Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction Method

Abstract

Let S be a closed convex set of matrices and C be a given matrix. The matrix nearness problem considered in this paper is to find a matrix X in the set S at which max {|𝑥_𝑖𝑗−𝑐_𝑖𝑗|} reaches its minimum value. In order to solve the matrix nearness problem, the problem is reformulated to a min-max problem firstly, then the relationship between the min-max problem and a monotone linear variational inequality (LVI) is built. Since the matrix in the LVI problem has a special structure, a projection and contraction method is suggested to solve this LVI problem. Moreover, some implementing details of the method are presented in this paper. Finally, preliminary numerical results are reported, which show that this simple algorithm is promising for this matrix nearness problem.

1. Introduction

Let 𝐶=(𝑐_𝑖𝑗)∈𝑅^𝑛×𝑛 be a given symmetric matrix and 𝑆^𝑛_Λ=𝐻∈𝑅^𝑛×𝑛∣𝐻^𝑇=𝐻,𝜆_min𝐼⪯𝐻⪯𝜆_max𝐼, (1.1) where 𝜆_min,𝜆_max are given scalars and 𝜆_min<𝜆_max, 𝐼 is the identity matrix, and 𝐴⪯𝐵 denotes that 𝐵−𝐴 is a positive semidefinite matrix. It is clear that 𝑆^𝑛_Λ is a nonempty closed convex set. The problem considered in this paper is min‖𝑋−𝐶‖_inf∣𝑋=𝑥_𝑖𝑗∈𝑆^𝑛_Λ, (1.2) where ‖𝑋−𝐶‖_inf=max𝑥_𝑖𝑗−𝑐_𝑖𝑗∣𝑖=1,…,𝑛,𝑗=1,…,𝑛. (1.3) Throughout the paper we assume that the solution set of problem (1.2) is nonempty.

Note that when 𝜆_min=0 and 𝜆_max=+∞, the set 𝑆^𝑛_Λ reduces to the semidefinite cone 𝑆^𝑛₊=𝐻∈𝑅^𝑛×𝑛∣𝐻^𝑇=𝐻,𝐻⪰0. (1.4) Using the terminology in interior point methods, 𝑆^𝑛₊ is called semidefinite cone, and thus the related problem belongs to the class of semidefinite programming [1].

Problem (1.2) can be viewed as a type of matrix nearness problem, that is, the problem of finding a matrix that satisfies some property and is nearest to a given one. A survey on matrix nearness problems can be found in [2]. The matrix nearness problems have many applications especially in finance, statistics, and compressive sensing. For example, one application in statistics is to make adjustments to a symmetric matrix so that it is consistent with prior knowledge or assumptions, and is a valid covariance matrix [3–8]. Paper [9] discusses a new class of matrix nearness problems that measure approximation error using a directed distance measure called a Bregman divergence and proposes a framework for studying these problems, discusses some specific matrix nearness problems, and provides algorithms for solving them numerically.

Note that a different norm is used in this paper than in these published papers [3–7] and this makes the objective function of problem (1.2) be nonsmooth, and the problem (1.2) cannot be solved very easily. In the next section, we will find that the problem (1.2) can be converted into a linear variational inequality and thus can be solved effectively with the projection and contraction (PC) method which is extremely simple both in theoretical analysis and numerical implementations [10–15].

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 2. In Section 3, some preliminaries on variational inequalities are summarized. The projection and contraction method for the LVI associated with the considered problem is suggested. In Section 4, the implementing details for applying the projection and contraction method to the matrix optimization problems are studied. Preliminary numerical results and some concluding remarks are reported in Sections 5 and 6, respectively.

2. Reformulating the Problem to a Monotone LVI

For any 𝑑∈𝑅^𝑚, we have ‖𝑑‖_∞=max_𝜉∈𝐵1𝜉^𝑇𝑑, (2.1) where 𝐵₁={𝜉∈𝑅^𝑚|‖𝜉‖₁≤1} and 𝜉^𝑇𝑑 is the Euclidean inner-product of 𝜉 and 𝑑.

In order to simplify the following descriptions, let vec(𝐴) be a linear transformation which converts the matrix 𝐴∈𝑅^𝑙×𝑛 into a column vector in 𝑅^𝑙𝑛 obtained by stacking the columns of the matrix 𝐴 on top of one another, that is, vec(𝐴)=𝑎₁₁,…,𝑎_𝑙1,𝑎₁₂,…,𝑎_𝑙2,…,𝑎_1𝑛,…,𝑎_𝑙𝑛^𝑇, (2.2) and let mat(vec(𝐴)) be the original matrix 𝐴, that is, mat(vec(𝐴))=𝐴. (2.3)

Based on (2.1) and the fact that the matrix 𝑋 and 𝐶 are symmetric, problem (1.2) can similarly be rewritten as the following min-max problem: min_{𝑋∈𝑆𝑛Λ}max_𝑍∈ℬvec(𝑍)^𝑇(vec(𝑋)−vec(𝐶)), (2.4) where ℬ=𝑍∈𝑅^𝑛×𝑛∣𝑍=𝑍^𝑇,^𝑛𝑛_𝑖=1𝑗=1𝑧_𝑖𝑗≤1. (2.5)

Remark 2.1. Since 𝑋 and 𝐶 are both symmetric matrices, we can restrict the matrices in set ℬ to be symmetric.

Let Ω₁=𝑥=vec(𝑋)∣𝑋∈𝑆^𝑛_Λ,Ω₂={𝑧=vec(𝑍)∣𝑍∈ℬ}. (2.6)(2.7) Since 𝑆^𝑛_Λ and ℬ are both convex sets, it is easy to prove that Ω₁ and Ω₂ are also convex sets.

Let (𝑋^∗,𝑍^∗)∈𝑆^𝑛_Λ×ℬ be any solution of (2.4) and 𝑥^∗=vec(𝑋^∗),𝑧^∗=vec(𝑍^∗). Then 𝑧^𝑇𝑥^∗−𝑐≤𝑧^∗𝑇𝑥^∗−𝑐≤𝑧^∗𝑇(𝑥−𝑐),∀(𝑥,𝑧)∈Ω, (2.8) where 𝑐=vec(𝐶) and Ω=Ω₁×Ω₂. Thus, (𝑥^∗,𝑧^∗) is a solution of the following variational inequality: find (𝑥^∗,𝑧^∗)∈Ω such that 𝑥−𝑥^∗𝑇𝑧^∗≥0,𝑧−𝑧^∗𝑇−𝑥^∗∀()+𝑐≥0,𝑥,𝑧∈Ω. (2.9) For convenience of coming analysis, we rewrite the linear variational inequality (2.9) in the following compact form: find 𝑢^∗∈Ω such that 𝑢−𝑢^∗𝑇𝑀𝑢^∗+𝑞≥0,∀𝑢∈Ω, (2.10) where 𝑢^∗=𝑥^∗𝑧^∗𝑥𝑧0𝑐,𝑢=,𝑀=0𝐼−𝐼0,𝑞=. (2.11) In the following, we denote the linear variational inequality (2.10)-(2.11) by LVI(Ω,𝑀,𝑞).

Remark 2.2. Since 𝑀 is skew-symmetric, the linear variational inequality LVI(Ω,𝑀,𝑞) is monotone.

3. Projection and Contraction Method for Monotone LVIs

In this section, we summarize some important concepts and preliminary results which are useful in the coming analysis.

3.1. Projection Mapping

Let Ω be a nonempty closed convex set of 𝑅^𝑚. For a given 𝑣∈𝑅^𝑚, the projection of 𝑣 onto Ω, denoted by 𝑃_Ω(𝑣), is the unique solution of the following problem: min_𝑢{‖𝑢−𝑣‖∣𝑢∈Ω}, (3.1) where ‖⋅‖ is the Euclidian norm. The projection under an Euclidean norm plays an important role in the proposed method. A basic property of the projection mapping on a closed convex set is 𝑣−𝑃_Ω(𝑣)^𝑇𝑢−𝑃_Ω(𝑣)≤0,∀𝑣∈𝑅^𝑚,∀𝑢∈Ω. (3.2)

In many cases of practical applications, the closed convex set Ω has a simple structure, and the projection on Ω is easy to carry out. For example, let 𝑒 be a vector whose each element is 1, and ℬ_∞=𝜉∈𝑅^𝑚∣𝜉_∞≤1. (3.3) Then the projection of a vector 𝑑∈𝑅^𝑚 on ℬ_∞ can be obtained by 𝑃_ℬ∞(𝑑)=max{−𝑒,min{𝑑,𝑒}}, (3.4) where the min and max are component wise.

3.2. Preliminaries on Linear Variational Inequalities

We denote the solution set of LVI(Ω,𝑀,𝑞) (2.10) by Ω^∗ and assume that Ω^∗≠∅. Since the early work of Eaves [16], it is well known that the variational inequality LVI(Ω,𝑀,𝑞) problem is equivalent to the following projection equation: 𝑢=𝑃_Ω𝑢−𝑀𝑢+𝑞. (3.5) In other words, to solve LVI(Ω,𝑀,𝑞) is equivalent to finding a zero point of the continuous residue function 𝑒(𝑢)∶=𝑢−𝑃_Ω𝑢−𝑀𝑢+𝑞. (3.6) Hence, 𝑒(𝑢)=0⟺𝑢∈Ω^∗. (3.7) In the literature of variational inequalities, ‖𝑒(𝑢)‖ is called the error bound of LVI. It quantitatively measures how much 𝑢 fails to be in Ω^∗.

3.3. The Projection and Contraction Method

Let 𝑢^∗∈Ω^∗ be a solution. For any 𝑢∈𝑅^𝑚, because 𝑃_Ω[𝑢−(𝑀𝑢+𝑞)]∈Ω, it follows from (2.10) that 𝑃_Ω𝑢−𝑀𝑢+𝑞−𝑢^∗𝑇𝑀𝑢^∗+𝑞≥0,∀𝑢∈𝑅^𝑚. (3.8) By setting 𝑣=𝑢−(𝑀𝑢+𝑞) and 𝑢=𝑢^∗ in (3.2), we have 𝑃_Ω𝑢−𝑀𝑢+𝑞−𝑢^∗𝑇𝑢−𝑀𝑢+𝑞−𝑃_Ω𝑢−𝑀𝑢+𝑞≥0,∀𝑢∈𝑅^𝑚. (3.9) Adding the above two inequalities, and using the notation of 𝑒(𝑢), we obtain 𝑢−𝑢^∗−𝑒(𝑢)^𝑇𝑒(𝑢)−𝑀𝑢−𝑢^∗≥0,∀𝑢∈𝑅^𝑚. (3.10) For positive semi-definite (not necessary symmetric) matrix 𝑀, the following theorem follows from (3.10) directly.

Theorem 3.1 (Theorem  1 in [11]). For any 𝑢^∗∈Ω^∗, we have 𝑢−𝑢^∗𝑇𝑑(𝑢)≥‖𝑒(𝑢)‖²,∀𝑢∈𝑅^𝑚, (3.11) where 𝑑(𝑢)=𝑀^𝑇𝑒(𝑢)+𝑒(𝑢). (3.12)

For 𝑢∈Ω⧵Ω^∗, it follows from (3.11)-(3.12) that −𝑑(𝑢) is a descent direction of the unknown function (1/2)‖𝑢−𝑢^∗‖². Some practical PC methods for LVI based on direction 𝑑(𝑢) are given in [11].

Algorithm 3.2 (Projection and Contraction Method for LVI Problem (2.10)). Given 𝑢⁰∈Ω. for 𝑘=0,1…, if 𝑢^𝑘∉Ω^∗, then do the following 𝑢^𝑘+1=𝑢^𝑘−𝛾𝛼𝑢^𝑘𝑑𝑢^𝑘, (3.13) where 𝛾∈(0,2), 𝑑(𝑢) is defined in (3.12), and 𝛼(𝑢)=‖𝑒(𝑢)‖²‖𝑑(𝑢)‖². (3.14)

Remark 3.3. In fact, 𝛾 is a relaxation factor and it is recommended to be taken from [1,2). In practical computation, usually we take 𝛾∈[1.2,1.9]. The method was first mentioned in [11]. Among the PC methods for asymmetric LVI(Ω,𝑀,𝑞) [10–13], this method makes just one projection in each iteration.
For completeness sake, we include the theorem for LVI (2.10) and its proofs.

Theorem 3.4 (Theorem  2 in [11]). The method (3.13)-(3.14) produces a sequence {𝑢^𝑘}, which satisfies 𝑢^𝑘+1−𝑢^∗2≤𝑢^𝑘−𝑢^∗2−𝛾2−𝛾𝛼𝑢^𝑘𝑒𝑢^𝑘2. (3.15)

Proof. It follows from (3.11) and (3.14) that 𝑢^𝑘+1−𝑢^∗2=𝑢^𝑘−𝑢^∗−𝛾𝛼𝑢^𝑘𝑑𝑢^𝑘2=𝑢^𝑘−𝑢^∗2−2𝛾𝛼𝑢^𝑘𝑢^𝑘−𝑢^∗𝑇𝑑𝑢^𝑘+𝛾²𝛼²𝑢^𝑘𝑑𝑢^𝑘2≤𝑢^𝑘−𝑢^∗2−2𝛾𝛼𝑢^𝑘𝑒𝑢^𝑘2+𝛾²𝛼²𝑢^𝑘𝑑𝑢^𝑘2=𝑢^𝑘−𝑢^∗2−𝛾2−𝛾𝛼𝑢^𝑘𝑒𝑢^𝑘2. (3.16) Thus the theorem is proved.

The method used in this paper is called projection and contraction method because it makes projections in each iteration and the generated sequence is Fejér monotone with respect to the solution set.

For skew-symmetric 𝑀 in LVI (2.10), it is easy to prove that 𝛼(𝑢)≡1/2. Thus, the contraction inequality (3.15) can be simplified to𝑢^𝑘+1−𝑢^∗2≤𝑢^𝑘−𝑢^∗2−𝛾2−𝛾2𝑒𝑢^𝑘2. (3.17) Following (3.17), the convergence results of the sequence {𝑢_𝑘} can be found in [11] or be proved similarly to that in [17].

Since the above inequality is true for all 𝑢^∗∈Ω^∗, we have dist²𝑢^𝑘+1,Ω^∗≤dist²𝑢^𝑘,Ω^∗−𝛾2−𝛾2𝑒𝑢^𝑘2, (3.18) where dist𝑢,Ω^∗=min𝑢−𝑢^∗∣𝑢^∗∈Ω^∗. (3.19) The above inequality states that we get a “great” profit from the 𝑘th iteration, if ‖𝑒(𝑢^𝑘)‖ is not too small; conversely, if we get a very small profit from the 𝑘th iteration, then ‖𝑒(𝑢^𝑘)‖ is already very small and 𝑢^𝑘 is a “sufficiently good” approximation of a 𝑢^∗∈Ω^∗.

4. Implementing Details of Algorithm 3.2 for Solving Problem (1.2)

We use the Algorithm 3.2 to solve the linear variational inequality (2.10) arising from problem (1.2). For a given 𝑢=(𝑥,𝑧)∈Ω, the process of computing a new iterate is listed as follows: 𝑧,𝑒(𝑢)=𝑒𝑀𝑢+𝑞=𝑐−𝑥_𝑥(𝑢)𝑒_𝑧(𝑢)=𝑥−𝑃_Ω1[𝑥−𝑧]𝑧−𝑃_Ω2[(,𝑑(𝑢)=𝑑𝑧+𝑥−𝑐)]_𝑥(𝑢)𝑑_𝑧(𝑢)=𝑀^𝑇𝑒(𝑢)(𝑢)=𝑒+𝑒^𝑥(𝑢)−𝑒_𝑧(𝑢)𝑒_𝑥(𝑢)+𝑒_𝑧(𝑢),𝑢^new=𝑥^new𝑧^new=𝑥−𝛾𝑑_𝑥(𝑢)2𝑧−𝛾𝑑_𝑧(𝑢)2,𝛾∈(0,2). (4.1) The key operations here are to compute 𝑃_Ω1[𝑥−𝑧] and 𝑃_Ω2[𝑧+(𝑥−𝑐)], where mat(𝑥),mat(𝑧),andmat(𝑧+(𝑥−𝑐)) are symmetric matrices. In the following, we first focus on the computing method of 𝑃_Ω1[𝑣], where mat(𝑣) is a symmetric matrix.

Since mat(𝑣) is a symmetric matrix, we have 𝑃_Ω1[𝑣]=argmin_𝑥‖𝑥−𝑣‖²∣𝑥∈Ω₁=vecargmin_𝑋‖𝑋−𝑉‖²_𝐹∣𝑋∈𝑆^𝑛_Λ, (4.2) where 𝑉=mat(𝑣). It is known that the optimal solution of problem min‖𝑋−𝑉‖²_𝐹∣𝑋∈𝑆^𝑛_Λ (4.3) is given by 𝑄Λ𝑄^𝑇, (4.4) where 𝑉𝑄=𝑄Λ,Λ=diag𝜆₁₁,…,𝜆_𝑛𝑛,Λ=diag𝜆₁₁,…,𝜆_𝑛𝑛,𝜆_𝑖𝑖=minmax𝜆_𝑖𝑖,𝜆_min,𝜆_max, (4.5) and ‖⋅‖_𝐹 is the Frobenius norm. Thus, we have 𝑃_Ω1[𝑣]=vec𝑄Λ𝑄^𝑇. (4.6) Now we move to consider the technique of computing 𝑃_Ω2[𝑣], where mat(𝑣) is a symmetric matrix.

Lemma 4.1. If mat(𝑣) is a symmetric matrix and 𝑥^∗ is the solution of the problem min_𝑥‖𝑥−𝑣‖∣^𝑚_𝑖=1𝑥_𝑖≤1 (4.7) then we have mat𝑥^∗=mat𝑥^∗𝑇. (4.8)

Proof. Since argmin_𝑥‖𝑥−𝑣‖∣^𝑚_𝑖=1𝑥_𝑖≤1=vecargmin_𝑋‖𝑋−𝑉‖_𝐹∣^𝑛𝑛_𝑖=1𝑗=1𝑥_𝑖𝑗≤1 (4.9) and 𝑉=𝑉^𝑇, where 𝑉=mat(𝑣), we have that if 𝑥^∗ is the solution of problem (4.7), then vec(mat(𝑥^∗)^𝑇) is also the solution of problem (4.7). As it is known that the solution of problem (4.7) is unique. Thus, 𝑥^∗=vec(mat(𝑥^∗)^𝑇), and the proof is complete.

Lemma 4.2. If 𝑥^∗ is the solution of problem min_𝑥𝑥−sign(𝑣)∘𝑣∣^𝑚_𝑖=1𝑥_𝑖≤1,𝑥≥0, (4.10) where sign(𝑣)=(sign(𝑣₁),sign(𝑣₂),…,sign(𝑣_𝑚))^𝑇, sign(𝑣_𝑖) is the sign of real number 𝑣_𝑖, and sign(𝑣)∘𝑣=sign𝑣₁𝑣₁,sign𝑣₂𝑣₂,…,sign𝑣_𝑚𝑣_𝑚^𝑇=|𝑣|, (4.11) then sign(𝑣)∘𝑥^∗ is the solution of the problem min_𝑥‖𝑥−𝑣‖∣^𝑚_𝑖=1𝑥_𝑖≤1. (4.12)

Proof . The result follows from ‖𝑥−𝑣‖=sign(𝑣)∘(𝑥−𝑣)_𝑚𝑖=1,sign𝑣_𝑖𝑥_𝑖=^𝑚_𝑖=1𝑥_𝑖≤1. (4.13)

Lemma 4.3. Let 𝑣≥0, 𝑇 be a permutation transformation sorting the components of 𝑣 in descending order, that is, the components of 𝑣=𝑇𝑣 are in descending order. Further, suppose that 𝑥^∗ is the solution of the problem min_𝑥‖𝑥−𝑣‖∣^𝑚_𝑖=1𝑥_𝑖≤1, (4.14) then 𝑥^∗=𝑇⁻¹𝑥^∗ is the solution of the problem min_𝑥‖𝑥−𝑣‖∣^𝑚_𝑖=1𝑥_𝑖≤1. (4.15)

Proof . Since 𝑇 is a permutation transformation, we have that 𝑇⁻¹=𝑇, (4.16) and the optimal values of objective function of problem (4.14) and (4.15) are equal.
Note that 𝑥^∗−𝑣=𝑇𝑥^∗−𝑣=𝑥^∗−𝑣,_𝑚𝑖=1𝑥^∗_𝑖=^𝑚_𝑖=1𝑥^∗_𝑖≤1. (4.17) Thus, 𝑥^∗ is the optimal solution of problem (4.15). And the proof is complete.

Remark 4.4. Suppose that mat(𝑣)∈𝑅^𝑛×𝑛 is a symmetric matrix for a given 𝑣∈𝑅^𝑚. Let 𝑣=𝑇(sign(𝑣)∘𝑣), where 𝑇 is a permutation transformation sorting the components of sign(𝑣)∘𝑣 in descending order. Lemmas 4.1–4.3 show that if 𝑥^∗ is the solution of the following problem min_𝑥12‖𝑥−𝑣‖²∣^𝑚_𝑖=1𝑥_𝑖≤1,𝑥≥0, (4.18) then 𝑃_Ω2[𝑣]=sign(𝑣)∘𝑇𝑥^∗. (4.19)

Hence, to solve the problem (4.18) is a key work to obtain the projection 𝑃_Ω2[𝑣]. Let 𝑒=(1,1,…,1)^𝑇∈𝑅^𝑚, then problem (4.18) can be rewritten as min_𝑥12‖𝑥−𝑣‖²∣𝑒^𝑇𝑥≤1,𝑥≥0. (4.20) The Lagrangian function for the constrained optimal problem (4.20) is defined as 𝐿=1𝑥,𝑦,𝑤2‖𝑥−𝑣‖²−𝑦1−𝑒^𝑇𝑥−𝑤^𝑇𝑥, (4.21) where scale 𝑦 and vector 𝑤∈𝑅^𝑚 are the Lagrange multipliers corresponding to inequalities 𝑒^𝑇𝑥≤1 and 𝑥≥0, respectively. By KKT condition, we have 𝑥−𝑣+𝑦𝑒−𝑤=0,𝑦≥0,1−𝑒^𝑇𝑥≥0,𝑦1−𝑒^𝑇𝑥=0,𝑤≥0,𝑥≥0,𝑤^𝑇𝑥=0, (4.22) that is, 𝑦≥0,1−𝑒^𝑇𝑥≥0,𝑦1−𝑒^𝑇𝑥=0,𝑥≥0,𝑥+𝑦𝑒−𝑣≥0,𝑥^𝑇𝑥+𝑦𝑒−𝑣=0. (4.23) It is easy to check that if 𝑒^𝑇𝑣≤1, then (𝑥^∗,𝑦^∗)=(𝑣,0) is the solution of problem (4.23). Now we assume that 𝑒^𝑇𝑣>1. In this case, let Δ𝑣=𝑣₁−𝑣₂,𝑣₂−𝑣₃,…,𝑣_𝑚−1−𝑣_𝑚,𝑣_𝑚^𝑇. (4.24) Note that Δ𝑣≥0 and ^𝑚_𝑖=1𝑖×Δ𝑣_𝑖=^𝑚_𝑖=1𝑣_𝑖>1. (4.25) Thus, there exists a least-integer 𝐾 such that ^𝐾_𝑖=1𝑖×Δ𝑣_𝑖≥1. (4.26) Since ^𝐾_𝑖=1𝑣_𝑖=^𝐾_𝑖∑₌₁𝑖×Δ𝑣_𝑖+𝐾×𝑣_𝐾+1,1≤𝐾<𝑚,_𝐾𝑖∑₌₁𝑖×Δ𝑣_𝑖,𝐾=𝑚, (4.27) we have that ^𝐾_𝑖=1𝑣_𝑖≥1. (4.28) Let 𝑦^∗=1𝐾^𝐾_𝑖=1𝑣_𝑖−1,𝑥^∗=𝑣₁−𝑦^∗,𝑣₂−𝑦^∗,…,𝑣_𝐾−𝑦^∗,0,…,0^𝑇∈𝑅^𝑚. (4.29)(4.30)

Theorem 4.5. Let 𝑥^∗ and 𝑦^∗ be given by (4.30) and (4.29), respectively, then (𝑥^∗,𝑦^∗) is the solution of problem (4.23), and thus 𝑃_Ω2[𝑣]=sign(𝑣)∘𝑇𝑥^∗. (4.31)

Proof . It follows from (4.26), (4.27), and (4.29) that 𝑦^∗=1𝐾^𝐾_𝑖=1𝑣_𝑖≥1−1𝐾^𝐾_𝑖=1𝑣_𝑖−𝑖×Δ𝑣_𝑖=𝑣_𝐾+1,1≤𝐾<𝑚,0,𝐾=𝑚,𝑦^∗=1𝐾^𝐾_𝑖=1𝑣_𝑖≤1−1𝐾^𝐾_𝑖=1𝑣_𝑖−^𝐾−1_𝑖=1𝑖×Δ𝑣_𝑖=1𝐾^𝐾−1_𝑖=1𝑣_𝑖−^𝐾−1_𝑖=1𝑖×Δ𝑣_𝑖+𝑣_𝐾=𝑣_𝐾. (4.32) Following from (4.32) it is easy to check that (𝑥^∗,𝑦^∗) is a solution of problem (4.23). Note that problem (4.18) is convex, thus (𝑥^∗,𝑦^∗) is the solution of problem (4.18). Further, according to Remark 4.4, we have 𝑃_Ω2[𝑣]=sign(𝑣)∘𝑇𝑥^∗. (4.33) The proof is complete.

Remark 4.6. Note that if 𝑋^∗ is the solution of problem 1min𝑋−𝛽𝐶_inf∣𝑋=𝑥_𝑖𝑗∈𝑆^𝑛_Λ, (4.34) where 𝛽>0 is a given scalar, and 𝑆^𝑛_Λ=𝐻∈𝑅^𝑛×𝑛∣𝐻^𝑇𝜆=𝐻,^min𝛽𝜆𝐼⪯𝐻⪯^max𝛽𝐼, (4.35) then 𝑋^∗=𝛽𝑋^∗ is the solution of problem (1.2). Thus, we can find the solution of problem (1.2) by solving problem (4.34).

Let Ω₁=𝑥=vec(𝑋)∣𝑋∈𝑆^𝑛_Λ. (4.36) Now, we are in the stage to describe the implementing process of Algorithm 3.2 for problem (1.2) in detail.

Algorithm 4.7 (projection and contraction method for problem (1.2)).
Step  1 (Initialization). Let 𝐶=(𝑐_𝑖𝑗)∈𝑅^𝑛×𝑛 be a given symmetric matrix, 𝜆_min,𝜆_max,𝛽>0 be given scalars. Choose arbitrarily an initial point 𝑥⁰=vec(𝑋⁰), 𝑧⁰=vec(𝑍⁰), and 𝑢⁰=^𝑥0_𝑧0, where 𝑋⁰∈Ω₁,𝑍⁰∈Ω₂, Ω₁, and Ω₂ are defined by (4.36) and (2.7), respectively. Let 𝛾∈(0,2), 𝑘=0, and 𝜀>0 be a prespecified tolerance.
Step  2 (Computation). Compute 𝑃Ω₁[𝑥^𝑘−𝑧^𝑘] and 𝑃_Ω2[𝑧^𝑘+(𝑥^𝑘−𝑐)] by using (4.6) and (4.31), respectively.
Let  𝑒_𝑥(𝑢^𝑘)=𝑥^𝑘Ω−𝑃₁[𝑥^𝑘−𝑧^𝑘], 𝑒_𝑧(𝑢^𝑘)=𝑧^𝑘−𝑃_Ω2[𝑧^𝑘+(𝑥^𝑘−𝑐)],  𝑑_𝑥(𝑢^𝑘)=𝑒_𝑥(𝑢^𝑘)−𝑒_𝑧(𝑢^𝑘), 𝑑_𝑧(𝑢^𝑘)=𝑒_𝑥(𝑢^𝑘)+𝑒_𝑧(𝑢^𝑘), where 𝑢^𝑘=^𝑥𝑘_𝑧𝑘.
Step  3. (Verification). If ‖𝑒(𝑢^𝑘)‖_∞<𝜀, then stop and output the approximate solution 𝑋=𝛽mat(𝑥^𝑘), where 𝑒(𝑢^𝑘)=^{𝑒𝑥(𝑢𝑘)}_{𝑒𝑧(𝑢𝑘)}.
Step  4. (Iteration). 𝑥^𝑘+1=𝑥^𝑘−𝛾𝑑_𝑥(𝑢^𝑘)/2, 𝑧^𝑘+1=𝑧^𝑘−𝛾𝑑_𝑧(𝑢^𝑘)/2, 𝑘∶=𝑘+1, goto Step  2.

5. Numerical Experiments

In this section, some examples are provided to illustrate the performance of Algorithm 4.7 for solving problem (1.2). In the following illustrative examples, the computer program for implementing Algorithm 4.7 is coded in Matlab and the program runs on IBM notebook (R51).

Example 5.1. Consider problem (1.2) with 𝐶=(𝐶+𝐶^𝑇)/2+eye(𝑛),𝜆_min=−50, and 𝜆_max=50, where 𝐶=rand(𝑛)−0.5, rand and eye are both the Matlab functions, and 𝑛 is the size of problem (1.2). Let 𝛾=1.5,𝜀=1×10⁻⁷, then we have 𝛽=√𝑛‖vec(𝐶)‖_∞,𝑥₀1=vec2𝜆₁+𝜆₂×eye(𝑛),𝑧₀=vec(zeros(𝑛)), (5.1) where zeros is also the Matlab function, 𝜆₁=max{𝜆₁/𝛽,𝜆_min/𝛽}, 𝜆₂=min{𝜆_𝑛/𝛽, 𝜆_max/𝛽}, and 𝜆₁,𝜆_𝑛 are the smallest and the largest eigenvalue of matrix 𝐶, respectively. Table 1 reports the numerical results of Example 5.1 solved by Algorithm 4.7, where 𝑘 is the number of iterations, the unit of time is second, and 𝑋 is the approximate solution of problem (1.2) obtained by Algorithm 4.7.

Table 1 

Numerical results of example 1.

Example 5.2. Consider problem (1.2) with 𝐶=𝐶𝐶^𝑇,𝜆_min=0, and 𝜆_max=20, where 𝐶=2×rand(𝑛)−1. In this test example, let 𝛾, 𝜀 be same as that in Example 5.1, 𝛽,𝑥₀, and 𝑧₀ be also given according to (5.1). Table 2 reports the numerical results of Example 5.2 and shows the numerical performance of Algorithm 4.7 for solving problem (1.2).

Table 2 

Numerical results of example 2.

6. Conclusions

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 4.7 properly. Thus, further studying of the effect of the parameters in Algorithm 4.7 maybe a very interesting work.

Acknowledgments

This research is financially supported by a research grant from the Research Grant Council of China (Project no. 10971095).