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We present a family of inexact Levenberg-Marquardt (LM) methods for the nonlinear equations which takes more general LM parameters and perturbation vectors. We derive an explicit formula of the convergence order of these inexact LM methods under the H derian local error bound condition and the H derian continuity of the Jacobian. Moreover, we develop a family of inexact LM methods with a nonmonotone line search and prove that it is globally convergent. Numerical results for solving the linear complementarity problem are reported. [ABSTRACT FROM AUTHOR]

We introduce a novel iterative algorithm, termed the Heavy-Ball-Based Hard Thresholding Pursuit for sparse phase retrieval problem (SPR-HBHTP), to reconstruct a sparse signal from a small number of magnitude-only measurements. Our algorithm is obtained via a natural combination of the Hard Thresholding Pursuit for sparse phase retrieval (SPR-HTP) and the classical Heavy-Ball (HB) acceleration method. The robustness and convergence for the proposed algorithm were established with the help of the restricted isometry property. Furthermore, we prove that our algorithm can exactly recover a sparse signal with overwhelming probability in finite steps whenever the initialization is in the neighborhood of the underlying sparse signal, provided that the measurement is accurate. Extensive numerical tests show that SPR-HBHTP has a markedly improved recovery performance and runtime compared to existing alternatives, such as the Hard Thresholding Pursuit for sparse phase retrieval problem (SPR-HTP), the SPARse Truncated Amplitude Flow (SPARTA), and Compressive Phase Retrieval with Alternating Minimization (CoPRAM). [ABSTRACT FROM AUTHOR]

In this paper, we propose a new smoothing Newton method to solve the circular cone programming (denoted by CCP). The proposed method is designed based on a non-monotone derivative-free line search scheme. We show that any accumulation point of the iteration sequence generated by this method is a solution of the CCP. Moreover, we prove that the proposed method is locally quadratically convergent without requiring strict complementarity conditions. Compared with existing smoothing Newton methods for solving the CCP, our method has three new features: (i) the generated iteration sequence is bounded; (ii) the value of the merit function converges to zero; (iii) the whole iteration sequence converges to an accumulation point if this point is isolated. Some numerical results are reported. [ABSTRACT FROM AUTHOR]

We consider the weighted linear complementarity problem (denoted by wLCP). Many numerical algorithms have been proposed for solving the monotone wLCP. In this paper, we propose a damped Gauss–Newton method to solve the non-monotone wLCP which is designed based on a derivative-free non-monotone line search. We show that the proposed method is well defined and it is globally convergent without any problem assumptions. Moreover, we analyze the local quadratic convergence of the proposed method under the non-singularity condition and the local error bound condition, respectively. Our method not only has encouraging local convergence properties but also can be used to solve non-monotone wLCPs. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]