Additive Schwarz algorithm for the nonlinear complementarity problem with M-function

In this paper, an additive Schwarz algorithm is considered for solving the finite-dimensional nonlinear complementarity problem with M-function. The monotone convergence of the algorithm is obtained with special choices of initial values. Moreover, the weighted max-norm bound is obtained for the iterative errors.     

A new pivoting algorithm for the linear complementarity problem allowing for an arbitrary starting point

The linear complementarity problem is to find nonnegative vectors which are affinely related and complementary. In this paper we propose a new complementary pivoting algorithm for solving the linear complementarity problem as a more efficient alternative to the algorithms proposed by Lemke and by Talman and Van der Heyden. The algorithm can start at an arbitrary nonnegative vector and converges under the same conditions as Lemke's algorithm.This research is part of the VF-program Competition and Cooperation.     

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.     

An algorithm for the linear complementarity problem with upper and lower bounds

In this paper, we adapt the octahedral simplicial algorithm for solving systems of nonlinear equations to solve the linear complementarity problem with upper and lower bounds. The proposed algorithm generates a piecewise linear path from an arbitrarily chosen pointz ⁰ to a solution point. This path is followed by linear programming pivot steps in a system ofn linear equations, wheren is the size of the problem. The starting pointz ⁰ is left in the direction of one of the 2 ⁿ vertices of the feasible region. The ray along whichz ⁰ is left depends on the sign pattern of the function value atz ⁰. The sign pattern of the linear function and the location of the points in comparison withz ⁰ completely govern the path of the algorithm.This research is part of the VF-Program Equilibrium and Disequilibrium in Demand and Supply, approved by the Netherlands Ministry of Education, Den Haag, The Netherlands.     

Interior hybrid proximal extragradient methods for the linear monotone complementarity problem

We present new infeasible path-following methods for linear monotone complementarity problems based on Auslender, Teboulle and Ben-Tiba’s log-quadratic barrier functions. The central paths associated with these barriers are always well defined and, for those problems which have a solution, convergent to a pair of complementary solutions. Starting points in these paths are easy to compute. The theoretical iteration-complexity of these new path-following methods is derived and improved by a strategy which uses relaxed hybrid proximal-extragradient steps to control the quadratic term. Encouraging preliminary numerical experiments are presented.     

Simplicial algorithm to solve the nonlinear complementarity problem onS n×R + m

In this paper, a simplicial algorithm is developed to solve the nonlinear complementarity problem onS ⁿ×R ₊ ^m . Furthermore, a condition for convergence is formulated. The triangulation which underlies the algorithm is a combination of the V-triangulation ofS ⁿ and the K-triangulation ofR ₊ ^m . Therefore, we will call it the VK-triangulation.The author wishes to thank Professor G. van der Laan for his valuable comments.     

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 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.     

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.     

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.     

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.     

On a “stability” in the linear complementarity problem

In this work we re-wrote the Linear Complementarity Problem in a formulation based on unknown projector operators. In particular, this formulation allows the introduction of a concept of “stability” that, in a certain way, might explain the way block pivotal algorithm performs.     

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.     

On a homogeneous algorithm for the monotone complementarity problem

Received May 15, 1995 / Revised version received May 20, 1998 Published online October 21, 1998     

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.     

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.     

Parallel pivotal algorithm for solving the linear complementarity problem

We propose a parallel implementation of the classical Lemke's algorithm for solving the linear complementarity problem. The algorithm is designed for a loosely coupled network of computers which is characterized by relatively high communication costs. We provide an accurate prediction of speedup based on a simple operation count. The algorithm produces speedup nearp, wherep is the number of processors, when tested on large problems as demonstrated by computational results on the CRYSTAL token-ring multicomputer and the Sequent Balance 21000 multiprocessor.This material is based on research supported by National Science Foundation Grants DCR-84-20963 and DCR-850-21228 and by Air Force Office of Scientific Research Grants AFSOR-86-0172 and AFSOR-86-0255 while the author was at the University of Wisconsin, Madison, Wisconsin.     

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas—Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.This paper is drawn largely from the dissertation research of the first author. The dissertation was performed at M.I.T. under the supervision of the second author, and was supported in part by the Army Research Office under grant number DAAL03-86-K-0171, and by the National Science Foundation under grant number ECS-8519058.     

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

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