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.     

Predictor-Corrector Smoothing Newton Method,Based on a New Smoothing Function,for Solving the Nonlinear Complementarity Problem with a P 0 Function

By smoothing a perturbed minimum function, we propose in this paper a new smoothing function. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P ₀ function are discussed. We investigate the boundedness of the iteration sequence generated by noninterior continuation/smoothing methods under the assumption that the solution set of the NCP is nonempty and bounded. Based on the new smoothing function, we present a predictor-corrector smoothing Newton algorithm for solving the NCP with a P ₀ function, which is shown to be globally linearly and locally superlinearly convergent under suitable assumptions. Some preliminary computational results are reported.     

A Null Space Approach for Solving Nonlinear Complementarity Problems

In this work, null space techniques are employed to tackle nonlinear complementarity problems （NCPs）. NCP conditions are transform into a nonlinear programming problem, which is handled by null space algorithms, The NCP conditions are divided into two groups, Some equalities and inequalities in an NCP are treated as constraints, While other equalities and inequalities in an NCP are to be regarded as objective function. Two groups are all updated in every step. Null space approaches are extended to nonlinear complementarity problems. Two different solvers are employed for all NCP in an algorithm.     

On construction of new NCP functions

We report a new method to construct complementarity functions for the nonlinear complementarity problem (NCP). Basic properties related to growth behavior, convexity and semismoothness of the newly discovered NCP functions are proved. We also present some variants, generalizations and other transformations of these NCP functions. Finally, we propose some interesting research directions that can be explored in the NCP research.     

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.     

Two Classes of Merit Functions for Infinite-Dimensional Second Order Complimentary Problems

In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.     

The prediction–correction approach to nonlinear complementarity problems

This paper presents a prediction–correction approach to solving the nonlinear complementarity problem (NCP). Each iteration of the new method consists of a prediction and a correction. The predictor is produced by an inexact Logarithmic-Quadratic Proximal method; and then it is corrected by the Proximal Point Algorithm. Convergence of the new method is proved under mild assumptions. Comparison to existing methods shows the superiority of the new method. Numerical experiments including the application to traffic equilibrium problems demonstrate that the new method is attractive in practice.     

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．     

New NCP-Functions and Their Properties

Recently, Luo and Tseng proposed a class of merit functions for the nonlinear complementarity problem (NCP) and showed that it enjoys several interesting properties under some assumptions. In this paper, adopting a similar idea to that of Luo and Tseng, we present new merit functions for the NCP, which can be decomposed into component functions. We show that these merit functions not only share many properties with the one proposed by Luo and Tseng but also enjoy additional favorable properties owing to their decomposable structure. In particular, we present fairly mild conditions under which these merit functions have bounded level sets.     

Uniqueness and differentiability of solutions of parametric nonlinear complementarity problems

We derive conditions for the local uniqueness of solutions of nonlinear complementarity problems (NCP). We then prove the existence, continuity, and directional differentiability of a locally unique parametric solution of the parametric NCP under stronger assumptions. In the absence of degeneracy this parametric solution is also shown to be continuously differentiable.     

A new smoothing and regularization Newton method for <Emphasis Type="Italic">P</Emphasis><Subscript>0</Subscript>-NCP

The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization Newton method for solving nonlinear complementarity problem with P ₀-function (P ₀-NCP). Without requiring strict complementarity assumption at the P ₀-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.     

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.     

非线性互补问题的一种不可行非内点连续算法

基于Chen-Harker—Kanzow-Smale光滑函数，对单调非线性互补问题NCP(f)给出了一种不可行非内点连续算法，该算法在每次迭代时只需求解一个线性等式系统，执行一次线搜索，算法在NCP(f)的解处不需要严格互补的条件下，具有全局线性收敛性和局部二次收敛性．     

Properties and construction of NCP functions

The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions.     

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．     

On the resolution of monotone complementarity problems

A reformulation of the nonlinear complementarity problem (NCP) as an unconstrained minimization problem is considered. It is shown that any stationary point of the unconstrained objective function is a solution of NCP if the mapping F involved in NCP is continuously differentiable and monotone, and that the level sets are bounded if F is continuous and strongly monotone. A descent algorithm is described which uses only function values of F. Some numerical results are given.     

A penalty method for a finite-dimensional obstacle problem with derivative constraints

We propose a power penalty method for an obstacle problem arising from the discretization of an infinite-dimensional optimization problem involving differential operators in both its objective function and constraints. In this method we approximate the mixed nonlinear complementarity problem (NCP) arising from the KKT conditions of the discretized problem by a nonlinear penalty equation. We then show the solution to the penalty equation converges exponentially to that of the mixed NCP. Numerical results will be presented to demonstrate the theoretical convergence rates of the method.     

Smoothing inexact Newton methods for NCP with various nonmonotone techniques

Various iterative methods for solving nonlinear complementarity problems (NCP) are developed in recent years. In this paper we propose Jacobian smoothing inexact Newton methods for NCP with different nonmonotone strategies. The methods are based on semismooth equation reformulation of NCP by Fischer-Burmeister function. Nonmonotone line-search techniques are used for globalization procedure. Numerical performance of algorithms are compared. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two classes of merit functions for the second-order cone complementarity problem

Recently Tseng (Math Program 83:159–185, 1998) extended a class of merit functions, proposed by Luo and Tseng (A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, pp. 204–225, 1997), for the nonlinear complementarity problem (NCP) to the semidefinite complementarity problem (SDCP) and showed several related properties. In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases. In addition, we study another class of merit functions which are based on a slight modification of the aforementioned class of merit functions. Both classes of merit functions provide an error bound for the SOCCP and have bounded level sets.Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.     

The non-interior continuation methods for solving theP 0 function nonlinear complementarity problem

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving theP ₀ function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving theP ₀ function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP