A projection-filter method for solving nonlinear complementarity problems

The Josephy-Newton method attacks nonlinear complementarity problems which consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. Utilizing the filter method, we presented a new globalization strategy for this Newton method applied to nonlinear complementarity problem without any merit function. The strategy is based on the projection-proximal point and filter methodology. Our linesearch procedure uses the regularized Newton direction to force global convergence by means of a projection step which reduces the distance to the solution of the problem. The resulting algorithm is globally convergent to a solution. Under natural assumptions, locally superlinear rate of convergence was established.     

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.     

A smoothing-type algorithm for solving nonlinear complementarity problems with a non-monotone line search

The smoothing-type algorithm has been successfully applied to solve various optimization problems. In general, the smoothing-type algorithm is designed based on some monotone line search. However, in order to achieve better numerical results, the non-monotone line search technique has been used in the numerical computations of some smoothing-type algorithms. In this paper, we propose a smoothing-type algorithm for solving the nonlinear complementarity problem with a non-monotone line search. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are also reported.     

非对称线性互补问题的并行二级多分裂迭代法

提出了求解非对称线性互补问题的并行二级多分裂迭代算法,并证明了该算法的收敛性,最后通过数值实验验证了算法的有效性和可行性.     

New multiplier method for solving linear complementarity problems

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. By introducing a Lagrangian of LCP(q, M), a new smooth merit function ϑ(x, λ) for LCP(q, M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When M is a P-matrix, the sequence {ϑ(x ^k, λ ^k)} (where {(x ^k, λ ^k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations if the solution is degenerate. Numerical results suggest that the method is highly efficient and promising. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(2): 162–171     

A hybrid inexact Logarithmic-Quadratic Proximal method for nonlinear complementarity problems

Inspired by the Logarithmic-Quadratic Proximal method [A. Auslender, M. Teboulle, S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12 (1999) 31-40], we present a new prediction-correction method for solving the nonlinear complementarity problems. In our method, an intermediate point is produced by approximately solving a nonlinear equation system based on the Logarithmic-Quadratic Proximal method; and the new iterate is obtained by convex combination of the previous point and the one generated by the improved extragradient method at each iteration. The proposed method allows for constant relative errors and this yields a more practical Logarithmic-Quadratic Proximal type method. The global convergence is established under mild conditions. Preliminary numerical results indicate that the method is effective for large-scale nonlinear complementarity problems.     

A semismooth equation approach to the solution of nonlinear complementarity problems

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.     

A smoothing conic trust region filter method for the nonlinear complementarity problem

This paper discusses nonlinear complementarity problems; its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. Filter methods are extensively studied to handle nonlinear complementarity problem. Because of good numerical results, filter techniques are attached. By means of a filter strategy, we present a new trust region method based on a conic model for nonlinear complementarity problems. Under a proper condition, the superlinear convergence of the algorithm is established without the strict complementarity condition.     

A new modified one-step smoothing Newton method for solving the general mixed complementarity problem

In last decades, there has been much effort on the solution and the analysis of the mixed complementarity problem (MCP) by reformulating MCP as an unconstrained minimization involving an MCP function. In this paper, we propose a new modified one-step smoothing Newton method for solving general (not necessarily P₀) mixed complementarity problems based on well-known Chen-Harker-Kanzow-Smale smooth function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established.     

INEXACT DAMPED NEWTON METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

In this paper, we propose an inexact clamped Newton method for solving nonlinear complementarity problems based on the equivalent B-differentiable equations.Global convergence and locally quadratic convergence are obtained,and numerical results are given.     

An algorithm for a class of nonlinear complementarity problems with non-Lipschitzian functions

In this paper, we focus on solving a class of nonlinear complementarity problems with non-Lipschitzian functions. We first introduce a generalized class of smoothing functions for the plus function. By combining it with Robinson's normal equation, we reformulate the complementarity problem as a family of parameterized smoothing equations. Then, a smoothing Newton method combined with a new nonmonotone line search scheme is employed to compute a solution of the smoothing equations. The global and local superlinear convergence of the proposed method is proved under mild assumptions. Preliminary numerical results obtained applying the proposed approach to nonlinear complementarity problems arising in free boundary problems are reported. They show that the smoothing function and the nonmonotone line search scheme proposed in this paper are effective.     

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.     

随机R0张量互补问题的投影Levenberg-Marquardt方法

本文考虑一类离散型随机$R_0$张量互补问题,利用Fischer-Burmeister函数将问题转化为约束优化问题,并用投影Levenberg-Marquardt方法对其进行了求解。在一般的条件下得到了该方法的全局收敛性,相关的数值实验表明了该方法的有效性。     

Improving the convergence of non-interior point algorithms for nonlinear complementarity problems

Recently, based upon the Chen-Harker-Kanzow-Smale smoothing function and the trajectory and the neighbourhood techniques, Hotta and Yoshise proposed a noninterior point algorithm for solving the nonlinear complementarity problem. Their algorithm is globally convergent under a relatively mild condition. In this paper, we modify their algorithm and combine it with the superlinear convergence theory for nonlinear equations. We provide a globally linearly convergent result for a slightly updated version of the Hotta-Yoshise algorithm and show that a further modified Hotta-Yoshise algorithm is globally and superlinearly convergent, with a convergence -order , under suitable conditions, where is an additional parameter.

     

Integral global optimization method for solution of nonlinear complementarity problems

The mapping in a nonlinear complementarity problem may be discontinuous. The integral global optimization algorithm is proposed to solve a nonlinear complementarity problem with a robust piecewise continuous mapping. Numerical examples are given to illustrate the effectiveness of the algorithm.     

An improved LQP-based method for solving nonlinear complementarity problems

The well-known logarithmic-quadratic proximal (LQP)method has motivated a number of efficient numerical algorithms for solving nonlinear complementarity problems (NCPs). In this paper,we aim at improving one of them, i.e., the LQP-based interior prediction-correction method proposed in [He, Liao and Yuan, J. Comp. Math., 2006, 24(1): 33–44], via identifying more appropriate step-sizes in the correction steps. Preliminary numerical results for solving some NCPs arising in traffic equilibrium problems are reported to verify the theoretical assertions.     

An inexact NE/SQP method for solving the nonlinear complementarity problem

In this paper, we present an extension to the NE/SQP method; the latter is a robust algorithm that we proposed for solving the nonlinear complementarity problem in an earlier article. In this extended version of NE/SQP, instead of exactly solving the quadratic program subproblems, approximate solutions are generated via an inexact rule.Under a proper choice for this rule, this inexact method is shown to inherit the same convergence properties of the original NE/SQP method. In addition to developing the convergence theory for the inexact method, we also present numerical results of the algorithm tested on two problems of varying size.     

A smoothing inexact Newton method for nonlinear complementarity problems

In this article, we propose a new smoothing inexact Newton algorithm for solving nonlinear complementarity problems (NCP) base on the smoothed Fischer-Burmeister function. In each iteration, the corresponding linear system is solved only approximately. The global convergence and local superlinear convergence are established without strict complementarity assumption at the NCP solution. Preliminary numerical results indicate that the method is effective for large-scale NCP.     

A smoothing self-adaptive Levenberg-Marquardt algorithm for solving system of nonlinear inequalities

In this paper, we consider the smoothing self-adaptive Levenberg-Marquardt algorithm for the system of nonlinear inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations H(x) = 0. A smoothing self-adaptive Levenberg-Marquardt algorithm is proposed for solving the system of nonlinear inequalities based on the new smoothing function. The Levenberg-Marquardt parameter μ_k is chosen as the product of μ_k = ∥H^k∥^δ with δ ∈ (0, 2] being a positive constant. We will show that if ∥H^k∥^δ provides a local error bound, which is weaker than the non-singularity, the proposed method converges superlinearly to the solution for δ ∈ (0, 1), while quadratically for δ ∈ [1, 2]. Numerical results show that the new method performs very well for system of inequalities.     

非线性互补约束规划问题的一个新的QP-free算法

本文研究了非线性互补约束均衡问题.利用互补函数以及光滑近似法,把非线性互补约束均衡问题转化为一个光滑非线性规划问题,得到了超线性收敛速度,数值实验结果表明本文提出的算法是可行的.