly constrained variational inequality

This paper proposes a new cooperative projection neural network （CPNN）, which combines automatically three individual neural network models with a common projection term. As a special case, the proposed CPNN can include three recent recurrent neural networks for solving monotone variational inequality problems with limit or linear constraints, respectively. Under the monotonicity condition of the corresponding Lagrangian mapping, the proposed CPNN is theoretically guaranteed to solve monotone variational inequality problems and a class of nonmonotone variational inequality problems with linear and nonlinear constraints. Unlike the extended projection neural network, the proposed CPNN has no limitation on the initial point for global convergence. Compared with other related cooperative neural networks and numerical optimization algorithms, the proposed CPNN has a low computational complexity and requires weak convergence conditions. An application in real-time grasping force optimization and examples demonstrate good performance of the proposed CPNN.

An Affine Scaling Interior Trust Region Method via Optimal Path for Solving Monotone Variational Inequality Problem with Linear Constraints

Based on a differentiable merit function proposed by Taji et al. in ＂Math. Prog. Stud., 58, 1993, 369-383＂, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors form an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.

Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem.We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h.The error estimates derived show that if we choose h=O(|logh|^(1/2)H^(3)),then the two-level method we provide has the same H1 and L^(2) convergence orders of the velocity and the pressure as the one-level stabilized method.However,the L^(2) convergence order of the velocity is not consistent with that of one-level stabilized method.Finally,we give the numerical results to support the theoretical analysis.

Modified Inertial Projection Method for Solving Pseudomonotone Variational Inequalities with Non-Lipschitz in Hilbert Spaces

This paper deals with a class of inertial gradient projection methods for solving a vari-ational inequality problem involving pseudomonotone and non-Lipschitz mappings in Hilbert spaces.The proposed algorithm incorporates inertial techniques and the projection and contraction method.The weak convergence is proved without the condition of the Lipschitz continuity of the mappings.Meanwhile,the linear convergence of the algorithm is established under strong pseudomonotonicity and Lipschitz continuity assumptions.The main results obtained in this paper extend and improve some related works in the literature.

An LQP-Based Two-Step Method for Structured Variational Inequalities

t The logarithmic quadratic proximal(LQP)regularization is a popular and powerful proximal regularization technique for solving monotone variational inequalities with nonnegative constraints.In this paper,we propose an implementable two-step method for solving structured variational inequality problems by combining LQP regularization and projection method.The proposed algorithm consists of two parts.The first step generates a pair of predictors via inexactly solving a system of nonlinear equations.Then,the second step updates the iterate via a simple correction step.We establish the global convergence of the new method under mild assumptions.To improve the numerical performance of our new method,we further present a self-adaptive version and implement it to solve a traffic equilibrium problem.The numerical results further demonstrate the efficiency of the proposed method.

Cutting the Cake of the Emission Cap in the Peer-to-peer Multi-energy Sharing Market

With the development of communication technology and distributed energy resources,trading of carbon emission rights and peer-to-peer energy transactions have become popular research directions on the end-user side.Therefore,a cap-andtrade emission framework with peer-to-peer energy trading is employed in this paper.The emission cap decomposition problem is solved under the circumstances of a multi-energy peer-topeer energy trading market.First,the multi-energy system is introduced in the peer-to-peer energy sharing model.The interaction between the prosumers and the system operator is defined.Then,the total emission cap,set by the operator,is modeled as a constraint.The decomposition of the emission cap is modeled as a cake-cutting game.Finally,the existence and uniqueness of the cake-cutting solution is proven by modeling the game to an equivalent monotone variational inequality problem.The complementary characteristics of multi energy in the market can ensure the utility of prosumers while reducing the total cap.

The Developments of Proximal Point Algorithms

The problem of finding a zero point of a maximal monotone operator plays a central role in modeling many application problems arising from various fields,and the proximal point algorithm(PPA)is among the fundamental algorithms for solving the zero-finding problem.PPA not only provides a very general framework of analyzing convergence and rate of convergence of many algorithms,but also can be very efficient in solving some structured problems.In this paper,we give a survey on the developments of PPA and its variants,including the recent results with linear proximal term,with the nonlinear proximal term,as well as the inexact forms with various approximate criteria.

Proximal Methods with Bregman Distances to Solve VIP on Hadamard Manifolds with Null Sectional Curvature

We present an extension of the proximal point method with Bregman distances to solve variational inequality problems(VIP)on Hadamard manifolds with null sectional curvature.Under some natural assumptions,as for example,the existence of solutions of the VIP and the monotonicity of the multivalued vector field,we prove that the sequence of the iterates given by the method converges to a solution of the problem.Furthermore,this convergence is linear or superlinear with respect to the Bregman distance.

Two-Level Defect-CorrectionMethod for Steady Navier-Stokes Problem with Friction Boundary Conditions

In this paper,we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions,which results in a variational inequality problem of the second kind.Based on Taylor-Hood element,we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh.The error estimates for the velocity in the H1 norm and the pressure in the L^(2) norm are derived.Finally,the numerical results are provided to confirm our theoretical analysis.