Levenberg-Marquardt method for absolute value equation associated with second-order cone

<p style='text-indent:20px;'>In this paper, we suggest the Levenberg-Marquardt method with Armijo line search for solving absolute value equations associated with the second-order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. We analyze the convergence of the proposed algorithm. For numerical reports, we not only show the efficiency of the proposed method, but also present numerical comparison with smoothing Newton method. It indicates that the proposed algorithm could also be a good choice for solving the SOCAVE.</p>

Smoothing Newton method for $ \ell^0 $-$ \ell^2 $ regularized linear inverse problem

<p style='text-indent:20px;'>Sparse regression plays a very important role in statistics, machine learning, image and signal processing. In this paper, we consider a high-dimensional linear inverse problem with <inline-formula><tex-math id="M3">\begin{document}$ \ell^0 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ \ell^2 $\end{document}</tex-math></inline-formula> penalty to stably reconstruct the sparse signals. Based on the sufficient and necessary condition of the coordinate-wise minimizers, we design a smoothing Newton method with continuation strategy on the regularization parameter. We prove the global convergence of the proposed algorithm. Several numerical examples are provided, and the comparisons with the state-of-the-art algorithms verify the effectiveness and superiority of the proposed method.</p>

A Smoothing Newton Method with a Mixed Line Search for Monotone Weighted Complementarity Problems

In this paper, we present a smoothing Newton method for solving the monotone weighted complementarity problem (WCP). In each iteration of our method, the iterative direction is achieved by solving a system of linear equations and the iterative step length is achieved by adopting a line search. A feature of the line search criteria used in this paper is that monotone and nonmonotone line search are mixed used. The proposed method is new even when the WCP reduces to the standard complementarity problem. Particularly, the proposed method is proved to possess the global convergence under a weak assumption. The preliminary experimental results show the effectiveness and robustness of the proposed method for solving the concerned WCP.