A Globally Convergent Filter-Type Trust Region Method for Semidefinite Programming

Abstract

When using interior methods for solving semidefinite programming (SDP), one needs to solve a system of linear equations at each iteration. For problems of large size, solving the system of linear equations can be very expensive. In this paper, based on a semismooth equation reformulation using Fischer's function, we propose a filter method with trust region for solving large-scale SDP problems. At each iteration we perform a number of conjugate gradient iterations, but do not need to solve a system of linear equations. Under mild assumptions, the convergence of this algorithm is established. Numerical examples are given to illustrate the convergence results obtained.

1. Introduction

Semidefinite programming (SDP) is convex programming over positive semidefinite matrices. For early application, SDP has been widely used in control theory and combinatorial optimization (see, e.g., [1–3]). Since some algorithms for linear optimization can be extended to many general SDP problems, that aroused much interest in SDP. In the past decade, many algorithms have been proposed for solving SDP, including interior-point methods (IPMs) [4–7], augmented methods [8–10], new Newton-type methods [11], modified barrier methods [12], and regularization approaches [13].

For small and medium sized SDP problems, IPMs are generally efficient. But for large-scale SDP problems, IPMs become very slow. In order to improve this shortcoming, [9, 14] proposed inexact IPMs using an iterative solver to compute a search direction at each iteration. More recently, [13] applied regularization approaches to solve SDP problems. All of these methods are first-order based on a gradient, or inexact second-order based on an approximation of Hessian matrix methods [15].

In this paper, we will extend filter-trust-region methods for solving linear (or nonlinear) programming [16] to large-scale SDP problems and use Lipschitz continuity. Furthermore, the accuracy of this method is controlled by a forcing parameter. It is shown that, under mild assumptions, this algorithm is convergent.

The paper is organized as follows. Some preliminaries are introduced in Section 2. In Section 3, we propose a filter-trust-region method for solving SDP problems, and we study the convergence of this method in Section 4. In Section 5, some numerical examples are presented to demonstrate the convergence results obtained in this paper. Finally, we give some conclusions in Section 6.

In this paper, we use the following common notation for SDP problems: 𝒳^𝑛 and ℛ^𝑚 denote the space of 𝑛×𝑛 real symmetric matrices and the space of vectors with 𝑚 dimensions, respectively; 𝑋≽0(𝑋≻0) denotes that 𝑋∈𝒳^𝑛 is positive semidefinite (positive definite), and 𝑋⪯0(𝑋≺0) is used to indicate that 𝑋∈𝒳^𝑛 is negative semidefinite (negative definite). A superscript 𝑇 represents transposes of matrices or vectors. For 𝑋,𝑌∈𝒳^𝑛, the standard scalar product on the space of 𝒳^𝑛 is defined by ⟨𝑋,𝑌⟩∶=𝑋•𝑌=trace(𝑋𝑌)=^𝑛_𝑖_,𝑗=1𝑋_𝑖,𝑗𝑌_𝑖,𝑗.(1.1) If 𝑋∈𝒳^𝑛 and 𝑥∈ℛ^𝑚, we denote that ‖𝑋‖_𝐹 is the Frobenius norm of 𝑋, that is, ‖𝑋‖_𝐹=√⟨𝑋,𝑋⟩=∑^{𝑛𝑖,𝑗=1}𝑋^2𝑖,𝑗 and ‖𝑥‖₂ is the 2-norm of 𝑥, that is, ‖𝑥‖₂=√𝑥^𝑇𝑥=∑^𝑚𝑖=1𝑥^2𝑖, respectively. Let 𝑋 be a 𝑝×𝑞 matrix. Then we denote by Vec(𝑋) a 𝑝𝑞 vector made of columns of 𝑋 stacked one by one, and the operator Mat(⋅) is the inverse of Vec(⋅), that is, Mat(Vec(𝑋))=𝑋. We also denote that 𝐼 is identity matrix.

2. Preliminaries

We consider a SDP problem of the form smubinjectto𝐶𝒜•(𝑋𝑋)=𝑏,𝑋⪰0,(2.1) where 𝐶∈𝒳^𝑛, 𝐴^(𝑖)∈𝒳^𝑛, 𝑖=1,2,…,𝑚, and 𝑏=(𝑏₁,𝑏₂,…,𝑏_𝑚)^𝑇∈ℛ^𝑚 are given dates; 𝒜 is a linear map from 𝒳^𝑛 to ℛ^𝑚 given by 𝒜(𝑋)∶=⎡⎢⎢⎢⎢⎢⎢⎣𝐴⁽¹⁾𝐴•𝑋₍₂₎𝐴⋮•𝑋_(𝑚)•𝑋⎤⎥⎥⎥⎥⎥⎥⎦,𝑋∈𝒳^𝑛.(2.2) The dual to the problem (2.1) is given by smubaxjectto𝑏𝒜^𝑇_∗𝑦(𝑦)+𝑆=𝐶,𝑆⪰0,(2.3) where 𝒜^∗ is an adjoint operator of 𝒜∶ℛ^𝑚→𝒳^𝑛 given by 𝒜^∗(𝑦)=^𝑚_𝑖=1𝑦_𝑖𝐴^(𝑖),𝑦∈ℛ^𝑚.(2.4) Obviously, 𝑋∈𝒳^𝑛 and (𝑦,𝑆)∈ℛ^𝑚×𝒳^𝑛 are the primal and dual variables, respectively.

It is easily verified that the SDP problem (2.1) is convex. When (2.1) and (2.3) have strictly feasible points, then strong duality holds, see [5, 12]. In this case, a point (𝑋,𝑦,𝑆) is optimal for SDP problems (2.1) and (2.3) if and only if 𝒜(𝑋)=𝑏,𝒜^∗(𝑦)+𝑆=𝐶,𝑋⪰0,𝑆⪰0,⟨𝑋,𝑆⟩=0.(2.5) In the sense that (𝑋,𝑦,𝑆) solves SDP problems (2.1) and (2.3) if and only if (𝑋,𝑦,𝑆) solves (2.5) when both SDP problems (2.1) and (2.3) have strictly feasible points.

We now introduce some lemmas which will be used in the sequel.

Lemma 2.1 (see [17]). Let 𝐴,𝐵∈𝒳^𝑛 and let 𝐴≽0,𝐵≽0. Then ⟨𝐴,𝐵⟩=0 if and only if 𝐴𝐵=0.

For 𝑋,𝑆∈𝒳^𝑛, we define a mapping 𝜙∶𝒳^𝑛×𝒳^𝑛→𝒳^𝑛 given by 𝜙(𝑋,𝑆)∶=𝑋+𝑆−√𝑋²+𝑆²,(2.6) which is attributed by Fischer to Burmeister (see [18, 19]). This function is nondifferentiable and has a basic property.

Lemma 2.2 (see [20, Lemma 6.1]). Let 𝜙 be the Fischer-Burmeister function defined in (2.6). Then 𝜙(𝑋,𝑆)=0⟺𝑋⪰0,𝑆⪰0,𝑋𝑆=0.(2.7)

In addition, for 𝜏>0 and 𝑋,𝑆∈𝒳^𝑛, we define a mapping 𝜙_𝜏∶𝒳^𝑛×𝒳^𝑛→𝒳^𝑛 by 𝜙_𝜏(𝑋,𝑆)∶=𝑋+𝑆−√𝑋²+𝑆²+2𝜏²𝐼,(2.8) which is differentiable and has following results.

Lemma 2.3 (see [11, Proposition  2.3]). Let 𝜏>0 be any positive number and let 𝜙_𝜏 be defined by (2.8). Then 𝜙_𝜏(𝑋,𝑆)=0⟺𝑋≻0,𝑆≻0,𝑋𝑆=𝜏²𝐼.(2.9)

Lemma 2.4. Let 𝜏>0 be any positive number, and let 𝜙_𝜏 be defined by (2.8). If 𝜏→0, we would have 𝜙_𝜏(𝑋,𝑆)=0⟺𝑋⪰0,𝑆⪰0,𝑋𝑆=0.(2.10)

Proof. The proof can be obtained from Lemmas 2.2 and 2.3.

Lemma 2.5 (see [20, pages 170–171]). For any 𝐶≻0, define the linear operator 𝐿_𝐶 by 𝐿_𝐶[𝑋]∶=𝐶𝑋+𝑋𝐶,𝑋∈𝒳^𝑛.(2.11) Then 𝐿_𝐶 is strictly monotone and so has an inverse 𝐿^𝐶−1.

Lemma 2.6 (see [21, Lemma  2]). Let 𝑋,𝑆,𝑈,𝑉∈𝒳^𝑛, and let 𝜙_𝜏 be defined by (2.8). For any 𝜏>0, we have that 𝜙_𝜏 is Fréchet-differentiable and ∇𝜙_𝜏(𝑋,𝑆)(𝑈,𝑉)=𝑈+𝑉−𝐿^𝐶−1[𝑋𝑈+𝑈𝑋+𝑆𝑉+𝑉𝑆],(2.12) where 𝐶∶=√𝑋²+𝑆²+2𝜏²𝐼.

Lemma 2.7 (see [22, Corollary 2.7]). Let 𝐹 be a map from 𝒳^𝑛 to 𝒳^𝑛. If 𝐹 is locally Lipschitzian on 𝒳^𝑛, then 𝐹 is almost everywhere Fréchet-differentiable on 𝒳^𝑛.

3. The Algorithm

In this section, we will present a filter-trust-region method for solving SDP problems (2.1) and (2.3). Firstly, for a parameter 𝜏>0, we construct a function: 𝐻_𝜏(𝑋,𝑦,𝑆)∶=⎛⎜⎜⎜⎜⎜⎜⎝𝒜𝒜(𝑋𝜏)−𝑏_∗𝑋+𝑆−√(𝑦)+𝑆−𝐶𝑋²+𝑆²+2𝜏²𝐼⎞⎟⎟⎟⎟⎟⎟⎠,(3.1) where (𝑋,𝑦,𝑆)∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛.

According to Lemmas 2.1, 2.3 and 2.4, the following theorem is obvious.

Theorem 3.1. Let 𝜏>0 and let 𝐻_𝜏(𝑋,𝑦,𝑆) be defined by (3.1). If SDP problems (2.1) and (2.3) have strictly feasible points, then 𝐻_𝜏𝑋_∗,𝑦_∗,𝑆_∗=0⟹𝑋_∗,𝑦_∗,𝑆_∗solves(2.5).(3.2)

In what follows, we will study properties of the function 𝐻_𝜏(𝑋,𝑦,𝑆). For simplicity, in the remaining sections of this paper, we denote 𝑍∶=(𝑋,𝑦,𝑆), 𝑍_𝑘∶=(𝑋_𝑘,𝑦_𝑘,𝑆_𝑘) and Δ𝑍∶=(Δ𝑋,Δ𝑦,Δ𝑆).

Theorem 3.2. Let 𝐻_𝜏(𝑍) be defined by (3.1). For any 𝑍,Δ𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛 and 𝜏>0, then 𝐻_𝜏(𝑍) is Fréchet-differentiable and ∇𝐻_𝜏(𝑍)(Δ𝑍)=⎛⎜⎜⎜⎜⎜⎜⎝𝒜𝒜(ΔΔ𝑋𝜏)−𝑏_∗Δ𝑋+Δ𝑆−𝐿_𝐶−1(Δ𝑦)+Δ𝑆−𝐶[𝑋Δ𝑋+Δ𝑋𝑋+𝑆Δ𝑆+Δ𝑆𝑆]⎞⎟⎟⎟⎟⎟⎟⎠,(3.3) where Δ𝜏>0 and 𝐶∶=√𝑋²+𝑆²+2𝜏²𝐼.

Proof. For any 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, since 𝒜(𝑋)−𝑏 and 𝒜^∗(𝑦)+𝑆−𝐶 are linear functions and continuous differentiable, it follows that they are also locally Lipschitz continuous. Then, from Lemma 2.7, 𝒜(𝑋)−𝑏 and 𝒜^∗(𝑦)+𝑆−𝐶 are Fréchet-differentiable. Furthermore, 𝑋+𝑆−√𝑋²+𝑆²+2𝜏²𝐼 is Fréchet-differentiable from Lemma 2.6. Thus, 𝐻_𝜏(𝑍) is Fréchet-differentiable and has the form of (3.3). We complete the proof.

We endow the variable 𝑍 with the following norm: ‖𝑍‖=‖(𝑋,𝑦,𝑆)‖∶=‖𝑋‖^2𝐹+‖𝑦‖²²+‖𝑆‖^2𝐹^1/2.(3.4) In addition, we set ℎ(𝑍)=ℎ₁(𝑍),ℎ₂(𝑍),ℎ₃(𝑍),ℎ₄(𝑍)^𝑇,(3.5) where ℎ₂ℎ₁(𝑍)=‖𝒜(𝑋)−𝑏‖₂,ℎ₃(𝑍)=‖‖𝒜^∗(𝑦)+𝑆−𝐶‖‖_𝐹,(𝑍)=‖‖‖𝑋+𝑆−√𝑋²+𝑆²+2𝜏²𝐼‖‖‖_𝐹ℎ₄(𝑍)=|𝜏|.(3.6) We also define the function 𝐻_𝜏(𝑍) and the vector ℎ(𝑍) with the following norm: ‖‖𝐻_𝜏(𝑍)‖‖=‖ℎ(𝑍)‖=⁴_𝑖=1ℎ_𝑖(𝑍)²^1/2=‖𝒜(𝑋)−𝑏‖²²+‖‖𝒜^∗(𝑦)+𝑆−𝐶‖‖^2𝐹+‖‖‖𝑋+𝑆−√𝑋²+𝑆²+2𝜏²𝐼‖‖‖^2𝐹+𝜏²^1/2.(3.7)

Now, for any 𝜏>0, we define the merit function Ψ∶𝒳^𝑛×ℛ^𝑚×𝒳^𝑛→ℛ by Ψ_𝜏(𝑍)∶=12‖‖𝐻^𝜏(𝑍)‖‖².(3.8)

Lemma 3.3. For any 𝜏>0 and 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, if 𝑋 and 𝑆 are nonsingular, then Ψ_𝜏(𝑍) is locally Lipschitz continuous and twice Fréchet-differentiable at every 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛.

Proof. For any 𝜏>0, since Ψ_𝜏(𝑍) is convex and continuously differentiable, it follows that Ψ_𝜏(𝑍) is also locally Lipschitz continuous.
In addition, for any 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, from [20, pages 173–175], ℎ₃(𝑍)² is twice Fréchet-differentiable. Furthermore, ℎ₁(𝑍)², ℎ₂(𝑍)², and ℎ₄(𝑍)² are continuous at every 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛 when 𝜏>0, which, together with Lemma 2.7, Ψ_𝜏(𝑍) is twice Fréchet-differentiable. The proof is completed.

Lemma 3.4. Let 𝐻_𝜏(𝑍) and Ψ_𝜏(𝑍) be defined by (3.1) and (3.8), respectively. For any 𝜏>0, we have Ψ_𝜏(𝑍)=0⟺𝐻_𝜏(𝑍)=0.(3.9)

Proof. The proof can be immediately obtained from the definition of 𝐻_𝜏(𝑍) and Ψ_𝜏(𝑍).

We follow the classical method for solving Ψ_𝜏(𝑍)=0, which consists some norm of the residual. For any 𝜏>0, we consider minΨ_𝜏(𝑍),(3.10) where 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛. Thus, for any 𝜏>0, we want to find a minimizer 𝑍^∗ of Ψ_𝜏(𝑍). Furthermore, if Ψ_𝜏(𝑍_∗)=0, then 𝑍_∗ is also a solution of 𝐻_𝜏(𝑍).

In order to state our method for solving (3.10), we consider using a filter mechanism to accept a new point. Just as [16, pages 19–20], the notation of filter is based on that of dominance.

Definition 3.5. For any 𝜏>0 and any 𝑍₁,𝑍₂∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, a point 𝑍₁ dominates a point 𝑍₂ if and only if ℎ_𝑖𝑍₁≤ℎ_𝑖𝑍₂∀𝑖=1,2,3,4.(3.11)

Thus, if iterate 𝑍₁ dominates iterate 𝑍₂, the latter is of no real interest to us since 𝑍₁ is at least as good as 𝑍₂ for each of the components of ℎ(𝑍). All we need to do is remember iterates that are no dominated by other iterates by using a structure called a filter.

Definition 3.6. Let 𝐹^(𝑘) be a set of 4-tuples of the following form: ℎ₁𝑍_𝑘,ℎ₂𝑍_𝑘,ℎ₃𝑍_𝑘,ℎ₄𝑍_𝑘.(3.12) We define 𝐹^(𝑘) as a filter if ℎ(𝑍_𝑘) and ℎ(𝑍_𝑙) belong to 𝐹^(𝑘), when 𝑘≠𝑙, then ℎ_𝑖𝑍_𝑘<ℎ_𝑖𝑍_𝑙foratleastone𝑖∈{1,2,3,4}.(3.13)

Definition 3.7. A new point 𝑍^+𝑘 is acceptable for the filter 𝐹^(𝑘) if and only if ∀ℎ𝑍_𝑘∈𝐹^(𝑘)∃𝑖∈{1,2,3,4}∶ℎ_𝑖𝑍^+𝑘≤ℎ_𝑖𝑍_𝑘−𝛼‖‖ℎ𝑍_𝑘‖‖,(3.14) where 𝛼∈(0,1/√4) is a small constant.

Now, we formally present our trust region algorithm by using filter techniques.

Algorithm 3.8. The Filter-Trust-Region Algorithm
Step 0. Choose an initial point 𝑍₀=(𝑋₀,𝑦₀,𝑆₀)∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, 𝜀>0, 0<𝛼<1/√4 and 𝜏₀=⟨𝑋₀,𝑆₀⟩/2𝑛. The constants 𝜂₁, 𝜂₂, 𝜂₃, 𝜇, 𝛾, 𝛾₁, and 𝛾₂ are also given and satisfy 0<𝜂₁≤𝜂₂≤𝜂₃<1,0<𝜇<1,0<𝛾<𝛾₁<1≤𝛾₂.(3.15)
Compute Ψ_𝜏0(𝑍₀), set Δ₀=0.5‖∇Ψ_𝜏0(𝑍₀)‖, 𝑘=0 and only (𝜇,−∞,𝜇,𝜇) in the filter 𝐹⁽⁰⁾.
Step 1. If ∇Ψ_𝜏𝑘(𝑍_𝑘)<𝜀, stop.
Step 2. Compute Δ𝑍_𝑘 by solving the following problem: sm.t.in𝜑‖_𝑘Δ(𝑍Δ𝑍‖)≤Δ_𝑘,(3.16) where 𝜑_𝑘(Δ𝑍)=12‖‖𝐻^𝜏𝑘𝑍_𝑘+∇𝐻_𝜏𝑘𝑍_𝑘(Δ𝑍)‖‖²=+Ψ1_𝜏𝑘𝑍_𝑘+𝐻_𝜏𝑘𝑍_𝑘^𝑇∇𝐻_𝜏𝑘𝑍_𝑘(Δ𝑍)2(Δ𝑍)^𝑇∇𝐻_𝜏𝑘𝑍_𝑘^𝑇∇𝐻_𝜏𝑘𝑍_𝑘(Δ𝑍).(3.17)
If ‖Δ𝑍_𝑘‖<𝜀, stop.
Otherwise, computer the trial point 𝑍^+𝑘=𝑍_𝑘+Δ𝑍_𝑘.
Step 3. Compute Ψ_𝜏𝑘(𝑍^+𝑘) and define the following ratio: 𝑟_𝑘=Ψ^𝜏𝑘𝑍_𝑘−Ψ_𝜏𝑘𝑍^+𝑘𝜑_𝑘(0)−𝜑_𝑘Δ𝑍_𝑘.(3.18)
Step 4. If 𝑟_𝑘≥𝜂₁, set 𝑍_𝑘+1=𝑍^+𝑘.
If 𝑟_𝑘<𝜂₁ but 𝑍^+𝑘 satisfies (3.14), then add ℎ(𝑍^+𝑘) to the filter 𝐹^(𝑘) and remove all points from 𝐹^(𝑘) dominated by ℎ(𝑍^+𝑘). At the same time, set 𝑍_𝑘+1=𝑍^+𝑘.
Else, set 𝑍_𝑘+1=𝑍_𝑘.
Step 5. Update 𝜏_𝑘 by choosing 𝜏_𝑘+1∈𝛾𝜏^𝑘𝜏_𝑘if𝑍_𝑘+1=𝑍^+𝑘,else;(3.19) and update trust-region radius Δ_𝑘 by choosing Δ_𝑘+1∶=⎧⎪⎨⎪⎩𝛾Δ^𝑘𝛾₁,if𝑟_𝑘<𝜂₁,Δ_𝑘Δ_𝑘,if𝑟_𝑘∈𝜂₁,𝜂₂,𝛾₂,if𝑟_𝑘∈𝜂₂,𝜂₃,Δ_𝑘,if𝑟_𝑘≥𝜂₃.(3.20)
Step 6. Set 𝑘∶=𝑘+1 and go to Step 1.

Remark 3.9. Algorithm 3.8 can be started any 𝜏>0. In fact, in order to increase the convergent speed greatly, we always choose 𝜏₀=⟨𝑋₀,𝑆₀⟩/2𝑛. In addition, in this algorithm, we fix 𝜏 at first, then search 𝑍 for Ψ_𝜏(𝑍)=0 to update 𝑍. At last we update 𝜏 and repeat.

The following lemma is a generalized case of Proposition 3.1 in [23].

Lemma 3.10. Algorithm 3.8 is well defined, that is, the inner iteration (Step 2) terminates finitely.

For the purpose of our analysis, in the sequence of points generated by Algorithm 3.8, we denote 𝒜={𝑘∣𝑟_𝑘≥𝜂₁},ℬ={𝑘|ℎ(𝑍^+𝑘)isaddedtothefilter𝐹^(𝑘)}, and 𝒞={𝑘|𝑍_𝑘+1=𝑍_𝑘+Δ𝑍_𝑘}. It is clear that, 𝒞=𝒜⋃ℬ.

Remark 3.11. Lemma 3.3 implies that there exists a constant 0<𝑀≤1 such that ℎ_𝑖𝑍_𝑘≤𝑀,‖‖∇²ℎ_𝑖𝑍_𝑘‖‖≤𝑀,‖‖∇²𝜑_𝑘(Δ𝑍)‖‖≤𝑀(3.21) for all 𝑘∈𝒞 and 𝑖∈{1,2,3,4}. The second of above inequalities ensures that the constant 0<𝑀≤1 can also be chosen such that ‖‖∇²Ψ_𝜏𝑘𝑍_𝑘‖‖≤𝑀.(3.22)

4. Convergence of Analysis

In this section, we present a proof of global convergence of Algorithm 3.8. First, we make the following assumptions.

Some lemmas will be presented to be used in the subsequent analysis. (S1)𝜑_𝑘(0)−𝜑_𝑘(Δ𝑍_𝑘)≥1/2‖∇Ψ_𝜏𝑘(𝑍_𝑘)‖min{Δ_𝑘,‖∇Ψ_𝜏𝑘(𝑍_𝑘)‖/‖∇𝐻_𝜏𝑘(𝑍_𝑘)^𝑇∇𝐻_𝜏𝑘(𝑍_𝑘)‖}, where Δ𝑍_𝑘 is a solution of (3.16). (S2)  The iterations generated by Algorithm 3.8 remain in a close, bounded domain.

Lemma 4.1 (see [24]). Let assumptions (S1) and (S2) hold. If there exists 𝑙₀>0 such that ‖∇Ψ_𝜏𝑘(𝑍_𝑘)‖≥𝑙₀>0 for all 𝑘; then there exists 𝑙₁>0 such that Δ_𝑘≥𝑙₁.

Lemma 4.2. Let {𝜏_𝑘} be the infinite sequence generated by the Algorithm 3.8. Then lim_𝑘→∞𝜏_𝑘=0.(4.1)

Proof. Since |𝒞|=|𝒜|=+∞, from Steps 4 and 5 of Algorithm 3.8, 𝜏_𝑘+1=𝛾𝜏_𝑘 and 0<𝛾<𝜏₀<1. Therefore, 𝜏_𝑘+1=𝛾^𝑘𝜏₀. Moreover, lim_𝑘→∞𝜏_𝑘=lim_𝑘→∞𝛾^𝑘𝜏₀=0(4.2) for 0<𝛾<𝜏₀<1, which completes the proof.

Theorem 4.3. Let |𝒞|<+∞, assumptions (S1) and (S2) hold. Then there exists 𝑘∈𝒞 such that ∇Ψ_𝜏𝑘𝑍_𝑘=0.(4.3)

Proof. Suppose that ∇Ψ_𝜏𝑘(𝑍_𝑘)≠0 for all 𝑘∈𝒞. Then there exists 𝜔₀>0 such that ‖‖∇Ψ_𝜏𝑘𝑍_𝑘‖‖≥𝜔₀>0.(4.4) From Lemma 4.1, there exists 𝜔₁>0 such that Δ_𝑘≥𝜔₁>0.(4.5)
On the other hand, |𝒞|<+∞, let 𝑁 be the last successful iteration, then 𝑍_𝑁+1=𝑍_𝑁+2=⋯=𝑍_𝑁+𝑗(𝑗≥1) are unsuccessful iterations. From Steps 4 and 5 of Algorithm 3.8, 𝑟_𝑁+𝑗<𝜂₁, for sufficiently large 𝑁, we have lim_𝑁→∞Δ_𝑁+𝑗=0,(4.6) which contradicts (4.5). The proof is completed.

We now consider what happens if the set 𝒜 is infinite in the course of Algorithm 3.8.

Theorem 4.4. Suppose that |𝒞|=|𝒜|=+∞, assumptions (S1) and (S2) hold. For any 𝜏>0 and 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, if 𝑋 and 𝑆 are nonsingular, then each accumulation point of the infinite sequences generated by Algorithm 3.8 is a stationary point of Ψ_𝜏(𝑍).

Proof. The proof is by contradiction. Suppose that {𝑍_𝑘} is an infinite sequence generated by Algorithm 3.8, and any accumulation point of {𝑍_𝑘} is not a stationary point of Ψ_𝜏(𝑍). Suppose furthermore that 𝑍_∗ and 𝜏_∗ are the accumulation points of {𝑍_𝑘} and {𝜏_𝑘}, respectively. Since 𝑍_∗ is not a stationary point of Ψ_𝜏(𝑍), then ∇Ψ_𝜏∗𝑍_∗≠0(4.7) and there exists 𝜖₀>0 such that ‖‖∇Ψ_𝜏∗𝑍_∗‖‖>𝜖₀>0.(4.8) For some 𝜖_∗>0, let 𝒩(𝑍_∗,𝜖_∗) be a neighborhood of 𝑍_∗. From (4.8), there exists {𝑍_𝑘}_𝑘∈𝐾∈𝒩(𝑍_∗,𝜖_∗) such that ‖‖∇Ψ_𝜏𝑘𝑍_𝑘‖‖≥𝜖₀>0,(4.9) where 𝐾⊆𝒜.
For 𝑚,𝑚+𝜈∈𝐾, because Ψ_𝜏𝑘𝑍_𝑘−Ψ_𝜏𝑘+1𝑍_𝑘+1≥𝜂₁𝜑_𝑘(0)−𝜑_𝑘Δ𝑍_𝑘,(4.10) we obtain that Ψ_𝜏𝑚𝑍_𝑚−Ψ_{𝜏𝑚+𝜈}𝑍_𝑚+𝜈=^𝑚+𝜈≥𝜂_{𝑖=𝑚∈𝐾1𝑚+𝜈}Ψ^𝜏_𝑖𝑍_𝑖−Ψ_𝜏𝑖+1𝑍_𝑖+1_𝑖=_𝑚∈𝐾𝜑_𝑘≥𝜂_1𝑚+𝜈(0)−𝜑_𝑘Δ𝑍_𝑘_𝑖=_𝑚∈𝐾12‖‖∇Ψ^𝜏𝑘𝑍_𝑘‖‖min⎧⎪⎨⎪⎩Δ_𝑘,‖‖∇Ψ^𝜏𝑘𝑍_𝑘‖‖≥𝜂_1𝑚+𝜈‖‖∇𝐻_𝜏𝑘𝑍_𝑘^𝑇∇𝐻_𝜏𝑘𝑍_𝑘‖‖⎫⎪⎬⎪⎭_𝑖=_𝑚∈𝐾12𝜖⁰min⎧⎪⎨⎪⎩Δ_𝑘,𝜖⁰‖‖∇𝐻_𝜏𝑘𝑍_𝑘^𝑇∇𝐻_𝜏𝑘𝑍_𝑘‖‖⎫⎪⎬⎪⎭.(4.11)
From (4.10), we know that Ψ_𝜏𝑘(𝑍_𝑘) is monotone decreasing and bounded below, which implies that Ψ_𝜏𝑚(𝑍_𝑚)−Ψ_{𝜏𝑚+𝑛}(𝑍_𝑚+𝜈)→0 for 𝑚→∞,𝑚∈𝐾. Thus, 𝜂_{1𝑚+𝜈𝑖=}_𝑚∈𝐾12𝜖⁰min⎧⎪⎨⎪⎩Δ_𝑘,𝜖⁰‖‖∇𝐻_𝜏𝑘𝑍_𝑘^𝑇∇𝐻_𝜏𝑘𝑍_𝑘‖‖⎫⎪⎬⎪⎭⟶0.(4.12) As a result, we have lim_{𝑘→∞,𝑘∈𝐾}Δ_𝑘=0.(4.13) By the update rule of Δ_𝑘, there exists an infinite subsequence 𝐾^⋆⊆𝐾, and we have that 𝑟_𝑖≤𝜂₁,lim_𝑖→∞Δ_𝑖=0,𝑖∈𝐾^⋆.(4.14) which contradicts 𝑘∈𝐾⊆𝒜. This completes the proof.

In what follows, we investigate the case where the number of iterations added to the filter 𝐹^(𝑘) in the course of Algorithm 3.8 is infinite.

Theorem 4.5. Suppose that |𝒞|=|ℬ|=+∞ but |𝐴|<+∞, SDP problems (2.1) and (2.3) have strictly feasible points. Suppose furthermore that assumptions (S1) and (S2) hold. For any 𝜏>0 and 𝑍∈𝒳^𝑛×ℛ^𝑚×𝒳^𝑛, if 𝑋 and 𝑆 are nonsingular, then lim_𝑘→∞‖‖𝐻_𝜏𝑍_𝑘‖‖=lim_𝑘→∞‖‖∇Ψ_𝜏𝑘𝑍_𝑘‖‖=0.(4.15)

Proof. First let {𝜏_𝑘} be the sequence generated by Algorithm 3.8. From Lemma 4.2, we have lim_𝑘→∞𝜏_𝑘=0,(4.16) which, together with assumption (S2), the desired result follows from [16, Lemma  3.1].

5. Numerical Experiments

In this section, we describe the results of some numerical experiments with the Algorithm 3.8 for the random sparse SDP considered in [13]. All programs are written in Matlab code and all computations are tested under Matlab 7.1 on Pentium 4.

In addition, in the computations, the following values are assigned to the parameters in the Algorithm: 𝜂₁=0.1, 𝜂₂=0.5, 𝜂₃=0.8, 𝜇=0.1, 𝛾=0.2, 𝛾₁=0.5, and 𝛾₂=2. We also use the stopping criteria is being of 𝜀=10⁻⁸.

In the following Table 1, the first two columns give the size of the matrix 𝐶 and the dimension of the variable 𝑦. In the middle columns, “𝐹-time” denotes the computing time (in seconds), “𝐹-it.” denotes the numbers iteration, and “𝐹-obj.” defines the value of Ψ_𝜏𝑘(𝑍_𝑘) when our stopping criteria is satisfied. Some numerical results of [13] are shown in the last two columns.

Table 1 

As shown in Table 1, all test problems have been solved just few iterations compared with [13]. Furthermore, this algorithm is less sensitive to the size of SDP problems. Comparatively speaking, our method is attractive and suitable for solving large-scale SDP problems.

6. Conclusions

In this paper, we have proposed a filter-trust-region method for SDP problems. Such a method offers a trade-off between the accuracy of solving the subproblems and the amount of work for solving them. Furthermore, numerical results show that our algorithm is attractive for large-scale SDP problems.

Acknowledgments

The authors would like to thank Professor Florian Jarre for his advice and guidance, Thomas David and Li Luo for their grateful help, and also the referees for their helpful comments. This work is supported by National Natural Science Foundation of China 10971162.