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Newton iterations in implicit time-stepping scheme for differential linear complementarity systems

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Abstract

We propose a generalized Newton method for solving the system of nonlinear equations with linear complementarity constraints in the implicit or semi-implicit time-stepping scheme for differential linear complementarity systems (DLCS). We choose a specific solution from the solution set of the linear complementarity constraints to define a locally Lipschitz continuous right-hand-side function in the differential equation. Moreover, we present a simple formula to compute an element in the Clarke generalized Jacobian of the solution function. We show that the implicit or semi-implicit time-stepping scheme using the generalized Newton method can be applied to a class of DLCS including the nondegenerate matrix DLCS and hidden Z-matrix DLCS, and has a superlinear convergence rate. To illustrate our approach, we show that choosing the least-element solution from the solution set of the Z-matrix linear complementarity constraints can define a Lipschitz continuous right-hand-side function with a computable Lipschitz constant. The Lipschitz constant helps us to choose the step size of the time-stepping scheme and guarantee the convergence.

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    Xiaojun Chen

  2. Department of Applied Mathematics and Software, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Shuhuang Xiang

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  1. Xiaojun Chen
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  2. Shuhuang Xiang
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Correspondence to Xiaojun Chen.

Additional information

Xiaojun Chen’s work is supported partly by Hong Kong Research Grant Council grant PolyU5003/09p. Shuhuang Xiang’s work is supported partly by NSF of China (No.10771218,11071260).

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Chen, X., Xiang, S. Newton iterations in implicit time-stepping scheme for differential linear complementarity systems. Math. Program. 138, 579–606 (2013). https://doi.org/10.1007/s10107-012-0527-x

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