In this article, based on M-identity tensor, we establish some parameterized S-type inclusion intervals for fourth-order partially symmetric tensors. The new inclusion intervals are tighter than some existing results. Furthermore, some new S-type upper bounds on the M-spectral radius of fourth-order partially symmetric tensors are obtained. As applications, the new S-type upper bounds as the parameter of WQZ-algorithm can make the algorithm more rapidly converge to the largest M-eigenvalue of fourth-order partially symmetric tensors. Finally, we propose two sufficient conditions for the M-positive definiteness of fourth-order partially symmetric tensors.

Sparsity is the essential characteristic of high-dimensional big data problems. Fused least absolute shrinkage and selection operator (FLASSO) model is an important tool for dealing with sparse problems in high-dimensional multivariate linear regression analysis. In this paper, a new linearized alternating direction method of multipliers (NLADMM) is proposed to solve the FLASSO. We present the range of multipliers of the adaptive step size and the relax factors, and then present an inexact version of NLADMM (INLADMM) to make the algorithm efficient and implement easily. Convergence analysis of the proposed algorithms is established under some mild conditions. Numerical results for solving FLASSO with fixed penalty factors show that INLADMM and NLADMM have better variable selection capabilities than the existing methods such as ADMM, LADMM and ELADMM. Especially, INLADMM also obtains solutions with higher accuracy in fewer time than ADMM, ELADMM and NLADMM.

In this paper, in order to find critical points of vector-valued functions with respect to the partial order induced by a closed, convex, and pointed cone with nonempty interior, we propose a nonlinear modified Polak-Ribiere-Polyak type conjugate gradient method with a nonnegative conjugate parameter. We show that the search direction in our method satisfies the sufficient descent condition independent of any line search. Furthermore, under mild assumptions, we obtain the results of global convergence with the standard Wolfe line search conditions as well as the standard Armijo line search strategy without convexity assumption of the objective functions. Computational experiments are given to show the effectiveness of the proposed method.

Noise-reduction methods are an area of intensive research in signal processing. In this article, a new conjugate gradient method is proposed for noise reduction in signal processing and image restoration. The superiority of this method lies in its employment of the ideas of accelerated conjugate gradient methods in conjunction with a new adaptive method for choosing the step size. In this work, using some assumptions, the weak convergence of the designed method was established. As example applications, we implemented our method to solve signal-processing and image-restoration problems. The results of our numerical simulations demonstrate the effectiveness and superiority of the new approach.