Algorithm For Nonlinear Complementarity Problems Research Articles

An interior-point derivative-free algorithm for nonlinear complementarity problems

An interior-point derivative-free algorithm for nonlinear complementarity problems

Penalized NCP-functions for nonlinear complementarity problems and a scaling algorithm

<p style='text-indent:20px;'>In this paper, we systematically study the properties of penalized NCP-functions in derivative-free algorithms for nonlinear complementarity problems (NCPs), and give some regular conditions for stationary points of penalized NCP-functions to be solutions of NCPs. The main contribution is to unify and generalize previous results. Based on one of above penalized NCP-functions, we analyze a scaling algorithm for NCPs. The numerical results show that the scaling can greatly improve the effectiveness of the algorithm.</p>

A Levenberg–Marquardt Method for Nonlinear Complementarity Problems Based on Nonmonotone Trust Region and Line Search Techniques

Using the FB function, we propose a new Levenberg–Marquardt algorithm for nonlinear complementarity problem. To obtain the global convergence, the algorithm uses the nonmonotone trust region and line search techniques under a convenient boundedness assumption. Furthermore, we get local superlinear/quadratic convergence of the algorithm under a nonsingularity condition. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.

A nonmonotone Levenberg–Marquardt method for nonlinear complementarity problems under local error bound

In this paper, we propose a new Levenberg–Marquardt algorithm for nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem using the FB function. To obtain the global convergence, we use a modified nonmonotone line search rule. Under the local error bound assumption, which is weaker than the nonsingularity condition, we get the local superlinear/quadratic convergence of the algorithm. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.

Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function

This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.

A nonmonotone derivative-free algorithm for nonlinear complementarity problems based on the new generalized penalized Fischer–Burmeister merit function

In this paper, we propose a new generalized penalized Fischer---Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455---471, 2009), which confirm the superior behaviour of the new merit function.

A Mixed Algorithm for Nonlinear Complementarity Problems

Combining nonmonotone trust region algorithms and PSO methods, we propose a mixed method for nonlinear complementarity problems. The iterative formula of μ k is very simple. When the determinant of B k +λ k I is very large or small, the PSO method will obtain a new point which gets us better astringency. Numerical test results show the effectiveness of this algorithm.

Parallel Two-Grid Semismooth Newton-Krylov-Schwarz Method for Nonlinear Complementarity Problems

We develop scalable parallel domain decomposition algorithms for nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems. Semismooth Newton is a popular approach for such problems, however, the method is not suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the grid size; i.e., when the grid is refined, the number of Newton iterations often increases drastically. In this paper, we introduce a family of Newton-Krylov-Schwarz methods based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is totally scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity often associated with obstacle problems. We present numerical results for several large scale calculations obtained on machines with hundreds of processors.

An interior proximal point algorithm for nonlinear complementarity problems

In this paper, we propose a new method for solving nonlinear complementarity problems (NCP), where the underlying function F is pseudomonotone and continuous. The method can be viewed as an extension of the method of Noor and Bnouhachem (2006) [13], by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. We prove the global convergence of the proposed method under some suitable conditions. Some numerical results are given to illustrate the efficiency and the implementation of the new proposed method.

An [formula omitted]-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized Fischer–Burmeister merit function

In the paper [J.-S. Chen, S. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications, 40 (2008) 389–404], the authors proposed a derivative-free descent algorithm for nonlinear complementarity problems (NCPs) by the generalized Fischer–Burmeister merit function: ψ p ( a , b ) = 1 2 [ ‖ ( a , b ) ‖ p − ( a + b ) ] 2 , and observed that the choice of the parameter p has a great influence on the numerical performance of the algorithm. In this paper, we analyze the phenomenon theoretically for a derivative-free descent algorithm which is based on a penalized form of ψ p and uses a different direction from that of Chen and Pan. More specifically, we show that the algorithm proposed is globally convergent and has a locally R -linear convergence rate, and furthermore, its convergence rate will become worse when the parameter p decreases. Numerical results are also reported for the test problems from MCPLIB, which further verify the theoretical results obtained.

A feasible directions algorithm for nonlinear complementarity problems and applications in mechanics

Complementarity problems are involved in mathematical models of several applications in engineering, economy and different branches of physics. We mention contact problems and dynamics of multiple bodies systems in solid mechanics. In this paper we present a new feasible direction algorithm for nonlinear complementarity problems. This one begins at an interior point, strictly satisfying the inequality conditions, and generates a sequence of interior points that converges to a solution of the problem. At each iteration, a feasible direction is obtained and a line search performed, looking for a new interior point with a lower value of an appropriate potential function. We prove global convergence of the present algorithm and present a theoretical study about the asymptotic convergence. Results obtained with several numerical test problems, and also application in mechanics, are described and compared with other well known techniques. All the examples were solved very efficiently with the present algorithm, employing always the same set of parameters.

A derivative-free filter algorithm for nonlinear complementarity problem

Recently, so-called derivative-free methods have attracted much attention, which do not require computation of derivatives of function and are particularly suitable for problems which the derivatives are not available or are extremely expensive to compute. This paper presents a new derivative-free algorithm for nonlinear complementarity problem, which makes use of the efficiency of the filter technique. The algorithm is shown, under the monotonicity assumption, to globally converge to the solution of the nonlinear complementarity problem. The numerical results show the algorithm is feasible.

A positive interior-point algorithm for nonlinear complementarity problems

A new iterative method, which is called positive interior-point algorithm, is presented for solving the nonlinear complementarity problems. This method is of the desirable feature of robustness. And the convergence theorems of the algorithm is established. In addition, some numerical results are reported.

Traffic equilibrium problem with route-specific costs: formulation and algorithms

Using a new gap function recently proposed by Facchinei and Soares [Facchinei, F., Soares, J., 1995. Testing a new class of algorithms for nonlinear complementarity problems. In: Giannessi, F., Maugeri, A. (Eds.), Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York], we convert the nonlinear complementarity problem (NCP) formulation for the traffic equilibrium problem to an equivalent unconstrained optimization. This equivalent formulation uses both route flows and the minimum origin–destination travel costs as the decision variables. Two unique features of this formulation are that: (i) it can model the traffic assignment problem with a general route cost structure; (ii) it is smooth, unconstrained, and that every stationary point of the minimization corresponds to a global minimum. These properties permit a number of efficient algorithms for its solution. Two solution approaches are developed to solve the proposed formulation. Numerical results using a route-specific cost structure are provided and compared with the classic traffic equilibrium problem, which assumes an additive route cost function.

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 Q-order 1 + t, under suitable conditions, where t ∈ (0, 1) is an additional parameter.