Convergence properties of a non-interior-point smoothing algorithm for the P*NCP
Zheng-Hai Huang Shang-Wen Xu
In this paper, a non-interior-point smoothing algorithm is applied to solve the $P_*$ nonlinear complementarity problem (NCP). The algorithm is proved to be globally convergent under an assumption that the $P_*$ NCP has a nonempty solution set. In particular, the solution obtained by the algorithm is shown to be a maximally complementary solution of the $P_*$ NCP. The results we obtained strictly generalize the relative results appeared in the literature.
keywords: $P_*$ nonlinear complementarity problem maximally complementarity solution. non-interior-point smoothing algorithm
A smoothing Newton algorithm for mathematical programs with complementarity constraints
Zheng-Hai Huang Jie Sun
We propose a smoothing Newton algorithm for solving mathematical programs with complementarity constraints (MPCCs). Under some reasonable conditions, the proposed algorithm is shown to be globally convergent and to generate a $B$-stationary point of the MPCC. Preliminary numerical results on some MacMPEC problems are reported.
keywords: global convergence. mathematical program with complementarity constraints B-stationary point smoothing algorithm
A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function
Na Zhao Zheng-Hai Huang
In this paper, we propose a new class of smoothing functions which uniformly approximates the median function of three scalars. The proposed functions are the generalization of the smoothing function proposed by Li and Fukushima. Some favorable properties of the functions are investigated. By using the proposed functions, we reformulate the box constrained variational inequality problem (VIP) as a system of parameterized smooth equations, and then propose a smoothing Newton algorithm with a nonmonotone line search to solve the VIP. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results for test problems from MCPLIB are also reported, which demonstrate that the proposed smoothing functions are valuable and the proposed algorithm is effective.
keywords: smoothing function Box constrained variational inequality problem smoothing Newton method nonmonotone line search.
Positive definiteness of Diffusion Kurtosis Imaging
Shenglong Hu Zheng-Hai Huang Hong-Yan Ni Liqun Qi
Diffusion Kurtosis Imaging (DKI) is a new Magnetic Resonance Imaging (MRI) model to characterize the non-Gaussian diffusion behavior in tissues. In reality, the term $bD_{app}-\frac{1}{6}b^2D_{app}^2K_{app}$ in the extended Stejskal and Tanner equation of DKI should be positive for an appropriate range of $b$-values to make sense physically. The positive definiteness of the above term reflects the signal attenuation in tissues during imaging. Hence, it is essential for the validation of DKI.
    In this paper, we analyze the positive definiteness of DKI. We first characterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical computation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.
keywords: conic linear programming. Diffusion kurtosis tensor positive definiteness
Global and global linear convergence of smoothing algorithm for the Cartesian $P_*(\kappa)$-SCLCP
Zheng-Hai Huang Nan Lu
In this paper, we consider the linear complementarity problem over Euclidean Jordan algebras with a Cartesian $P_*(\kappa)$-transformation, which is denoted by the Cartesian $P_*(\kappa)$-SCLCP. A smoothing algorithm is extended to solve the Cartesian $P_*(\kappa)$-SCLCP. We show that the algorithm is globally convergent if the problem concerned has a solution. In particular, we show that the algorithm is globally linearly convergent under a weak assumption.
keywords: smoothing algorithm. symmetric cone Complementarity problem Euclidean Jordan algebra
Test of copositive tensors
Li Li Xinzhen Zhang Zheng-Hai Huang Liqun Qi

In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

keywords: Symmetric tensor polynomial optimization SDP relaxation
Global error bounds for the tensor complementarity problem with a P-tensor
Mengmeng Zheng Ying Zhang Zheng-Hai Huang

As a natural extension of the linear complementarity problem, the tensor complementarity problem has been studied recently; and many theoretical results have been obtained. In this paper, we investigate the global error bound for the tensor complementarity problem with a P-tensor. We give two global error bounds for this class of complementarity problems with the help of two positively homogeneous operators defined by a P-tensor. When the order of the involved tensor reduces to 2, the results obtained in this paper coincide exactly with the one for the linear complementarity problem.

keywords: Tensor complementarity problem global error bound P-tensor positively homogeneous operator linear complementarity problem

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