Abstract
In this paper, a double nonmonotone quasi-Newton method is proposed for the nonlinear complementarity problem. By using 3-1 piecewise and 4-1 piecewise nonlinear complementarity functions, the nonlinear complementarity problem is reformulated into a smooth equation. By a double nonmonotone line search, a smooth Broyden-like algorithm is proposed, where a single solution of a smooth equation at each iteration is required with the reduction in the scale of the calculation. Under suitable conditions, the global convergence of the algorithm is proved, and numerical results with some practical applications are given to show the efficiency of the algorithm.
Highlights
In this paper, we consider the following nonlinear complementarity problem (NCP): find x ∈ Rn such that x ≥ 0, F(x) ≥ 0, xTF(x) 0. (1)where F: Rn ⟶ Rn is continuously differentiable and the superscript T denotes the transpose operator
Nonlinear complementarity problems arisen in many practical applications, for example, the KKT systems of mathematical programming problem, the economic equilibrium, the engineering design problem, can be reformulated into the NCP [1,2,3]
Based on the piecewise NCP functions, the nonlinear complementarity problem is transformed into the smooth equation
Summary
We consider the following nonlinear complementarity problem (NCP): find x ∈ Rn such that x ≥ 0, F(x) ≥ 0, xTF(x) 0. Motivated by the 3-1 piecewise NCP function, Su and Yang [17, 18] developed smooth-based Newton algorithms with nonmonotone line search for nonlinear complementarity and generalized nonlinear complementarity problems. Different from the Mathematical Problems in Engineering previous methods, the authors introduced independent variable quantities to simplify the algorithm, reducing the amount of calculation without using the smoothing parameter. Smoothing procedure allows one to use successful quasiNewton approaches, and there are many quasi Newton methods available for the nonlinear complementarity problems based on some smoothing functions [19,20,21,22,23,24,25,26]. We will construct a 3-1 piecewise and 4-1 piecewise NCP functions and develop a double nonmonotone quasi Newton method to solve the nonlinear complementarity problems. Based on the piecewise NCP functions, the nonlinear complementarity problem is transformed into the smooth equation.
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