A simplicial algorithm for the nonlinear complementarity problem

A triangulation of the nonnegative orthant and a special labeling of the vertices lead to a combinatorial procedure for seeking solutions or approximate solutions to the nonlinear complementarity problem under coercive-like assumptions on the problem functions. Derivatives are not required. Convergence is proved, computational considerations are discussed, and some preliminary applications to convex programming and saddle point computation, along with numerical results, are presented.     

A penalty method for a mixed nonlinear complementarity problem

We propose a power penalty method for a mixed nonlinear complementarity problem (MNCP) and show that the solution to the penalty equation converges to that of the MNCP exponentially as the penalty parameter approaches infinity, provided that the mapping involved in the MNCP is both continuous and ξ-monotone. Furthermore, a convergence theorem is established when the monotonicity assumption on the mapping is removed. To demonstrate the usefulness and the convergence rates of this method, we design a non-trivial test MNCP problem arising in shape-preserving bi-harmonic interpolation and apply our method to this test problem. The numerical results confirm our theoretical findings.     

A modified SQP-filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.     

一种求解非线性互补问题的filter内点算法

利用Armijio条件和信赖域方法,构造新的价值函数.首次将内点算法与filter技术结合起来,提出一种求解非线性互补问题的新算法,即filter内点算法.在主算法中使用Armijio型线搜索求取步长,在修复算法中使用信赖域方法进行适当控制以保证算法的收敛性.文章还讨论了算法的全局收敛性.最后用数值实验表明了该方法是有效的.     

Remarks on a nonlinear complementarity problem

A class of nonlinear complementarity problems defined by monotone operators is considered in the spaceL _p [a,b], and an existence theorem is established using Tarski's fixed-point theorem.     

Predictor-corrector method for nonlinear complementarity problem

Recently, Ye et al. proved that the predictor-corrector method proposed by Mizuno et al. maintains -iteration complexity while exhibiting the quadratic convergence of the dual gap to zero under very mild conditions. This impressive result becomes the best-known in the interior point methods. In this paper, we modify the predictor-corrector method and then extend it to solving the nonlinear complementarity problem. We prove that the new method has a -iteration complexity while maintaining the quadratic asymptotic convergence.     

A smoothing Newton-type method for generalized nonlinear complementarity problem

By using a new type of smoothing function, we first reformulate the generalized nonlinear complementarity problem over a polyhedral cone as a smoothing system of equations, and then develop a smoothing Newton-type method for solving it. For the proposed method, we obtain its global convergence under milder conditions, and we further establish its local superlinear (quadratic) convergence rate under the BD-regular assumption. Preliminary numerical experiments are also reported in this paper.     

A smoothing least square method for nonlinear complementarity problem

By using the smoothing functions and the least square reformulation, in this paper, we present a smoothing least square method for the nonlinear complementarity problem. The method can overcome the difficulty of the non‐smooth method and a major drawback of some existed equation‐based methods. Under the standard assumptions, we obtain the global convergence of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds

Ideas of a simplicial variable dimension restart algorithm to approximate zero points onR ⁿ developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds. The algorithm can be considered as a modification of the2n-ray zero point finding algorithm onR ⁿ . It appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n-ray algorithm applied to an equivalent zero point problem. This is caused by the fact that the algorithm utilizes the complementarity conditions on the variables. This work is part of the VF-program “Equilibrium and Disequilibrium in Demand and Supply,” which has been approved by the Netherlands Ministry of Education and Sciences.     

On asymptotic approximation of a solution of a boundary-value problem for a nonlinear nonautonomous equation

On the basis of periodic Ateb functions, in the resonance and nonresonance cases, we construct the asymptotic approximation of one-frequency solutions of a boundary-value problem for a nonlinear nonautonomous equation.     

Inexact Newton methods for the nonlinear complementarity problem

An exact Newton method for solving a nonlinear complementarity problem consists of solving a sequence of linear complementarity subproblems. For problems of large size, solving the subproblems exactly can be very expensive. In this paper we study inexact Newton methods for solving the nonlinear, complementarity problem. In such an inexact method, the subproblems are solved only up to a certain degree of accuracy. The necessary accuracies that are needed to preserve the nice features of the exact Newton method are established and analyzed. We also discuss some extensions as well as an application. This research was based on work supported by the National Science Foundation under grant ECS-8407240.     

Parallel Newton methods for the nonlinear complementarity problem

In this paper, we discuss how the basic Newton method for solving the nonlinear complementarity problem can be implemented in a parallel computation environment. We propose some synchronized and asynchronous Newton methods and establish their convergence.This work was based on research supported by the National Science Foundation under grant ECS-8407240 and by a University Research and Development grant from Cray Research Inc. The research was initiated when the authors were with the University of Texas at Dallas.     

A kind of nonmonotone filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. But the QP subproblem may be inconsistent. In this paper, we propose a kind nonmonotone filter method in which the QP subproblem is consistent. By means of nonmonotone filter, this method has no demand on the penalty parameter which is difficult to obtain. Moreover, the restoration phase is not needed any more. Under reasonable conditions, we obtain the global convergence of the algorithm. Some numerical results are presented.     

A two-step modulus-based multisplitting iteration method for the nonlinear complementarity problem

In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for the nonlinear complementarity problem. And we prove its convergence when the system matrix is an $H$-matrix with positive diagonal elements. Numerical experiments show that the proposed method is efficient.     

On Tamir's algorithm for solving the nonlinear complementarity problem

Some comments concerning Tamir's algorithm for solving the nonlinear complementarity problem are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lower-dimensional linear complementarity problem approaches to the solution of a bi-obstacle problem

A globally convergent Broyden-like method for solving a bi-obstacle problem is proposed based on its equivalent lower-dimensional linear complementarity problem. A suitable line search technique is introduced here. The global and superlinear convergence of the method is verified under appropriate assumptions.