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JIMO

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.

JIMO

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.

JIMO

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.

IPI

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.

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.

JIMO

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.

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