We consider the problem of seeking a symmetric positive semidefinite matrix in a closed convex set to approximate a given matrix. This problem may arise in several areas of numerical linear algebra or come from finance industry or statistics and thus has many applications. For solving this class of matrix optimization problems, many methods have been proposed in the literature. The proximal alternating direction method is one of those methods which can be easily applied to solve these matrix optimization problems. Generally, the proximal parameters of the proximal alternating direction method are greater than zero. In this paper, we conclude that the restriction on the proximal parameters can be relaxed for solving this kind of matrix optimization problems. Numerical experiments also show that the proximal alternating direction method with the relaxed proximal parameters is convergent and generally has a better performance than the classical proximal alternating direction method.
1. Introduction
This paper concerns the following problem:
(1)minX{12∥X-C∥F2∣X∈S+n∩SB},
where C∈Rn×n is a given symmetric matrix,
(2)S+n={X∈Rn×n∣XT=X,X≽0},SB={X∈Rn×n∣Tr(AiX)=bi,i=1,2,…,p,Tr(GjX)≤dj,j=1,2,…,m},
matrices Ai∈Rn×n and Gj∈Rn×n are symmetric and scalars, bi and dj are the problem data, X≽0 denotes that X is a positive semidefinite matrix, Tr denotes the trace of a matrix, and ∥·∥F denotes the Frobenius norm; that is,
(3)∥X∥F=(Tr(XTX))1/2=(∑i,j=1nXij2)1/2,
and S+n∩SB is nonempty. Throughout this paper, we assume that the Slater’s constraint qualification condition holds so that there is no duality gap if we use Lagrangian techniques to find the optimal solution to problem (1).
Problem (1) is a type of matrix nearness problem, that is, the problem of finding a matrix that satisfies some properties and is nearest to a given one. Problem (1) can be called the least squares covariance adjustment problem or the least squares semidefinite programming problem and solved by many methods [1–4]. In a least squares covariance adjustment problem, we make adjustments to a symmetric matrix so that it is consistent with prior knowledge or assumptions and a valid covariance matrix [2, 5, 6]. The matrix nearness problem has many applications especially in several areas of numerical linear algebra, finance industry, and statistics in [6]. A recent survey of matrix nearness problems can be found in [7]. It is clear that the matrix nearness problem considered here is a convex optimization problem. It thus follows from the strict feasibility and coercivity of the objective function that the minimum of (1) is attainable and unique.
In the literature of interior point algorithms, S+n is called the semidefinite cone and the related problem (1) belongs to the class of semidefinite programming (SDP) and second-order cone programming (SOCP) [8]. In fact, it is possible to reformulate problem (1) into a mixed SDP and SOCP as in [3, 9]:
(4)mints.t.〈Ai,X〉=bi,i=1,2,…,p,s.t.〈Gj,X〉≤dj,j=1,2,…,m,s.t.t≥∥X-C∥F,s.t.X∈S+n,
where 〈X,Y〉=Tr(XTY).
Thus, problem (1) can be efficiently solved by standard interior-point methods such as SeDuMi [10] and SDPT3 [11] when the number of variables (i.e., entries in the matrix X) is modest, say under 1000 (corresponds to n around 32) and the number of equality and inequality constraints is not too large (say 5,000) [2, 3, 12].
Specially, let
(5)SB={X∈Rn×n∣Diag(X)=e},
where Diag(X) is the vector of diagonal elements of X and e is the vector of 1s. Then problem (1) can be viewed as the nearest correlation matrix problem. For the nearest correlation matrix problem, a quadratically convergent Newton algorithm was presented recently by Qi and Sun [13], and improved by Borsdorf and Higham [1]. For problem (1) with equality and inequality constraints, one difficulty in finding an efficient method for solving this problem is the presence of the inequality constraints. In [3], Gao and Sun overcome this difficulty by reformulating the problem as a system of semismooth equations with two level metric projection operators and then design an inexact smoothing Newton method to solve the resulting semismooth system. For the problem (1) with large number of equality and inequality constraints, the numerical experiments in [14] show that the alternating direction method (hereafter alternating direction method is abbreviated as ADM) is more efficient in computing time than the inexact smoothing Newton method which additionally requires solving a large system of linear equations at each iteration. The ADM has many applications in solving optimization problems [15, 16]. Papers written by Zhang, Han, Li, Yuan, and Bauschke and Borwein show that the ADM can be applied to solve convex feasibility problems [17–19].
The proximal ADM is a class of ADM type methods which can also be easily applied to solve the matrix optimization problems. Generally, the proximal parameters (i.e., the parameters r and s in (14) and (15)) of the proximal ADM are greater than zero. In this paper, we will show that the restriction on the proximal parameters can be relaxed while the proximal ADM is used to solve problem (1). Numerical experiments also show that the proximal ADM with the relaxed proximal parameters generally has a better performance than the classical proximal ADM.
The paper is organized as follows. In Section 2, we give some preliminaries about the proximal alternating direction method. In Section 3, we convert the problem (1) to a structured variational inequality and apply the proximal ADM to solve it. The basic analysis and convergent results of the proximal ADM with relaxed proximal parameters are built in Section 4. Preliminary numerical results are reported in Section 5. Finally, we give some conclusions in Section 6.
2. Proximal Alternating Direction Method
In order to introduce the proximal ADM, we first consider the following structured variational inequality problem which includes two separable subvariational inequality problems: find (x,y)∈Ω such that
(6)(x′-x)Tf(x)≥0,(y′-y)Tg(y)≥0,∀(x′,y′)∈Ω,
where
(7)Ω={(x,y)∣Ax+By=b,x∈𝒳,y∈𝒴},f:Rn1→Rn1 and g:Rn2→Rn2 are monotone; that is,
(8)(x~-x)T(f(x~)-f(x))≥0,∀x~,x∈Rn1,(y~-y)T(g(y~)-g(y))≥0,∀y~,y∈Rn2,A∈Rl×n1, B∈Rl×n2, and b∈Rl; 𝒳⊂Rn1 and 𝒴⊂Rn2 are closed convex sets. Studies of such variational inequality can be found in Glowinski [20], Glowinski and Le Tallec [21], Eckstein and Fukushima [22–24], He and Yang [25], He et al. [26], and Xu [27].
By attaching a Lagrange multiplier vector λ∈Rl to the linear constraint Ax+By=b, problem (6)-(7) can be explained as the following form (see [20, 21, 24]): find w=(x,y,λ)∈𝒲 such that
(9)(x′-x)T[f(x)-ATλ]≥0(y′-y)T[g(y)-BTλ]≥0,∀w′=(x′,y′,λ′)∈𝒲,Ax+By-b=0,
where
(10)𝒲=𝒳×𝒴×Rl.
For solving (9)-(10), Gabay [28] and Gabay and Mercier [29] proposed the ADM method. In the classical ADM method, the new iterate wk+1=(xk+1,yk+1,λk+1)∈𝒲 is generated from a given triple wk=(xk,yk,λk)∈𝒲 via the following procedure.
First, xk+1 is found by solving the following problem:
(11)(x′-x)T{f(x)-AT[λk-β(Ax+Byk-b)]}≥0,∀x′∈𝒳,
where x∈𝒳. Then, yk+1 is obtained by solving
(12)(y′-y)T{g(y)-BT[λk-β(Axk+1+By-b)]}≥0,∀y′∈𝒴,
where y∈𝒴. Finally, the multiplier is updated by
(13)λk+1=λ-β(Axk+1+Byk+1-b),
where β>0 is a given penalty parameter for the linearly constraint Ax+By-b=0. Most of the existing ADM methods require that the subvariational inequality problems (11)-(12) should be solved exactly at each iteration. Note that the involved subvariational inequality problem (11)-(12) may not be well-conditioned without strongly monotone assumptions on f and g. Hence, it is difficult to solve these subvariational inequality problems exactly in many cases. In order to improve the condition of solving the subproblem by the ADM, some proximal ADMs were proposed (see, e.g., [26, 27, 30–34]). The classical proximal ADM is one of the attractive ADMs. From a given triple wk=(xk,yk,λk)∈𝒲, the classical proximal ADM produces the new iterate wk+1=(xk+1,yk+1,λk+1)∈𝒲 by the following procedure.
First, xk+1 is obtained by solving the following variational inequality problem:
(14)(x′-x)T{f(x)-AT[λk-β(Ax+Byk-b)]+r(x-xk)}≥0,∀x′∈𝒳,
where r>0 is the given proximal parameter and x∈𝒳. Then, yk+1 is found by solving
(15)(y′-y)T{(y-yk)g(y)-BT[λk-β(Axk+1+By-b)]+s(y-yk)}≥0,∀y′∈𝒴,
where s>0 is the given proximal parameter and y∈𝒴. Finally, the multiplier is updated by
(16)λk+1=λk-β(Axk+1+Byk+1-b).
In this paper, we will conclude that problem (1) can be solved by the proximal ADM and the restriction on the proximal parameters r>0, s>0 can be relaxed as r>-1/2, s>-1/2 when the proximal ADM is applied to solve problem (1). Our numerical experiments later also show that the numerical performance of the proximal ADM with smaller value of proximal parameters is generally better than the proximal ADM with comparatively larger value of proximal parameters.
3. Converting Problem (<xref ref-type="disp-formula" rid="EEq1">1</xref>) to a Structured Variational Inequality
In order to solve the problem (1) with proximal ADM, we convert problem (1) to the following equivalent one:
(17)minX,Y12∥X-C∥F2+12∥Y-C∥F2s.t.X-Y=0,s.tX∈S+n,Y∈SB.
Following the KKT condition of (17), the solution to (17) can be found by finding w=(X,Y,Λ)∈𝒲 such that
(18)〈X′-X,(X-C)-Λ〉≥0,〈Y′-Y,(Y-C)+Λ〉≥0,∀w′=(X′,Y′,Λ′)∈𝒲,X-Y=0,
where
(19)𝒲=S+n×SB×Rn×n.
It is easy to see that problem (18)-(19) is a special case of the structured variational inequality (9)-(10) and thus can be solved by proximal ADM. For given wk=(Xk,Yk,Λk)∈𝒲, it is fortunate that the wk+1=(Xk+1,Yk+1,Λk+1) can be exactly obtained by the proximal ADM in the following way:
(20)Xk+1=PS+n{11+β+r(C+rXk+βYk+Λk)},(21)Yk+1=PSB{11+β+s(C+βXk+1+sYk-Λk)},(22)Λk+1=Λk-β(Xk+1-Yk+1),
where the projection of v on a nonempty closed convex set S of Rm×n under Frobenius norm, denoted by PS(v), is the unique solution to the following problem; that is,
(23)PS(v)=argminu{∥u-v∥F2∣u∈S}.
It follows that the solution to
(24)min{12∥Z-X∥F2∣Z∈S+n}
is called the projection of X on S+n and denoted by PS+n(X). Using the fact that matrix Frobenius norm is invariant under unitary transform, it is known (see [35]) that
(25)PS+n(X)=QΛ~QT,
where
(26)QTXQ=diag(λ1,…,λn)
is the symmetric Schur decomposition of X (Q=(q1,…,qn) is an orthogonal matrix whose column vector qi, i=1,…,n, is the eigenvector of X, and λi, i=1,…,n, is the related eigenvalue),
(27)Λ~=diag(λ~1,…,λ~n),λ~i=max(λi,0).
In order to obtain the projection PSB(X), we need to solve the following quadratic program:(28)minZ12∥Z-X∥F2s.t.Tr(AiZ)=bi,i=1,2,…,p,s.t.Tr(GjZ)≤dj,j=1,2,…,m.
The dual problem of (28) can be written as
(29)minv12vTHv+qTvs.t.v∈Rp×R+m,
where H is positive semidefinite and H and q have the following form, respectively:(30)H=(Tr(A1A1T)⋯Tr(A1ApT)Tr(A1G1T)⋯Tr(A1GmT)⋮⋯⋮⋮⋯⋮Tr(ApA1T)⋯Tr(ApApT)Tr(ApG1T)⋯Tr(ApGmT)Tr(G1A1T)⋯Tr(G1ApT)Tr(G1G1T)⋯Tr(G1GmT)⋮⋯⋮⋮⋯⋮Tr(GmA1T)⋯Tr(GmApT)Tr(GmG1T)⋯Tr(GmGmT)),q=(b1-Tr(A1X)⋮bp-Tr(ApX)d1-Tr(G1X)⋮dm-Tr(GmX)).
Problem (29) is often a medium-scale quadratic programming (QP) problem. A variety of methods for solving the QP are commonly used, including interior-point methods and active set algorithm (see [36, 37]).
Particularly, if SB is the following special case:
(31)SB={X∈Rn×n∣XT=X,HL≤X≤HU},
where H≥0 expresses that each element of H is nonnegative, HL and HU are given n×n symmetric matrices, and X≤HU means that HU-X≥0; then PSB(X) is easy to be carried out and is given by
(32)PSB(X)=min(max(X,HL),HU),
where max(X,Y) and min(X,Y) compute the element-wise maximum and minimum of matrix X and Y, respectively.
4. Main Results
Let {wk} be the sequence generated by applying the procedure (14)–(16) to problem (18)-(19); then for any w′=(X′,Y′,Λ′)∈𝒲, we have that
(33)〈X′-Xk+1,Xk+1-C-Λk+1-β(Yk-Yk+1)+r(Xk+1-Xk)〉≥0,〈Y′-Yk+1,Yk+1-C+Λk+1+s(Yk+1-Yk)〉≥0,Λk+1=Λk-β(Xk+1-Yk+1).
Further, letting
(34)F(wk+1)=(Xk+1-C-Λk+1Yk+1-C+Λk+1Xk+1-Yk+1),d1(wk,wk+1)=(rIn000(s+β)In0001βIn)(Xk-Xk+1Yk-Yk+1Λk-Λk+1),
where In∈Rn×n is the unit matrix, and
(35)d2(wk,wk+1)=F(wk+1)-β(In-In0)(Yk-Yk+1),
then we can get the following lemmas.
Lemma 1.
Let {wk} be the sequence generated by applying the proximal ADM to problem (18)-(19) and let w*∈𝒲* be any solution to problem (18)-(19); then one has
(36)〈wk+1-w*,d2(wk,wk+1)〉≥-〈Λk-Λk+1,Yk-Yk+1〉+∥Xk+1-X*∥F2+∥Yk+1-Y*∥F2.
Proof.
From (22) and (35), we have
(37)〈wk+1-w*,d2(wk,wk+1)〉=-〈Λk-Λk+1,Yk-Yk+1〉+〈wk+1-w*,F(wk+1)〉.
Since (9) and w* are a solution to problem (18)-(19) and Xk+1∈S+n, Yk+1∈SB, we have
(38)〈wk+1-w*,F(w*)〉≥0.
From (38), it follows that
(39)〈wk+1-w*,F(wk+1)-F(wk+1)+F(w*)〉≥0.
Thus, we have
(40)〈wk+1-w*,F(wk+1)〉≥〈wk+1-w*,F(wk+1)-F(w*)〉=〈Xk+1-X*,Xk+1-X*-(Λk+1-Λ*)〉+〈Yk+1-Y*,Yk+1-Y*+(Λk+1-Λ*)〉+〈Λk+1-Λ*,Xk+1-X*-(Yk+1-Y*)〉=〈Xk+1-X*,Xk+1-X*〉+〈Yk+1-Y*,Yk+1-Y*〉=∥Xk+1-X*∥F2+∥Yk+1-Y*∥F2.
Substituting (40) into (37), we get the assertion of this lemma.
Lemma 2.
Let {wk} be the sequence generated by applying the proximal ADM to problem (18)-(19) and let w*∈𝒲* be any solution to problem (18)-(19); then one has
(41)〈wk-w*,G0(wk-wk+1)〉≥〈wk-wk+1,G0(wk-wk+1)〉-〈Λk-Λk+1,Yk-Yk+1〉+∥Xk+1-X*∥F2+∥Yk+1-Y*∥F2,
where
(42)G0=(rIn000(s+β)In0001βIn).
Proof.
It follows from (33) that
(43)〈w′-wk+1,d2(wk,wk+1)-d1(wk,wk+1)〉≥0,∀w′∈𝒲.
Thus, we have
(44)〈wk+1-w*,d1(wk,wk+1)〉≥〈wk+1-w*,d2(wk,wk+1)〉≥-〈Λk-Λk+1,Yk-Yk+1〉+∥Xk+1-X*∥F2+∥Yk+1-Y*∥F2.
From the above inequality, we get
(45)〈wk-w*,G0(wk-wk+1)〉≥〈wk-wk+1,G0(wk-wk+1)〉-〈Λk-Λk+1,Yk-Yk+1〉+∥Xk+1-X*∥F2+∥Yk+1-Y*∥F2.
Hence, (41) holds and the proof is completed.
Theorem 3.
Let {wk} be the sequence generated by applying the proximal ADM to problem (18)-(19) and let w*∈𝒲* be any solution to problem (18)-(19); then one has
(46)∥wk+1-w*∥G2≤∥wk-w*∥G2-〈wk-wk+1,M(wk-wk+1)〉,
where
(47)G=((r+1)In000(1+s+β)In0001βIn),M=((12+r)In000(12+s+β)In-In0-In1βIn),
and ∥w∥G2=〈w,Gw〉.
Proof.
From (41), we have
(48)∥wk+1-w*∥G02=∥wk-w*-(wk-wk+1)∥G02≤∥wk-w*∥G02-2∥wk-wk+1∥G02+2〈Λk-Λk+1,Yk-Yk+1〉-2∥Xk+1-X*∥F2-2∥Yk+1-Y*∥F2+∥wk-wk+1∥G02=∥wk-w*∥G02-∥wk-wk+1∥G02+2〈Λk-Λk+1,Yk-Yk+1〉-2∥Xk+1-X*∥F2-2∥Yk+1-Y*∥F2.
Rearranging the inequality above, we find that(49)∥wk+1-w*∥G2≤∥wk-w*∥G2-〈wk-wk+1,(rIn000(s+β)In-In0-In1βIn)(wk-wk+1)〉-(∥Xk+1-X*∥F2+∥Xk-X*∥F2)-(∥Yk+1-Y*∥F2+∥Yk-Y*∥F2).Using the Cauchy-Schwarz Inequality on the last term of the right-hand side of (49), we obtain
(50)∥Xk+1-X*∥F2+∥Xk-X*∥F2≥12∥Xk+1-Xk∥F2,∥Yk+1-Y*∥F2+∥Yk-Y*∥F2≥12∥Yk+1-Yk∥F2.
Substituting (50) into (49), we get
(51)∥wk+1-w*∥G2≤∥wk-w*∥G2-〈wk-wk+1,M(wk-wk+1)〉.
Thus, the proof is completed.
Based on the Theorem 3, we get the following lemma.
Lemma 4.
Let {wk} be the sequence generated by applying proximal ADM to problem (18)-(19), w*∈𝒲* any solution to problem (18)-(19), r>-1/2, and s>-1/2; then one has the following.
The sequence {∥wk-w*∥G2} is nonincreasing;
The sequence {wk} is bounded;
limk→∞∥wk+1-wk∥F2=0;
G and M are both symmetric positive-definite matrices.
Proof.
Since
(52)|(12+s+β)In-In-In1βIn|=((1/2)+s)β,
it is easy to check that if r>-1/2, s>-1/2, then G and M are symmetric positive-definite matrices.
Let τ>0 be the smallest eigenvalue of matrix M. Then, from (46), we have
(53)∥wk+1-w*∥G2≤∥wk-w*∥G2-τ∥wk-wk+1∥F2.
Following (53), we immediately have that ∥wk-w*∥G2 is nonincreasing and thus the sequence {wk} is bounded. Moreover, we have
(54)∥wk+1-w*∥G2≤∥w0-w*∥G2-τ∑j=0k∥wj-wj+1∥F2.
So, we get
(55)∑j=0k∥wj-wj+1∥F2<∞,∀k>0,
then
(56)limk→∞∥wk-wk+1∥F2=0.
Thus, the proof is completed.
Following Lemma 4, now we are in the stage of giving the main convergence results of proximal ADM with r>-1/2 and s>-1/2 for problem (18)-(19).
Theorem 5.
Let {wk} be the sequence generated by applying proximal ADM to problem (18)-(19), r>-1/2, and s>-1/2; then {wk} converges to a solution point of (18)-(19).
Proof.
Since the sequence {wk} is bounded (see point (2) of Lemma 4), it has at least one cluster point. Let w∞ be a cluster point of {wk} and the subsequence {wkj} converges to w∞. It follows from (33) that
(57)limj→∞〈X′-Xkj+1,Xkj+1-C-Λkj+1-β(Ykj-Ykj+1)+r(Xkj+1-Xkj)〉≥0,limj→∞〈Y′-Ykj+1,Ykj+1-C+Λkj+1+s(Ykj+1-Ykj)〉≥0,∀w′∈𝒲,limj→∞Λkj+1=Λkj-β(Xkj+1-Ykj+1).
Following point (3) of Lemma 4, we have
(58)〈X′-X∞,X∞-C-Λ∞〉≥0,〈Y′-Y∞,Y∞-C+Λ∞〉≥0,∀w′∈𝒲,X∞-Y∞=0.
This means that w∞ is a solution point of (18)-(19). Since {wkj} converges to w∞, we have that, for any given ε>0, there exists an integer N>0 such that
(59)∥wkj-w∞∥G2<ε,∀kj≥N.
Furthermore, using the inequality (53), we have
(60)∥wk-w∞∥G2<∥wkj-w∞∥G2,∀k≥kj.
Combining (59) and (60), we get that
(61)∥wk-w∞∥G2<ε,∀k>N.
This implies that the sequence {wk} converges to w∞. So the proof is completed.
5. Numerical Experiments
In this section, we implement the proximal ADM to solve the problem (1) and show the numerical performances of proximal ADM with different proximal parameters. Additionally, we compare the classical ADM (i.e., the proximal ADM with proximal parameters r=0 and s=0) with the alternating projections method proposed by Higham [6] numerically and show that the alternating projections method is not equivalent to proximal ADM with zero proximal parameters. All the codes were written in Matlab 7.1 and run on IBM notebook PC R400.
Example 6.
In the first numerical experiment, we set the C1 as an n×n matrix whose entries are generated randomly in [-1,1]. Let C=(C1+C1T)/2 and further let the diagonal elements of C be 1 that is, Cii=1, i=1,2,…,n. In this test example, we simply let SB be in the form of (31) and
(62)HL=(lij)∈Rn×n,lij={-0.5,i≠j1,i=j,i,j=1,2,…,n,HU=(uij)∈Rn×n,uij={0.5,i≠j1,i=j,i,j=1,2,…,n.
Moreover, let X0=eye(n), Y0=eye(n), Λ0=zeroes(n), β=4, and ε=10-6, where eye(n) and zeroes(n) are both the Matlab functions. For different problem size n and different proximal parameters r and s, Table 1 shows the computational results. There, we report the number of iterations (It.) and the computing time in seconds (CPU.) it takes to reach convergence. The stopping criterion of the proximal ADM is
(63)∥wk+1-wk∥max<ε,
where ∥X∥max=max(max(abs(X))) is the maximum absolute value of the elements of the matrix X.
Numerical results of Example 6.
n
r=-0.3, s=-0.3
r=0, s=0
r=3, s=3
It.
CPU.
It.
CPU.
It.
CPU.
100
31
0.292
34
0.331
72
0.764
200
33
1.346
39
1.570
84
3.364
300
38
4.265
41
5.746
90
9.991
400
40
9.872
43
9.919
94
22.03
500
39
15.83
45
18.39
98
39.91
Remark 7.
Note that if the proximal parameters are equal to zero, that is, r=0 and s=0, then the proximal ADM is the classical ADM.
Example 8.
All the data are the same as in Example 6 except that C1 is an n×n matrix whose entries are generated randomly in [-1000,1000],
(64)HL=(lij)∈Rn×n,lij={-500,i≠j1000,i=j,i,j=1,2,…,n,HU=(uij)∈Rn×n,uij={500,i≠j1000,i=j,i,j=1,2…,n.
The computational results are reported in Table 2.
Numerical results of Example 8.
n
r=-0.3, s=-0.3
r=0, s=0
r=3, s=3
It.
CPU.
It.
CPU.
It.
CPU.
100
49
0.476
54
0.551
116
1.837
200
51
2.197
57
2.334
128
5.430
300
59
6.614
61
8.108
136
15.25
400
56
12.74
63
14.51
140
31.65
500
58
23.90
66
26.90
147
59.98
Example 9.
Let SB be in the form of (31) and lij=0, uij=+∞, i,j=1,2,…,n. Assume that C, X0, Y0, Λ0, β, ε, and the stopping criterion are the same as those in Example 6, but the diagonal elements of matrix C are replaced by
(65)Cii=α+(1-α)×rand,i=1,2,…,n,
where α∈(0,1) is a given number, rand is the Matlab function generating a number randomly in [0,1]. In the following numerical experiments, we let α=0.2. For different problem size n and different proximal parameters r and s, Table 3 shows the number of iterations and the computing time in seconds it takes to reach convergence.
Numerical results of Example 9.
n
r=-0.3, s=-0.3
r=0, s=0
r=3, s=3
It.
CPU.
It.
CPU.
It.
CPU.
100
32
0.282
35
0.288
70
0.566
200
33
1.295
36
1.397
72
4.006
300
34
3.745
37
4.156
73
8.285
400
34
7.885
37
8.571
73
16.73
500
34
14.07
37
15.42
74
29.87
Example 10.
All the data are the same as in Example 9 except that α=0. The computational results are reported in Table 4.
Numerical results of Example 10.
5
r=-0.3, s=-0.3
r=0, s=0
r=3, s=3
It.
CPU.
It.
CPU.
It.
CPU.
100
32
0.259
35
0.300
70
0.557
200
33
1.306
36
1.424
72
2.880
300
33
3.750
37
4.087
72
7.958
400
34
7.799
37
8.546
74
16.98
500
34
13.96
37
16.10
74
30.77
Example 11.
Let C1 be an n×n matrix whose entries are generated randomly in [-0.5,0.5], C=(C1+C1T)/2, and let the diagonal elements of C be 1. And let(66)SB={X∈Rn×n∣X=XT,Xij=eij,(i,j)∈ℬe,Xij≥lij,(i,j)∈ℬl,Xij≤uij,(i,j)∈ℬuRn×n},
where ℬe,ℬl,ℬu are subsets of {(i,j)∣1≤i,j≤n} denoting the indexes of such entries of X that are constrained by equality, lower bounds, and upper bounds, respectively. In this test example, we let the index sets ℬe,ℬl, and ℬu be the same as in Example 5.4 of [3]; that is, ℬe={(i,i)∣i=1,2,…,n} and ℬl,ℬu⊂{(i,j)∣1≤i<j≤n} consist of the indices of min(n^r,n-i) randomly generated elements at the ith row of X, i=1,2,…,n with n^r=5 and n^r=10, respectively. We take eii=1 for (i,i)∈ℬe, lij=-0.1 for (i,j)∈ℬl, and uij=0.1 for (i,j)∈ℬu.
Moreover, let X0, Y0, Λ0, β, ε, and the stopping criterion be the same as those in Example 6. For different problem size n, different proximal parameters r and s, and different values of n^r, Tables 5(a) and 5(b) show the number of iterations and the computing time in seconds it takes to reach convergence, respectively.
Numerical experiments show that the proximal ADM with relaxed parameters is convergent. Moreover, we draw the conclusion that the proximal ADM with smaller value of proximal parameters generally converges more quickly than the proximal ADM with comparatively larger value of proximal parameters to solve the problem (1).
(a) Numerical results of Example 11 with n^r=5. (b) Numerical results of Example 11 with n^r=10.
n
r=-0.3,
s=-0.3
r=0,
s=0
r=1,
s=1
It.
CPU.
It.
CPU.
It.
CPU.
100
22
0.293
25
0.354
34
0.448
200
25
2.119
28
2.425
40
3.436
300
27
7.141
30
8.024
44
11.64
400
29
17.40
31
18.59
46
27.32
500
30
34.17
33
37.45
48
53.84
n
r=-0.3,
s=-0.3
r=0,
s=0
r=1,
s=1
It.
CPU.
It.
CPU.
It.
CPU.
100
23
0.309
25
0.342
33
0.439
200
24
2.029
27
2.305
38
3.162
300
27
7.150
29
7.801
42
11.29
400
28
16.68
31
18.47
45
26.60
500
29
32.73
32
36.37
47
53.06
Example 12.
In this test example, we apply the proximal ADM with r=0, s=0 (i.e., the classical ADM) to solve the nearest correlation matrix problem, that is, problem (1) with SB in the form of (5), and compare the classical ADM numerically with the alternating projections method (APM) [6]. The APM computes the nearest correlation matrix to a symmetric C∈Rn×n by the following process:
ΔS0=0,
Y0=C;
for k=1,2,…
Rk=Yk-1-ΔSk-1;
Xk=PS+n(Rk);
ΔSk=Xk-Rk;
Yk=PSB(Xk);
end.
In this numerical experiment, the stopping criterion of the APM is
(67)max{∥Xk-Xk-1∥max,∥Yk-Yk-1∥max,∥Xk-Yk∥max}<ε.
Let the matrix C and the initial parameters of classical ADM be the same as those in Example 6. Table 6(a) reports the numerical performance of proximal ADM and the APM for computing the nearest correlation matrix to C.
Further, let C1 be an n×n matrix whose entries are generated randomly in [0,1] and C=(C1+C1T)/2. The other data are the same as above. Table 6(b) reports the numerical performance of the classical ADM and the APM for computing the nearest correlation matrix to the matrix C. Numerical experiments show that the classical ADM generally exhibits a better numerical performance than the APM for the test problems above.
(a) Numerical results of Example 12. (b) Numerical results of Example 12.
n
ADM
APM
It.
CPU.
It.
CPU.
100
28
0.381
47
0.743
200
33
2.878
59
5.443
300
36
9.462
70
20.68
400
38
22.50
81
54.38
500
39
43.32
89
114.7
n
ADM
APM
It.
CPU.
It.
CPU.
100
27
0.634
42
0.582
200
30
2.590
59
5.428
300
32
8.524
65
19.36
400
34
20.34
75
50.79
500
35
39.43
86
111.6
6. Conclusions
In this paper, we apply the proximal ADM to a class of matrix optimization problems and find that the restriction of proximal parameters can be relaxed. Moreover, numerical experiments show that the proximal ADM with relaxed parameters generally has a better numerical performance in solving the matrix optimization problem than the classical proximal alternating direction method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the referees very sincerely for their valuable suggestions and careful reading of their paper. This research is financially supported by a research Grant from the Research Grant Council of China (Project no. 10971095).
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