A family of NCP functions and a descent method for the nonlinear complementarity problem (original) (raw)
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A New Derivative-Free Descent Method for the Nonlinear Complementarity Problem
Applied Optimization, 2000
Recently, much eort has been made in solving and analyzing the nonlinear complementarity problem (NCP) by means of a reformulation of the problem as an equivalent unconstrained optimization problem involving a merit function. In this paper, we propose a new merit function for the NCP and show several favorable properties of the proposed function. In particular, we give conditions under which the function provides a global error bound for the NCP and conditions under which its level sets are bounded. Moreover, we propose a new derivative-free descent algorithm for solving the NCP based on this function. We show that any accumulation point generated by the algorithm is a solution of the NCP under the monotonicity assumption on the problem. Also, we prove that the sequence generated by the algorithm converges linearly to the solution under the strong monotonicity assumption. The most interesting feature of the algorithm is that the search direction and the stepsize are adjusted simultaneously during the linesearch process, whereas a xed search direction is used in the linesearch process in earlier derivative-free algorithms proposed by Geiger and Kanzow, Jiang, Mangasarian and Solodov, and Yamashita and Fukushima. Making use of this particular feature, we can establish the linear convergence of the algorithm under more practical assumptions compared with the linearly convergent derivative-free algorithm recently proposed by Mangasarian and Solodov.
Journal of Industrial and Management Optimization, 2012
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.
Information Sciences, 2010
In this paper, we consider a neural network model for solving the nonlinear complementarity problem (NCP). The neural network is derived from an equivalent unconstrained minimization reformulation of the NCP, which is based on the generalized Fischer-Burmeister function / p ða; bÞ ¼ kða; bÞk p À ða þ bÞ. We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability as well as exponential stability. It was found that a larger p leads to a better convergence rate of the trajectory. Numerical simulations verify the obtained theoretical results.
On the Resolution of the Generalized Nonlinear Complementarity Problem
SIAM Journal on Optimization, 2002
Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems. Theoretical results that relate stationary points of the function that is minimized to the solutions of the GNCP are presented. Perturbations of the GNCP are also considered, and results are obtained related to the resolution of GNCPs with very general assumptions on the data. These theoretical results show that local methods for box-constrained optimization applied to the associated problem are efficient tools for solving the GNCP. Numerical experiments are presented that encourage the use of this approach.
2006
] where an NCP-function and a descent method were proposed for the nonlinear complementarity problem. An unconstrained reformulation was formulated due to a merit function based on the proposed NCP-function. We continue to explore properties of the merit function in this paper. In particular, we show that the gradient of the merit function is globally Lipschitz continuous which is important from computational aspect. Moreover, we show that the merit function is SC 1 function which means it is continuously differentiable and its gradient is semismooth. On the other hand, we provide an alternative proof, which uses the new properties of the merit function, for the convergence result of the descent method considered in [
Solving complementarity problems by means of anew smooth constrained nonlinear solverRoberto
1999
Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 and F (x) = 0 is considered. For solving this problem an algorithm of Inexact-Newton type is de-ned. Global and local convergence proofs are presented. As a practical application, the Horizontal Nonlinear Complementarity Problem is introduced. It is shown that the Inexact-Newton algorithm can be applied to this problem. Numerical experiments are performed and commented.