A non-monotone smoothing Newton algorithm for solving the system of generalized absolute value equations

TL;DR

This work tackles solving the generalized absolute value equations $A x + B |x| - b = 0$ by introducing a smoothing function and a non-monotone smoothing Newton algorithm (NSNA). The authors reformulate GAVE as a smooth system $H(z)=0$ with $z=(\mu,x)$ using the smoothing function $\phi(\mu,x)$, and develop NSNA with a merit function $\mathcal{M}(z)=||H(z)||^2$ and non-monotone line search. They prove global convergence and, under a column $\mathcal{W}$-property assumption, local quadratic convergence to a solution, with numerical experiments on HLCP-derived instances showing NSNA can outperform existing monotone smoothing Newton methods. The results provide a robust, efficient approach for solving GAVE, with implications for related LCP/HLCP problems in optimization.

Abstract

The system of generalized absolute value equations (GAVE) has attracted more and more attention in the optimization community. In this paper, by introducing a smoothing function, we develop a smoothing Newton algorithm with non-monotone line search to solve the GAVE. We show that the non-monotone algorithm is globally and locally quadratically convergent under a weaker assumption than those given in most existing algorithms for solving the GAVE. Numerical results are given to demonstrate the viability and efficiency of the approach.

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Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
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• Definition 2.3
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• Definition 2.5
• Lemma 2.1
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• Lemma 2.3