Method For Nonlinear Complementarity Problems Research Articles

A new predicto-corrector method for pseudomonotone nonlinear complementarity problems

Based on the Logarithmic-Quadratic Proximal (LQP) method 2, we propose a new prediction-correction method for nonlinear complementarity problem (NCP). We obtain the predictor through a simplified inexact logarithmic-quadratic proximal method under a relaxed inexact criterion. The corrector is obtained by the improved extragradient method. Under certain conditions, the global convergence of the proposed method is proved. Preliminary numerical results indicate the efficiency of the proposed method.

Globalizing a nonsmooth Newton method via nonmonotone path search

We give a framework for the globalization of a nonsmooth Newton method. In part one we start with recalling B. Kummer’s approach to convergence analysis of a nonsmooth Newton method and state his results for local convergence. In part two we give a globalized version of this method. Our approach uses a path search idea to control the descent. After elaborating the single steps, we analyze and prove the global convergence resp. the local superlinear or quadratic convergence of the algorithm. In the third part we illustrate the method for nonlinear complementarity problems.

The convergence of a smoothing damped Gauss–Newton method for nonlinear complementarity problem

The nonlinear complementarity problem (denoted by NCP( F )) can be reformulated as the least l 2 -norm solution of a possibly inconsistent nonsmooth system of equations. In this paper, by introducing a novel smoothing function, the least square problem is approximated by a sequence of parameterized optimization problems which are of twice continuously differentiable objective functions. A smoothing damped Gauss–Newton method is utilized to solve the parameterized optimization problems. The global convergence of the proposed algorithm is proved under mild conditions, and local quadratic convergence of the method is obtained with suitable assumptions.

Jacobian smoothing Brown’s method for NCP

Jacobian smoothing Brown’s method for nonlinear complementarity problems (NCP) is studied in this paper. This method is a generalization of classical Brown’s method. It belongs to the class of Jacobian smoothing methods for solving semismooth equations. Local convergence of the proposed method is proved in the case of a strictly complementary solution of NCP. Furthermore, a locally convergent hybrid method for general NCP is introduced. Some numerical experiments are also presented.

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for [formula omitted]-NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P 0 -function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φ p with p > 1 . We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p , for example p ∈ [ 1.1 , 2 ] , usually has better numerical performance, and the generalized FB functions φ p with p ∈ [ 1.1 , 2 ) can be used as the substitutions for the FB function φ 2 .

The quadratic convergence of a smoothing Levenberg–Marquardt method for nonlinear complementarity problem

The nonlinear complementarity problem (denoted by NCP( F)) can be reformulated as the solution of a possibly inconsistent nonsmooth system of equations. Based on the ideas developed in smoothing Newton methods, we approximated the problem of the least l 2-norm solution of the equivalent nonsmooth equations of NCP( F) with a family of parameterized optimization problem with twice continuously differentiable objective functions by making use of a new smoothing function. Then we presented a smoothing Levenberg–Marquardt method to solve the parameterized smooth optimization problem. By using the smooth and semismooth technique, the local quadratic convergence of the proposed method is proved under some suitable assumptions.

A new smoothing quasi-Newton method for nonlinear complementarity problems

A new smoothing quasi-Newton method for nonlinear complementarity problems is presented. The method is a generalization of Thomas’ method for smooth nonlinear systems and has similar properties as Broyden's method. Local convergence is analyzed for a strictly complementary solution as well as for a degenerate solution. Presented numerical results demonstrate quite similar behavior of Thomas’ and Broyden's methods.

Modified proximal-point method for nonlinear complementarity problems

In this paper, we suggest and analyze a new modified proximal-point method for solving nonlinear complementarity problems. Under suitable conditions, we prove that the new method is globally convergent. We report some preliminary computational results to illustrate the efficiency of the proposed method.

A new smoothing quasi-Newton method for nonlinear complementarity problems

The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing quasi-Newton method for the solution of the nonlinear complementarity problems by constructing a new smoothing approximation function. Global and local superlinear convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in the paper.

Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P 0-Functions

By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP with P 0 functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition.

Solving nonlinear complementarity problems with neural networks: a reformulation method approach

In this paper, we present a neural network approach for solving nonlinear complementarity problems. The neural network model is derived from an unconstrained minimization reformulation of the complementarity problem. The existence and the convergence of the trajectory of the neural network are addressed in detail. In addition, we also explore the stability properties, such as the stability in the sense of Lyapunov, the asymptotic stability and the exponential stability, for the neural network model. The theory developed here is also valid for neural network models derived from a number of reformulation methods for nonlinear complementarity problems. Simulation results are also reported.

The global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform P -functions

We propose an infeasible non-interior path-following method for nonlinear complementarity problems with uniform P-functions. This method is based on the smoothing techniques introduced by Kanzow. A key to our analysis is the introduction of a new notion of neighborhood for the central path which is suitable for infeasible non-interior path-following methods. By restricting the iterates in the neighborhood of the central path, we provide a systematic procedure to update the smoothing parameter and establish the global linear convergence of this method. Some preliminary computational results are reported.

A smoothing Newton method for general nonlinear complementarity problems

Smoothing Newton methods for nonlinear complementarity problems NCP(F) often require F to be at least a P0-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP(F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points.

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ 2n →ℜ 2n , u∈ℜ n is a parameter variable and x∈ℜ n is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z k =(u k ,x k )} such that the mapping H(·) is continuously differentiable at each z k and may be non-differentiable at the limiting point of {z k }. We prove that three most often used Gabriel-More smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising.

Continuation method for nonlinear complementarity problems via normal maps

In a recent paper by Chen and Mangasarian (C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications 2 (1996), 97–138) a class of parametric smoothing functions has been proposed to approximate the plus function present in many optimization and complementarity related problems. This paper uses these smoothing functions to approximate the normal map formulation of nonlinear complementarity problems (NCP). Properties of the smoothing function are investigated based on the density functions that defines the smooth approximations. A continuation method is then proposed to solve the NCPs arising from the approximations. Sufficient conditions are provided to guarantee the boundedness of the solution trajectory. Furthermore, the structure of the subproblems arising in the proposed continuation method is analyzed for different choices of smoothing functions. Computational results of the continuation method are reported.

On the local convergence of quasi-Newton methods for nonlinear complementarity problems

A family of Least-Change Secant-Update methods for solving nonlinear complementarity problems based on nonsmooth systems of equations is introduced. Local and superlinear convergence results for the algorithms are proved. Two different reformulations of the nonlinear complementarity problem as a nonsmooth system are compared, both from the theoretical and the practical point of view. A global algorithm for solving the nonlinear complementarity problem which uses the algorithms introduced here is also presented. Some numerical experiments show a good performance of this algorithm.

Beyond Monotonicity in Regularization Methods for Nonlinear Complementarity Problems

Regularization methods for the solution of nonlinear complementarity problems are standard methods for the solution of monotone complementarity problems and possess strong convergence properties. In this paper, we replace the monotonicity assumption by a P0 -function condition. We show that many properties of regularization methods still hold for this larger class of problems. However, we also provide some counterexamples which indicate that not all results carry over from monotone to P0 -function complementarity problems.

A Global Linear and Local Quadratic Noninterior Continuation Method for Nonlinear Complementarity Problems Based on Chen--Mangasarian Smoothing Functions

A noninterior continuation method is proposed for nonlinear complementarity problems. It improves the noninterior continuation methods recently studied by Burke and Xu [Math. Oper. Res., 23 (1998), pp. 719--734] and Xu [The Global Linear Convergence of an Infeasible Non-Interior Path-following Algorithm for Complementarity Problems with Uniform P-functions, Preprint, Department of Mathematics, University of Washington, Seattle, 1996]; the interior point neighborhood technique is extended to a broader class of smoothing functions introduced by Chen and Mangasarian [Comput. Optim. Appl., 5 (1996), pp. 97--138]. The method is shown to be globally linearly convergent following the methodology established by Burke and Xu. In addition, a local acceleration step is added to the method so that it is also locally quadratically convergent under suitable assumptions.

An inexact QP -based method for nonlinear complementarity problems

We consider a quadratic programming-based method for nonlinear complementarity problems which allows inexact solutions of the quadratic subproblems. The main features of this method are that all iterates stay in the feasible set and that the method has some strong global and local convergence properties. Numerical results for all complementarity problems from the MCPLIB test problem collection are also reported.

A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems

We introduce a new, one-parametric class of NCP-functions. This class subsumes the Fischer function and reduces to the minimum function in a limiting case of the parameter. This new class of NCP-functions is used in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. We present a detailed investigation of the properties of the equation operator, of the corresponding merit function as well as of a suitable semismooth Newton-type method. Finally, numerical results are presented for this method being applied to a number of test problems.