This paper presents modified halfspace-relaxation projection (HRP) methods for solving the split feasibility problem (SFP). Incorporating with the techniques of identifying the optimal step length with positive lower bounds, the new methods improve the efficiencies of the HRP method (Qu and Xiu (2008)). Some numerical results are reported to verify the computational preference.
1. Introduction
Let C and Q be nonempty closed convex sets in Rn and Rm, respectively, and A an m×n real matrix. The problem, to find x∈C with Ax∈Q if such x exists, was called the split feasibility problem (SFP) by Censor and Elfving [1].
In this paper, we consider an equivalent reformulation [2] of the SFP:
(1.1)minimizef(z)subjecttoz=(xy)∈Ω,
where
(1.2)f(z)=12‖Bz‖2=12‖y-Ax‖2,B=(-AI),Ω=C×Q.
For convenience, we only consider the Euclidean norm. It is obvious that f(z) is convex. If z=(xT,yT)T∈Ω and f(z)=0, then x solves the SFP. Throughout we assume that the solution set of the SFP is nonempty. And thus the solution set of (1.1), denoted by Ω*, is nonempty. In addition, in this paper, we always assume that the set Ω is given by
(1.3)Ω={z∈Rn+m∣c(z)≤0},
where c:Rn+m→R is a convex (not necessarily differentiable) function. This representation of Ω is general enough, because any system of inequalities {cj(z)≤0,j∈J}, where cj(z) are convex and J is an arbitrary index set, can be reformulated as the single inequality c(z)≤0 with c(z)=sup{cj(z)∣j∈J}. For any z∈Rn+m, at least one subgradient ξ∈∂c(z) can be calculated, where ∂c(z) is a subgradient of c(z) at z and is defined as follows:
(1.4)∂c(z)={ξ∈Rn+m∣c(u)≥c(z)+(u-z)Tξ,∀u∈Rn+m}.
Qu and Xiu [2] proposed a halfspace-relaxation projection method to solve the convex optimization problem (1.1). Starting from any z0∈Rn×Rm, the HRP method iteratively updates zk according to the formulae:
(1.5)z-k=PΩk[zk-αk∇f(zk)],(1.6)zk+1=zk-γk[zk-z-k-αk(∇f(zk)-∇f(z-k))],
where
(1.7)Ωk={z∈Rn+m∣c(zk)+(z-zk)Tξk≤0},ξk is an element in ∂c(zk), αk=γlmk and mk is the smallest nonnegative integer m such that
(1.8)αk(zk-z-k)T(∇f(zk)-∇f(z-k))≤(1-ρ)‖zk-z-k‖2,ρ∈(0,1),(1.9)γk=θρ‖zk-z-k‖2‖zk-z-k-αk(∇f(zk)-∇f(z-k))‖2,θ∈(0,2).
The notation PΩk(v) denotes the projection of v onto Ωk under the Euclidean norm, that is,
(1.10)PΩk(v)=arg min{‖u-v‖∣u∈Ωk}.
Here the halfspace Ωk contains the given closed convex set Ω and is related to the current iterative point zk. From the expressions of Ωk, the projection onto Ωk is simple to be computed (for details, see Proposition 3.3). The idea to construct the halfspace Ωk and replace PΩ by PΩk is from the halfspace-relaxation projection technique presented by Fukushima [3]. This technique is often used to design algorithms (see, e.g., [2, 4, 5]) to solve the SFP. The drawback of the HRP method in [2] is that the step length γk defined in (1.9) may be very small since limk→∞∥zk-z-k∥=0.
Note that the reformulation (1.1) is equivalent to a monotone variational inequality (VI):
(1.11)z*∈Ω,(z-z*)T∇f(z*)≥0,∀z∈Ω,
where
(1.12)∇f(z)=BTBz.
The forward-backward splitting method [6] and the extragradient method [7, 8] are considerably simple projection-type methods in the literature. They are applicable for solving monotone variational inequalities, especially for (1.11). For given zk, let
(1.13)z-k=PΩ[zk-αk∇f(zk)].
Under the assumption
(1.14)αk‖∇f(zk)-∇f(z-k)‖≤ν‖zk-z-k‖,ν∈(0,1),
the forward-backward (FB) splitting method generates the new iterate via
(1.15)zk+1=PΩ[z-k+αk(∇f(zk)-∇f(z-k))],
while the extra-gradient (EG) method generates the new iterate by
(1.16)zk+1=PΩ[zk-αk∇f(z-k)].
The forward-backward splitting method (1.15) can be rewritten as
(1.17)zk+1=PΩ[zk-γk(zk-z-k-αk(∇f(zk)-∇f(z-k)))],
where the descent direction -(zk-z-k-αk(∇f(zk)-∇f(z-k))) is the same as (1.6) and the step length γk along this direction always equals to 1. He et al. [9] proposed the modified versions of the FB method and EG method by incorporating the optimal step length γk along the descent directions -(zk-z-k-αk(∇f(zk)-∇f(z-k))) and -αk∇f(z-k), respectively. Here γk is defined by
(1.18)γk=θγk*,θ∈(0,2),γk*=(zk-z-k)T(zk-z-k-αk(∇f(zk)-∇f(z-k)))‖zk-z-k-αk(∇f(zk)-∇f(z-k))‖2.
Under the assumption (1.14), γk*≥1/2 is lower bounded.
This paper is to develop two kinds of modified halfspace-relaxation projection methods for solving the SFP by improving the HRP method in [2]. One is an FB type HRP method, the other is an EG type HRP method. The numerical results reported in [9] show that efforts of identifying the optimal step length usually lead to attractive numerical improvements. This fact triggers us to investigate the selection of optimal step length with positive lower bounds in the new methods to accelerate convergence. The preferences to the HRP method are verified by numerical experiments for the test problems arising in [2].
The rest of this paper is organized as follows. In Section 2, we summarize some preliminaries of variational inequalities. In Section 3, we present the new methods and provide some remarks. The selection of optimal step length of the new methods is investigated in Section 4. Then, the global convergence of the new methods is proved in Section 5. Some preliminary numerical results are reported in Section 6 to show the efficiency of the new methods, and the numerical superiority to the HRP method in [2]. Finally, some conclusions are made in Section 7.
2. Preliminaries
In the following, we state some basic concepts for the variational inequality VI(𝒮,F):
(2.1)s*∈S,(s-s*)TF(s*)≥0,∀s∈S,
where F is a mapping from RN into RN, and 𝒮⊆RN is a nonempty closed convex set. The mapping F is said to be monotone on RN if
(2.2)(s-t)T(F(s)-F(t))≥0,∀s,t∈RN.
Notice that the variational inequality VI(𝒮,F) is invariant when we multiply F by some positive scalar α. Thus VI(𝒮,F) is equivalent to the following projection equation (see [10]):
(2.3)s=PS[s-αF(s)],
that is, to solve VI(𝒮,F) is equivalent to finding a zero point of the residue function
(2.4)e(s,α):=s-PS[s-αF(s)].
Note that e(s,α) is a continuous function of s because the projection mapping is nonexpansive. The following lemma states a useful property of ∥e(s,α)∥.
Let F be a mapping from RN into RN. For any s∈RN and α>0, we have
(2.5)min{1,α}‖e(s,1)‖≤‖e(s,α)‖≤max{1,α}‖e(s,1)‖.
Remark 2.2.
Let 𝒮~⊇𝒮 be a nonempty closed convex set and let e𝒮~(s,α) be defined as follows:
(2.6)eS~(s,α)=s-PS~[s-αF(s)].
Inequalities (2.5) still hold for e𝒮~(s,α).
Some fundamental inequalities are listed below without proof, see, for example, [10].
Lemma 2.3.
Let 𝒮~ be a nonempty closed convex set. Then the following inequalities always hold
(2.7)(t-PS~(t))T(PS~(t)-s)≥0,∀t∈RN,∀s∈S~,(2.8)‖PS~(t)-s‖2≤‖t-s‖2-‖t-PS~(t)‖2,∀t∈RN,s∈S~.
The next lemma lists some inequalities which will be useful for the following analysis.
Lemma 2.4.
Let 𝒮~⊇𝒮 be a nonempty closed convex set, s* a solution of the monotone VI(𝒮,F) (2.1) and especially F(s*)=0. For any s∈RN and α>0, one has
(2.9)α(s-s*)TF(PS~[s-αF(s)])≥αeS~(s,α)TF(PS~[s-αF(s)]),(2.10)(s-s*)T{eS~(s,α)-α[F(s)-F(PS~[s-αF(s)])]}≥eS~(s,α)T{eS~(s,α)-α[F(s)-F(PS~[s-αF(s)])]}.
Proof.
Under the assumption that F is monotone we have
(2.11){αF(PS~[s-αF(s)])-αF(s*)}T{PS~[s-αF(s)]-s*}≥0,∀s∈RN.
Using F(s*)=0 and the notation of e𝒮~(s,α), from (2.11) the assertion (2.9) is proved. Setting t=s-αF(s) and s=s* in the inequality (2.7) and using the notation of e𝒮~(s,α), we obtain
(2.12){eS~(s,α)-αF(s)}T{PS~[s-αF(s)]-s*}≥0,∀s∈RN.
Adding (2.11) and (2.12), and using F(s*)=0, we have (2.10). The proof is complete.
Note that the assumption F(s*)=0 in Lemma 2.4 is reasonable. The following proposition and remark will explain this.
For the optimization problem (1.1), the following two statements are equivalent:
z*∈Ω and f(z*)=0,
z*∈Ω and ∇f(z*)=0.
Remark 2.6.
Under the assumption that the solution set of the SFP is nonempty, if z*=((x*)T,(y*)T)T is a solution of (1.1), then we have
(2.13)∇f(z*)=BTBz*=BT(y*-Ax*)=0.
This point z* is also the solution point of the VI (1.11).
The next lemma provides an important boundedness property of the subdifferential, see, for example, [11].
Lemma 2.7.
Suppose h:RN→R is a convex function, then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded on any bounded subset of RN.
In this section, we will propose two kinds of modified halfspace-relaxation projection methods—Algorithms 1 and 2. Algorithm 1 is an FB type HRP method and Algorithm 2 is an EG type HRP method. The relationship of these two methods is that they use the same optimal step length along different descent directions. The detailed procedures are presented as below.
The Modified Halfspace-Relaxation Projection Methods
Step 1. Let α0>0, 0<μ<ν<1, z0∈Rn+m, θ∈(0,2), ε>0 and k=0. (In practical computation, we suggest to take μ=0.3,ν=0.9 and θ=1.8).
Step 2. Set
(3.1)z-k=PΩk[zk-αk∇f(zk)],
where Ωk is defined in (1.7). If ∥zk-z-k∥≤ε, terminate the iteration with the iterate zk=((xk)T,(yk)T)T, and then xk is the approximate solution of the SFP. Otherwise, go to Step 3.
Step 3. If
(3.2)rk:=αk‖∇f(zk)-∇f(z-k)‖‖zk-z-k‖≤ν,
then set
(3.3)ek(zk,αk)=eΩk(zk,αk)=zk-z-k,gk(zk,αk)=αk∇f(z-k),(3.4)dk(zk,αk)=ek(zk,αk)-αk∇f(zk)+gk(zk,αk),(3.5)γk*=ek(zk,αk)Tdk(zk,αk)‖dk(zk,αk)‖2,γk=θγk*,(3.6)zk+1=PΩk[zk-γkdk(zk,αk)],(Algorithm1:FBtypeHRPmethod)
or
(3.7)zk+1=PΩk[zk-γkgk(zk,αk)],(Algorithm2:EGtypeHRPmethod)(3.8)αk:={32αk ifrk≤μ,αkotherwise,αk+1=αk,k=k+1,gotoStep2.
Step 4. Reduce the value of αk by αk:=(2/3)αk*min{1,1/rk},
(3.9)setz-k=PΩk[zk-αk∇f(zk)]andgotoStep3.
Remark 3.1.
In Step 3, if the selected αk satisfies 0<αk≤ν/L (L is the largest eigenvalue of the matrix BTB), then from (1.12), we have
(3.10)αk‖∇f(zk)-∇f(z-k)‖≤αkL‖zk-z-k‖≤ν‖zk-z-k‖,
and thus Condition (3.2) is satisfied. Without loss of generality, we can assume that infk{αk}=αmin>0.
Remark 3.2.
By the definition of subgradient, it is clear that the halfspace Ωk contains Ω. From the expressions of Ωk, the orthogonal projections onto Ωk may be directly calculated and then we have the following proposition (see [3, 12]).
Proposition 3.3.
For any z∈Rn+m,
(3.11)PΩk(z)={z-c(zk)+(z-zk)Tξk∥ξk∥2ξk,ifc(zk)+(z-zk)Tξk>0;z,otherwise,
where Ωk is defined in (1.7).
Remark 3.4.
For the FB type HRP method, taking
(3.12)zk+1=zk-γkdk(zk,αk)
as the new iterate instead of Formula (3.6) seems more applicable in practice. Since from Proposition 3.3 the projection onto Ωk is easy to be computed, Formula (3.6) is still preferable to generate the new iterate zk+1.
Remark 3.5.
The proposed methods and the HRP method in [2] can be used to solve more general convex optimization problem
(3.13)minimizef(z)subjecttoz∈Ω,
where f(z) is a general convex function only with the property that ∇f(z*)=0 for any solution point z* of (3.13), and Ω is defined in (1.3). The corresponding theoretical analysis is similar as these methods to solve (1.1).
4. The Optimal Step Length
This section concentrates on investigating the optimal step length with positive lower bounds in order to accelerate convergence of the new methods. To justify the reason of choosing the optimal step length γk in the FB type HRP method (3.6), we start from the following general form of the FB type HRP method:
(4.1)zFBk+1(γ)=PΩk[zPCk+1(γ)],
where
(4.2)zPCk+1(γ)=zk-γdk(zk,αk).
Let
(4.3)Θk(FBγ):=‖zk-z*‖2-‖zFBk+1(γ)-z*‖2,
which measures the progress made by the FB type HRP method. Note that Θk(FBγ) is a function of the step length γ. It is natural to consider maximizing this function by choosing an optimal parameter γ. The solution z* is not known, so we cannot maximize Θk(FBγ) directly. The following theorem gives an estimate of Θk(FBγ) which does not include the unknown solution z*.
Theorem 4.1.
Let z* be an arbitrary point in Ω*. If the step length in the general FB type HRP method is taken γ>0, then we have
(4.4)Θk(FBγ):=‖zk-z*‖2-‖zFBk+1(γ)-z*‖2≥Υk(γ),
where
(4.5)Υk(γ):=2γek(zk,αk)Tdk(zk,αk)-γ2‖dk(zk,αk)‖2.
Proof.
Since zFBk+1(γ)=PΩk[zPCk+1(γ)] and z*∈Ω⊆Ωk, it follows from (2.8) that
(4.6)‖zFBk+1(γ)-z*‖2≤‖zPCk+1(γ)-z*‖2-‖zPCk+1(γ)-zFBk+1(γ)‖2≤‖zPCk+1(γ)-z*‖2,
and consequently
(4.7)Θk(FBγ)≥‖zk-z*‖2-‖zPCk+1(γ)-z*‖2.
Setting α=αk, s=zk, s*=z* and 𝒮~=Ωk in the equality (2.10) and using the notation of ek(zk,αk) (see (3.3)) and dk(zk,αk) (see (3.4)), we have
(4.8)(zk-z*)Tdk(zk,αk)≥ek(zk,αk)Tdk(zk,αk).
Using this and (4.2), we get
(4.9)‖zk-z*‖2-‖zPCk+1(γ)-z*‖2=‖zk-z*‖2-‖zk-γdk(zk,αk)-z*‖2=2γ(zk-z*)Tdk(zk,αk)-γ2‖dk(zk,αk)‖2≥2γek(zk,αk)Tdk(zk,αk)-γ2‖dk(zk,αk)‖2,
and then from (4.7) the theorem is proved.
Similarly, we start from the general form of the EG type HRP method
(4.10)zEGk+1(γ)=PΩk[zk-γgk(zk,αk)].
to analyze the optimal step length in the EG type HRP method (3.7). The following theorem estimates the “progress” in the sense of Euclidean distance made by the new iterate and thus motivates us to investigate the selection of the optimal length γk in the EG type HRP method (3.7).
Theorem 4.2.
Let z* be an arbitrary point in Ω*. If the step length in the general EG type HRP method is taken γ>0, then one has
(4.11)Θk(EGγ):=‖zk-z*‖2-‖zEGk+1(γ)-z*‖2≥Υk(γ),
where Υk(γ) is defined in (4.5) and zPCk+1(γ) is defined in (4.2).
Proof.
Since zEGk+1(γ)=PΩk[zk-γgk(zk,αk)] and z*∈Ω⊆Ωk, it follows from (2.8) that
(4.12)‖zEGk+1(γ)-z*‖2≤‖zk-γgk(zk,αk)-z*‖2-‖zk-γgk(zk,αk)-zEGk+1(γ)‖2,
and consequently we get
(4.13)Θk(EGγ)≥‖zk-z*‖2-‖zk-z*-γgk(zk,αk)‖2+‖zk-zEGk+1(γ)-γgk(zk,αk)‖2=‖zk-zEGk+1(γ)‖2+2γ(zk-z*)Tgk(zk,αk)-2γ(zk-zEGk+1(γ))Tgk(zk,αk).
Setting α=αk, s=zk, s*=z* and 𝒮~=Ωk in the equality (2.9) and using the notation of ek(zk,αk) and gk(zk,αk) (see (3.3)), we have
(4.14)(zk-z*)Tgk(zk,αk)≥ek(zk,αk)Tgk(zk,αk).
From the above inequality, we obtain
(4.15)Θk(EGγ)≥‖zk-zEGk+1(γ)‖2+2γek(zk,αk)Tgk(zk,αk)-2γ(zk-zEGk+1(γ))Tgk(zk,αk).
Using gk(zk,αk)=dk(zk,αk)-[ek(zk,αk)-αk∇f(zk)] (see (3.4)), it follows that
(4.16)Θk(EGγ)≥‖zk-zEGk+1(γ)‖2+2γek(zk,αk)T{dk(zk,αk)-[ek(zk,αk)-αk∇f(zk)]}-2γ(zk-zEGk+1(γ))T{dk(zk,αk)-[ek(zk,αk)-αk∇f(zk)]},
which can be rewritten as
(4.17)Θk(EGγ)≥‖zk-zEGk+1(γ)-γdk(zk,αk)‖2-γ2‖dk(zk,αk)‖2+2γek(zk,αk)Tdk(zk,αk)+2γ(zk-zEGk+1(γ)-ek(zk,αk))T(ek(zk,αk)-αk∇f(zk))≥Υk(γ)+2γ(zk-zEGk+1(γ)-ek(zk,αk))T(ek(zk,αk)-αk∇f(zk)).
Now we consider the last term in the right-hand side of (4.17). Notice that
(4.18)zk-zEGk+1(γ)-ek(zk,αk)=PΩk[zk-αk∇f(zk)]-zEGk+1(γ).
Setting t:=zk-αk∇f(zk), s:=zEGk+1(γ) and 𝒮~=Ωk in the basic inequality (2.7) of the projection mapping and using the notation of ek(zk,αk), we get
(4.19){PΩk[zk-αk∇f(zk)]-zEGk+1(γ)}T{ek(zk,αk)-αk∇f(zk)}≥0,
and therefore
(4.20){zk-zEGk+1(γ)-ek(zk,αk)}T{ek(zk,αk)-αk∇f(zk)}≥0.
Substituting (4.20) in (4.17), it follows that
(4.21)Θk(EGγ)≥Υk(γ)
and the theorem is proved.
Theorems 4.1 and 4.2 provide the basis of the selection of the optimal step length of the new methods. Note that Υk(γ) is the profit-function since it is a lower-bound of the progress obtained by the new methods (both the FB type HRP method and EG type HRP method). This motivates us to maximize the profit-function Υk(γ) to accelerate convergence of the new methods. Since Υk(γ) a quadratic function of γ, it reaches its maximum at
(4.22)γk*:=ek(zk,αk)Tdk(zk,αk)‖dk(zk,αk)‖2.
Note that under Condition (3.2), using the notation of dk(zk,αk) we have
(4.23)ek(zk,αk)Tdk(zk,αk)=‖ek(zk,αk)‖2-αkek(zk,αk)T(∇f(zk)-∇f(z-k))≥(1-ν)‖ek(zk,αk)‖2.
In addition, since
(4.24)ek(zk,αk)Tdk(zk,αk)=‖ek(zk,αk)‖2-αkek(zk,αk)T(∇f(zk)-∇f(z-k))≥12‖ek(zk,αk)‖2-αkek(zk,αk)T(∇f(zk)-∇f(z-k))+12αk‖∇f(zk)-∇f(z-k)‖2=12∥dk(zk,αk)∥2,
we have
(4.25)γk*:=ek(zk,αk)Tdk(zk,αk)‖dk(zk,αk)‖2≥12.
From numerical point of view, it is necessary to attach a relax factor to the optimal step length γk* obtained theoretically to achieve faster convergence. The following theorem concerns how to choose the relax factor.
Theorem 4.3.
Let z* be an arbitrary point in Ω*, θ a positive constant and γk* defined in (4.22). For given zk∈Ωk, αk is chosen such that Condition (3.2) is satisfied. Whenever the new iterate zk+1(θγk*) is generated by
(4.26)zk+1(θγk*)=PΩk[zk-θγk*dk(zk,αk)]orzk+1(θγk*)=PΩk[zk-θγk*gk(zk,αk)],
we have
(4.27)‖zk+1(θγk*)-z*‖2≤‖zk-z*‖2-θ(2-θ)(1-ν)2‖ek(zk,αk)‖2.
Proof.
From Theorems 4.1 and 4.2 we have
(4.28)‖zk-z*‖2-‖zk+1(θγk*)-z*‖2≥Υk(θγk*).
Using (4.5), (4.23), and (4.25), we obtain
(4.29)Υk(θγk*)=2θγk*ek(zk,αk)Tdk(zk,αk)-(θγk*)2‖dk(zk,αk)‖2=2θγk*ek(zk,αk)Tdk(zk,αk)-θ2γk*ek(zk,αk)Tdk(zk,αk)=γk*θ(2-θ)ek(zk,αk)Tdk(zk,αk)≥θ(2-θ)(1-ν)2‖ek(zk,αk)‖2,
and the assertion is proved.
Theorem 4.3 shows theoretically that any θ∈(0,2) guarantees that the new iterate makes progress to a solution. Therefore, in practical computation, we choose γk=θγk* with θ∈(0,2) as the step length in the new methods. We need to point out that from numerical experiments, θ∈[1,2) is much preferable since it leads to better numerical performance.
5. Convergence
It follows from (4.27) that for both the FB type HRP method (3.6) and the EG type HRP method (3.7), there exists a constant τ>0, such that
(5.1)‖zk+1-z*‖2≤‖zk-z*‖2-τ‖ek(zk,αk)‖2,∀z*∈Ω*.
The convergence result of the proposed methods in this paper is based on the following theorem.
Theorem 5.1.
Let {zk} be a sequence generated by the proposed method (3.6) or (3.7). Then {zk} converges to a point z~, which belongs to Ω*.
Proof.
First, from (5.1) we get
(5.2)limk→∞‖ek(zk,αk)‖=0.
Note that
(5.3)ek(zk,αk)=zk-z-k,(see(3.3)).
We have
(5.4)limk→∞‖zk-z-k‖=0.
Again, it follows from (5.1) that the sequence {zk} is bounded. Let z~ be a cluster point of {zk} and the subsequence {zkj} converges to z~. We are ready to show that z~ is a solution point of (1.1).
First, we show that z~∈Ω. Since z-kj∈Ωkj, then by the definition of Ωkj, we have
(5.5)c(zkj)+(z-kj-zkj)Tξkj≤0,∀j=1,2,….
Passing onto the limit in this inequality and taking into account (5.4) and Lemma 2.7, we obtain that
(5.6)c(z~)≤0.
Hence, we conclude z~∈Ω.
Next, we need to show (z-z~)T∇f(z~)≥0, ∀z∈Ω. To do so, we first prove
(5.7)limj→∞‖ekj(zkj,1)‖=0.
It follows from Remark 3.1 in Section 3 that infj{αkj}≥infk{αk}=αmin>0. Then from Lemma 2.1, we have
(5.8)‖ekj(zkj,1)‖≤‖zkj-z-kj‖min{1,αkj},
which, together with (5.4), implies that
(5.9)limj→∞‖ekj(zkj,1)‖≤limj→∞‖zkj-z-kj‖min{1,αkj}≤limj→∞‖zkj-z-kj‖min{1,αmin}=0.
Setting t=zkj-∇f(zkj), 𝒮~=Ωkj in the inequality (2.7), for any z∈Ω⊆Ωkj, we obtain
(5.10)(zkj-∇f(zkj)-PΩkj(zkj-∇f(zkj)))T(PΩkj(zkj-∇f(zkj))-z)≥0.
From the fact that ekj(zkj,1)=zkj-PΩkj[zkj-∇f(zkj)], we have
(5.11)(ekj(zkj,1)-∇f(zkj))T(zkj-ekj(zkj,1)-z)≥0,∀z∈Ω,
that is,
(5.12)(z-zkj)T∇f(zkj)+ekj(zkj,1)T(zkj-ekj(zkj,1)-z+∇f(zkj))≥0,∀z∈Ω.
Letting j→∞, taking into account (5.7), we deduce
(5.13)(z-z~)T∇f(z~)≥0,∀z∈Ω,
which implies that z~∈Ω*. Then from (5.1), it follows that
(5.14)‖zk+1-z~‖2≤‖zk-z~‖2-τ‖ek(zk,αk)‖2.
Together with the fact that the subsequence {zkj} converges to z~, we can conclude that {zk} converges to z~. The proof is complete.
6. Numerical Results
In this section, we implement the proposed methods to solve some numerical examples arising in [2] and then report the results. To show the superiority of the new methods, we also compare them with the HRP method in [2]. The codes for implementing the proposed methods were written by Matlab 7.9.0 (R2009b) and run on an HP Compaq 6910p Notebook (2.00 GHz of Intel Core 2 Duo CPU and 2.00 GB of RAM). The stopping criterion is ∥ek(zk,αk)∥≤ε.
For the new methods, we take ε=10-10, α0=1, μ=0.3, ν=0.9 and θ=1.8. To compare with the HRP method and the new methods, we list the numbers of iterations, the computation times (CPU(Sec.)) and the approximate solutions in Tables 1, 2, 3, 4, 5, 6, 7, 8, and 9. For the HRP method in [2], we list the original numerical results in [2].
Results for Example 6.1 using the HRP method in [2].
Starting points
Number of iterations
CPU (Sec.)
Approximate solution
(1,2,3,0,0,0)T
43
0.0500
(0.3213,0.2815,0.1425)T
(1,1,1,1,1,1)T
67
0.0910
(0.8577,0.8577,1.3097)T
(1,2,3,4,5,6)T
85
0.1210
(1.1548,0.8518,1.8095)T
Results for Example 6.1 using FB type HRP method.
Starting points
Number of iterations
CPU (Sec.)
Approximate solution
(1,2,3,0,0,0)T
15
<0.0001
(0.7335,0.9309,1.2014)T
(1,1,1,1,1,1)T
0
<0.0001
(1.0000,1.0000,1.0000)T
(1,2,3,4,5,6)T
36
<0.0001
(2.5000,0.9572,1.0466)T
Results for Example 6.1 using EG type HRP method.
Starting points
Number of iterations
CPU (Sec.)
Approximate solution
(1,2,3,0,0,0)T
15
<0.0001
(0.6505,1.0000,1.3744)T
(1,1,1,1,1,1)T
0
<0.0001
(1.0000,1.0000,1.0000)T
(1,2,3,4,5,6)T
38
<0.0001
(2.5000,1.3024,1.3927)T
Results for Example 6.2 using the HRP method in [2].
Starting points
Number of iterations
CPU (Sec.)
Approximate solution
(1,2,3,0,0,0)T
1890
2.7740
(-0.1203,0.0285,0.0582)T
(1,1,1,1,1,1)T
2978
4.2860
(0.8603,-0.1658,-0.5073)T
(1,2,3,4,5,6)T
3317
4.8570
(3.6522,-0.1526,-2.3719)T
Results for Example 6.2 using FB type HRP method.
Starting points
Number of iterations
CPU (Sec.)
Approximate solution
(1,2,3,0,0,0)T
609
0.0620
(-1.2024,0.0724,0.5986)T
(1,1,1,1,1,1)T
630
0.0470
(-1.2039,0.0723,0.5993)T
(1,2,3,4,5,6)T
680
0.0620
(-1.1284,0.0759,0.5613)T
Results for Example 6.2 using EG type HRP method.
Starting points
Number of iterations
CPU (Sec.)
Approximate solution
(1,2,3,0,0,0)T
757
0.0620
(-1.1758,0.0737,0.5852)T
(1,1,1,1,1,1)T
567
0.0470
(-1.2161,0.0716,0.6055)T
(1,2,3,4,5,6)T
711
0.0620
(-1.1815,0.0734,0.5881)T
Results for Example 6.3 using the HRP method in [2].
n (dimension)
Number of iterations
CPU (Sec.)
10
36
0.0160
100
38
0.2970
1000
40
15.5000
5000
41
416.7340
Results for Example 6.3 using FB type HRP method.
n (dimension)
Number of iterations
CPU (Sec.)
10
15
<0.0001
100
16
0.0160
1000
17
0.1250
5000
17
0.3130
Results for Example 6.3 using EG type HRP method.
n(dimension)
Number of iterations
CPU (Sec.)
10
15
<0.0001
100
16
0.0160
1000
17
0.1250
5000
17
0.3130
Example 6.1 (a convex feasibility problem).
Let C={x∈R3∣x22+x32-4≤0}, Q={x∈R3∣x3-1-x12≤0}. Find some point x in C∩Q.
Obviously this example can be regarded as an SFP with A=I.
For Example 6.1, it is easy to verify that the point (1,1,1,1,1,1)T is a solution of (1.1). Therefore, the FB type and EG type HRP method only use 0 iteration when we choose the starting point z0=(1,1,1,1,1,1)T. While applying the HRP method in [2] to solve Example 6.1 and choosing the same starting point, the number of iterations is 67. This is the original numerical result listed in Table 1 of [2].
Example 6.2 (a split feasibility problem).
Let A=(2-13425202),C={x∈R3∣x1+x22+2x3≤0}, Q={x∈R3∣x12+x2-x3≤0}. Find some point x∈C with Ax∈Q.
Example 6.3 (a nonlinear programming problem).
Consider the problem
(6.1)Minimizef(z)=∑i=1nzi2subjectto∑i≠jzi2-zj-j≤0,j=1,…,n.
This example is a general nonlinear programming problem not the reformulation (1.1) for the SFP. Notice that it has a unique solution z*=(0,…,0)T and ∇f(z*)=2z*=(0,…,0)T. Then from Remark 3.5 in Section 3, the proposed methods and the HRP method in [2] can be used to find its solution.
The computational preferences to the HRP method (1.5)-(1.6) are revealed clearly in Tables 1–9. The numerical results demonstrate that the selection of optimal step length in both the FB type HRP method and the EG type HRP method reduces considerable computational load of the HRP method in [2].
7. Conclusions
In this paper we consider the split feasibility problem, which is a special case of the multiple-sets split feasibility problem [13–15]. With some new strategies for determining the optimal step length, this paper improves the HRP method in [2] and thus develops modified halfspace-relaxation projection methods for solving the split feasibility problem. Compared to the HRP method in [2], the new methods reduce the number of iterations moderately with little additional computation.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant no. 11001053) and the Research Award Fund for Outstanding Young Teachers in Southeast University.
CensorY.ElfvingT.A multiprojection algorithm using Bregman projections in a product spaceQuB.XiuN.A new halfspace-relaxation projection method for the split feasibility problemFukushimaM.A relaxed projection method for variational inequalitiesQuB.XiuN.A note on the CQ algorithm for the split feasibility problemYangQ.The relaxed CQ algorithm solving the split feasibility problemTsengP.A modified forward-backward splitting method for maximal monotone mappingsKhobotovE. N.A modification of the extragradient method for solving variational inequalities and some optimization problemsKorpelevičG. M.An extragradient method for finding saddle points and for other problemsHeB.YuanX.ZhangJ. J. Z.Comparison of two kinds of prediction-correction methods for monotone variational inequalitiesEavesB. C.On the basic theorem of complementarityRockafellarR. T.PolyakB. T.Minimization of unsmooth functionalsCensorY.ElfvingT.KopfN.BortfeldT.The multiple-sets split feasibility problem and its applications for inverse problemsXuH.-K.A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problemZhangW.HanD.LiZ.A self-adaptive projection method for solving the multiple-sets split feasibility problem