Norms of Dual Complex Vectors and Dual Complex Matrices

In this paper,we investigate some properties of dual complex numbers,dual complex vectors,and dual complex matrices.First,based on the magnitude of the dual complex number,we study the Young inequality,the Hölder inequality,and the Minkowski inequality in the setting of dual complex numbers.Second,we define the p-norm of a dual complex vector,which is a nonnegative dual number,and show some related properties.Third,we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices.In particular,we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors,and give expressions of three important operator norms of dual complex matrices.

Some ω-unique and ω-ΡProperties for Linear Transformations on Hilbert Spaces

给一真实(有限维或无限维) 有一个乔丹产品的 Hilbert 空间 H，我们在 H 上为线性转变介绍 ω-unique 和 ω-P 性质的概念，并且在这些概念之中调查一些互联。特别地，我们在 H 上为象 Lyapunov 一样转变讨论 ω-unique 和 ω-P 性质。在 Hilbert 空间的约旦产品和 Lorentz 锥的性质在我们的分析起重要作用。

Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, Btensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP;and hence, they are very easy to calculate.

AN ASYMPTOTICAL O((k + 1)n^3L) AFFINE SCALING ALGORITHM FOR THE P_*(k)-MATRIX LINEAR COMPLEMENTRITY PROBLEM

Based on the generalized Dikin-type direction proposed by Jansen et al in 1997, we give out in this paper a generalized Dikin-type affine scaling algorithm for solving the P-*(kappa)-matrix linear complementarity problem (LCP). Form using high-order correctors technique and rank-one updating, the iteration complexity and the total computational turn out asymptotically O((kappa + 1)root nL) and O((kappa + 1)n(3)L) respectively.

The Sparsest Solution to the System of Absolute Value Equations

On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.

An Introduction to the Computational Complexity of Matrix Multiplication

This article introduces the approach on studying the computational complexity of matrix multiplication by ranks of the matrix multiplication tensors.Basic results and recent developments in this area are reviewed.

A Fixed Point Method for the Linear Complementarity Problem Arising from American Option Pricing

For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over Relaxation （PSOR） has been widely applied to solve the resulted LCP. In this paper, we propose a fixed point iterative method to solve this type of LCPs, where the splitting technique of the matrix is used. We show that the proposed method is globally convergent under mild assumptions. The preliminary numerical results are reported, which demonstrate that the proposed method is more accurate than the PSOR for the problems we tested.

A Homogeneous Smoothing-type Algorithm for Symmetric Cone Linear Programs

In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program （（SCLP） for short） by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of （SCLP）. Furthermore, if （SCLP） has a solution, then the algorithm will generate a solution of （SCLP）, and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of （SCLP）.

Entropy Function-Based Algorithms for Solving a Class of Nonconvex Minimization Problems

Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy function for solving a class of l_(p)minimization problems,which includes the well-known unconstrained l_(2)-l_(p)problem as a special case.We show that any accumulation point of the sequence generated by the proposed algorithm is a stationary point of the l_(p)minimization problem,and derive a lower bound for the nonzero entries of the stationary point of the smoothing problem.We implement a specific version of the proposed algorithm which indicates that the entropy function-based algorithm is effective.

Fault-tolerant Concave Facility Location Problem with Uniform Requirements

In this paper, we consider the fault-tolerant concave facility location problem （FTCFL） with uniform requirements. By investigating the structure of the FTCFL, we obtain a modified dual-fitting bifactor approximation algorithm. Combining the scaling and greedy argumentation technique, the approximation factor is proved to be 1.52.