NCP Functions Applied to Lagrangian Globalization for the Nonlinear Complementarity Problem

Based on NCP functions, we present a Lagrangian globalization (LG) algorithm model for solving the nonlinear complementarity problem. In particular, this algorithm model does not depend on some specific NCP function. Under several theoretical assumptions on NCP functions we prove that the algorithm model is well-defined and globally convergent. Several NCP functions applicable to the LG-method are analyzed in details and shown to satisfy these assumptions. Furthermore, we identify not only the properties of NCP functions which enable them to be used in the LG method but also their properties which enable the strict complementarity condition to be removed from the convergence conditions of the LG method. Moreover, we construct a new NCP function which possesses some favourable properties.     

Properties of Restricted NCP Functions for Nonlinear Complementarity Problems

In this paper, we study restricted NCP functions which may be used to reformulate the nonlinear complementarity problem as a constrained minimization problem. In particular, we consider three classes of restricted NCP functions, two of them introduced by Solodov and the other proposed in this paper. We give conditions under which a minimization problem based on a restricted NCP function enjoys favorable properties, such as equivalence between a stationary point of the minimization problem and the nonlinear complementarity problem, strict complementarity at a solution of the minimization problem, and boundedness of the level sets of the objective function. We examine these properties for three restricted NCP functions and show that the merit function based on the restricted NCP function proposed in this paper enjoys favorable properties compared with those based on the other restricted NCP functions.     

Superlinear／Quadratic One-step Smoothing Newton Methodf or P0-NCP

We propose a one-step smoothing Newton method for solving the non-linear complementarity problem with P0-function (P0-NCP) based on the smoothing symmetric perturbed Fisher function(for short, denoted as the SSPF-function). The proposed algorithm has to solve only one linear system of equations and performs only one line search per iteration. Without requiring any strict complementarity assumption at the P0-NCP solution, we show that the proposed algorithm converges globally and superlinearly under mild conditions. Furthermore, the algorithm has local quadratic convergence under suitable conditions. The main feature of our global convergence results is that we do not assume a priori the existence of an accumulation point. Compared to the previous literatures, our algorithm has stronger convergence results under weaker conditions.     

A LQP BASED INTERIOR PREDICTION-CORRECTION METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

To solve nonlinear complementarity problems （NCP）, at each iteration, the classical proximal point algorithm solves a well-conditioned sub-NCP while the Logarithmic-Quadratic Proximal （LQP） method solves a system of nonlinear equations （LQP system）. This paper presents a practical LQP method-based prediction-correction method for NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed restriction, and the new iterate （the corrector） is computed directly by an explicit formula derived from the original LQP method. The implementations are very easy to be carried out. Global convergence of the method is proved under the same mild assumptions as the original LQP method. Finally, numerical results for traffic equilibrium problems are provided to verify that the method is effective for some practical problems.     

非线性互补问题的水平值估计算法

本文研究非线性互补问题（NCP）的求解算法，先将NCP转化为约束全局优化问题（CGOP），然后直接移植求解问题（CGOP）的水平值估计算法^[4,5]来求解问题（NCP）．文章证明了算法对于NCP是收敛的，数值实验说明了算法的有效性．     

A new class of smoothing functions and a smoothing Newton method for complementarity problems

In this paper, we introduce a new class of smoothing functions, which include some popular smoothing complementarity functions. We show that the new smoothing functions possess a system of favorite properties. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P ₀ function are discussed. The Jacobian consistency of this class of smoothing functions is analyzed. Based on the new smoothing functions, we investigate a smoothing Newton algorithm for the NCP and discuss its global and local superlinear convergence. Some preliminary numerical results are reported.     

P_0函数非线性互补问题的一步非内点连续方法的收敛性

本文对于Ｐ０函数非线性互补问题提出了一个基于Ｋａｎｚｏｗ光滑函数的一步非内点连续方法,在适当的假设条件下,证明了方法的全局线性及局部二次收敛性．特别,在方法的全局线性收敛性的分析中,不需要假定非线性互补问题的函数的Ｊａｃｏｂｉ阵是Ｌｉｐｓｃｈｉｔｚ连续的．文献中为了得到非内点连续方法的全局线性收敛性,这一假定是被广泛使用的．本文提出的方法在每一次迭代只须解一个线性方程式组．     

Nonlinear complementarity as unconstrained optimization

Several methods for solving the nonlinear complementarity problem (NCP) are developed. These methods are generalizations of the recently proposed algorithms of Mangasarian and Solodov (Ref. 1) and are based on an unconstrianed minimization formulation of the nonlinear complementarity problem. It is shown that, under certain assumptions, any stationary point of the unconstrained objective function is already a solution of NCP. In particulr, these assumptions are satisfied by the mangasarian and Soolodov implicit Lagranian functioin. Furthermore, a special Newton-type method is suggested, and conditions for its local quadratic convergence are given. Finally, some preliminary numerical results are presented.The author would like to thank Dr. Oswald Knoth (Leipzig) for pointing out that the equivalence of Lemma 2.2. is not true for complementarity problems which have no solutions. He is also grateful to the anonymous referencees for their helpful comments.     

An NE/SQP Method for the Bounded Nonlinear Complementarity Problem

NE/SQP (Refs. 2–3) is a recent algorithm that has proven quite effective for solving the nonlinear complementarity problem (NCP). NE/SQP is robust in the sense that its direction-finding subproblems are always solvable; in addition, the convergence rate of this method is q-quadratic. In this note, we consider a generalized version of NE/SQP, as first described in Ref. 4, which is suitable for the bounded NCP. We extend the work in Ref. 4 by demonstrating a stronger convergence result and present numerical results on test problems.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

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.     

Globally convergent Jacobian smoothing inexact Newton methods for NCP

A new smoothing algorithm for the solution of nonlinear complementarity problems (NCP) is introduced in this paper. It is based on semismooth equation reformulation of NCP by Fischer–Burmeister function and its related smooth approximation. In each iteration the corresponding linear system is solved only approximately. Since inexact directions are not necessarily descent, a nonmonotone technique is used for globalization procedure. Numerical results are also presented. Research supported by Ministry of Science, Republic of Serbia, grant No. 144006.     

On stationary points of the implicit Lagrangian for nonlinear complementarity problems

Mangasarian and Solodov have recently introduced an unconstrained optimization problem whose global minima are solutions of the nonlinear complementarity problem (NCP). In this paper, we show that, if the mapping involved in NCP has a positive-definite Jacobian, then any stationary point of the optimization problem actually solves NCP. We also discuss a descent method for solving the unconstrained optimization problem.The authors are indebted to a referee for a helpful suggestion that led them to develop the descent method described in Section 3. They are grateful to Professor F. Facchinei, who kindly pointed out an error in the proof of Theorem 2.3 in an earlier version of the paper. The also thank Professor P. Tseng for a discussion on Theorem 3.1.     

A new hybrid method for nonlinear complementarity problems

In this paper, a new hybrid method is proposed for solving nonlinear complementarity problems (NCP) with P ₀ function. In the new method, we combine a smoothing nonmonotone trust region method based on a conic model and line search techniques. We reformulate the NCP as a system of semismooth equations using the Fischer-Burmeister function. Using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing nonmonotone trust region algorithm of a conic model for solving the NCP with P ₀ functions. This is different from the classical trust region methods, in that when a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength is defined by a line search. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, the superlinear convergence of the algorithm is established without a strict complementarity condition.     

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. This work was partially supported by the Research Grant Council of Hong Kong and the National Natural Science Foundation of China (Grant 10171030).     

On the ERM formulation and a stochastic approximation algorithm of the stochastic-R_0 EVLCP

In this paper, a class of stochastic extended vertical linear complementarity problems is studied as an extension of the stochastic linear complementarity problem. The expected residual minimization (ERM) formulation of this stochastic extended vertical complementarity problem is proposed based on an NCP function. We study the corresponding properties of the ERM problem, such as existence of solutions, coercive property and differentiability. Finally, we propose a descent stochastic approximation method for solving this problem. A comprehensive convergence analysis is given. A number of test examples are constructed and the numerical results are presented.     

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 P ₀-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 Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998     

A GLOBALal DERIVTIVE-FREE DESCENT METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

1. IntroductionConsider the nonlinear complementarity problem (NCP for short), which is to findan x E M" such thatwhere F: Wu - ac and the inequalities are taken componentwise. This problem havemany important applications in various fields. [13, 7, 5].Due to the less storage in computation, derivative--free descent method, which meansthe search direction used does not involye the Jacobian matrix of F, is popular infinding solutions of nonlinear complementarity Problems. We briefly view som…     

A Globally Derivative-Free Descent Method for Nonlinear Complementarity Problems

Based on a class of functions, which generalize the squared Fischer-Burmeister NCP function and have many desirable properties as the latter function has, we reformulate nonlinear complementarity problem (NCP for short) as an equivalent unconstrained optimization problem, for which we propose a derivative-free descent method in monotone case. We show its global convergence under some mild conditions. If $F$, the function involved in NCP, is $R_0$-function, the optimization problems has bounded level sets. A local property of the merit function is discussed. Finally,we report some numerical results.