Let
Let
Note that when
Problem (
Note that a different norm is used in this paper than in these published papers [
The paper is organized as follows. The relationship between the matrix nearness problem considered in this paper and a monotone linear variational inequality (LVI) is built in Section
For any
In order to simplify the following descriptions, let
Based on (
Since
Let
Let
Since
In this section, we summarize some important concepts and preliminary results which are useful in the coming analysis.
Let
In many cases of practical applications, the closed convex set
We denote the solution set of LVI
Let
For any
For
Given
In fact,
For completeness sake, we include the theorem for LVI (
The method (
It follows from (
The method used in this paper is called
For skew-symmetric
Since the above inequality is true for all
We use the Algorithm
Since
If
Since
If
The result follows from
Let
Since
Note that
Suppose that
Hence, to solve the problem (
Let
It follows from (
Note that if
Let
Let
where
In this section, some examples are provided to illustrate the performance of Algorithm
Consider problem (
Numerical results of example 1.
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Consider problem (
Numerical results of example 2.
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In this paper, a relationship between the matrix nearness problem and the linear variational inequality has been built. The matrix nearness problem considered in this paper can be solved by applying an algorithm for the related linear variational inequality. Based on this point, a projection and contraction method is presented for solving the matrix nearness problem, and the implementing details are introduced in this paper. Numerical experiments show that the method suggested in this paper has a good performance, and the method can be improved by setting the parameters in Algorithm
This research is financially supported by a research grant from the Research Grant Council of China (Project no. 10971095).