# A Relaxed Self-Adaptive Projection Algorithm for Solving the Multiple-Sets Split Equality Problem

Authors: Haitao Che; Haibin Chen
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. Firstly, we transfer the original problem to the constrained multiple-sets split equality problem and a fixed point equation system is established. Then, we show the equivalence of the constrained multiple-sets split equality problem and the fixed point equation system. Secondly, we present a relaxed self-adaptive projection algorithm for the fixed point equation system. The advantage of the self-adaptive step size is that it could be obtained directly from the iterative procedure. Furthermore, we prove the convergence of the proposed algorithm. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.


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## Body

## 1. Introduction

LetH1,H2,andH3 be real Hilbert spaces. For i=1,2,⋯,t,j=1,2,⋯,r, Ci and Qj are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively, and assume that A:H1⟶H3,B:H2⟶H3 are two bounded linear operators. The multiple-sets split equality problem (MSSEP) is to find x and y satisfying the property
(1)x∈C=⋂i=1tCi,y∈Q=⋂j=1rQjsuchthatAx=By.WhenB=I, MSSEP (1) reduces to the multiple-sets split feasibility problem
(2)findapointx∈C=⋂i=1tCi,Ax∈Q=⋂j=1rQj,which is applied to intensity-modulated radiation therapy [1–11], signal processing [12–21], and image reconstruction [22–38]. Censor et al. [39] proposed the proximity function px to measure the distance of a point to all sets
(3)px=12∑i=1tlix−PCix2+12∑j=1rλjAx−PQjAx2,where li>0 for all i, and λj>0 for all j with ∑i=1tli+∑j=1rλj=1 . To solve (2), they considered the following constrained MSSEP:
(4)findapointx∈Ωsuchthatxsolves2,and then presented the projection method
(5)xk+1=PΩxk−s∇px,where s>0 and Ω is an auxiliary simple nonempty closed convex set with Ω∩S≠∅, and S denotes the solution set of (2). The convergence of the projection method was obtained under some mild conditions.Whent=r=1, MSSEP (1) reduces to the split equality problem which was introduced by Moudafi [40] as follows:
(6)findtwopointsx∈C,y∈QsuchthatAx=By,which is applied to the game theory [41] and optimal control and approximation theory [42]. The following alternating CQ algorithm (ACQ) was introduced by Moudafi [40] as follows:
(7)xk+1=PCxk−γkA∗Axk−Byk,yk+1=PQyk+βkB∗Axk+1−Byk,where γk,βk∈ε,min1/λA,1/λB−ε for small enough ε>0, A∗ and B∗ denote the adjoint of A and B, respectively. λA and λB are the spectral radiuses of A∗A and B∗B, respectively. Since the computation of PC and PQ onto a closed convex subset might be hard to be implemented, Fukushima [43] suggested a way to compute the projection onto a level set of a convex function by considering a sequence of projections onto half-spaces containing the original level set. Then, Moudafi [44] introduced the following relaxed alternating CQ algorithm (RACQ):
(8)xk+1=PCkxk−γkA∗Axk−Byk,yk+1=PQkyk+βkB∗Axk+1−Byk,where Ck and Qk are two sequences of closed convex sets.Recently, Dang et al. [45] gave the following relaxed two-point projection method to solve MSSEP (1):
(9)xk+1=PΩ1xk−γ∑i=1tαixk−PCi,kxk+ATAxk−Byk,yk+1=PΩ2yk−γ∑j=1rβjyk−PQj,kyk−BTAxk+1−Byk,where γ∈0,min{1/21/2,1/4A21/4A2,1/4B2, Ci,k,i=1,2,⋯,r and Qj,k,j=1,2,⋯,t are two sequences of closed convex sets corresponding to Ci and Qj, respectively. Ω1⊂H1 and Ω2⊂H2 are auxiliary simple sets. αi>0 for all i, and βj>0 for all j with ∑i=1tαi+∑j=1rβj=1. Under some mild conditions, the weak convergence of the algorithm (9) was obtained.Noting that the determination of the stepsizeγ of algorithm (9) depends on the operator (matrix) norms ∥A∥ and ∥B∥. This implies that if we implement the relaxed two-point projection method (9), one first need to calculate operator norms of A and B, which is in general not an easy work in practice. To overcome this weakness, Lopez et al. [46] and Zhao and Yang [47] introduced self-adaptive methods of which the advantage of the methods is that the stepsizes do not need prior knowledge of the operator norms. Motivated by them, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. First, we transfer the origin problem to the constrained multiple-sets split equality problem and establish the fixed point equation system. We show the equivalence of the constrained multiple-sets split equality problem and the fixed point equation system. Second, based on the fixed point equation system, we present a relaxed self-adaptive projection algorithm for solving the constrained multiple-sets split equality problem, and the convergence of the proposed algorithm is obtained. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.The remainder of this paper is organized as follows. Section2 shows some preliminaries and notations used for subsequent analysis. In Section 3, we transfer the origin problem to the constrained multiple-sets split equality problem and establish the fixed point equation system and propose a relaxed self-adaptive projection algorithm for solving the constrained multiple-sets split equality problem. The convergence of the proposed algorithm is obtained. In Section 4, several numerical results are shown to confirm the effectiveness of our algorithm.


## 2. Preliminaries

Throughout this paper, we use⟶ and ⇀ to denote the strong convergence and weak convergence, respectively. We write ωwxk=x:∃xkj⇀x to indicate the weak ω-limit set of xk. For any x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(10)∥x−PCx∥≤∥x−y∥,∀y∈C.It is well known thatPC is nonexpansive and firmly nonexpansive. Moreover, PC has the following well-known properties (see for example [48]).Lemma 1.
Let C⊂H be nonempty, closed and convex. Then for all x,y∈H and z∈C,
(i)
x−PCx,z−PCx≤0
(ii)
∥PCx−PCy∥2≤PCx−PCy,x−y;
(iii)
∥PCx−z∥2≤∥x−z∥2−∥PCx−x∥2.Definition 2.
Letf:H⟶Rbe convex. The subdifferential of f at xis defined as
(11)∂fx=ξ∈H∣fy≥fx+ξ,y−x,∀y∈H.
An element of∂fx is said to be a subgradient.Lemma 3.
Suppose f:H⟶R is a convex function, then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded set of H.


## 3. Algorithm and Its Convergence

In this section, we focus on a relaxed self-adaptive projection algorithm and obtain the convergence of the proposed algorithm. Following the idea of Censor et al. [39], we give two additional closed convex sets Ω1⊂H1 and Ω2⊂H2 and consider the constrained multiple-sets split equality problem
(12)findx∈Ω1,y∈Ω2suchthatx,ysolves1,where the sets Ci and Qj can be expressed by
(13)Ci=x∈H1∣cix≤0,Qj=y∈H2∣qjyk≤0,ci:H1⟶R and qj:H2⟶R are convex functions for all i=1,2,⋯,t and j=1,2,⋯,r, and Γ denotes the solution set of (32). Define
(14)Ci,k=x∈H1∣cixk+ξi,k,x−xk≤0,where ξi,k∈∂cixk and
(15)Qj,k=y∈H2∣qjyk+ηj,k,y−yk≤0,where ηj,k∈∂qjyk. It is easily seen that Ci⊂Ci,k and Qj⊂Qj,k for all k. Notice that Ci,k and Qj,k are half-spaces and thus the corresponding projections have closed-form expressions. Hence, we focus on the following multiple-sets split equality problem (CMSSEP):
(16)findx∈Ω1,y∈Ω2tosolvex∈C=⋂i=1tCi,k,y∈Q=⋂j=1rQj,ksuchthatAx=By.Now, we define the proximity functionpkx,y:
(17)pkx,y=12∑i=1tαix−PCi,kx2+12∑j=1rβjy−PQj,ky2+12Ax−By2,where αi>0 for all i, and βj>0 for all j with ∑i=1tαi+∑j=1rβj=1.Using the proximity functionpkx,y, we can obtain the following technical lemmas.Lemma 4.
Assume that (16) is consistent (i.e.,(16) has a solution) and denotes its solution set by Γ. If x,y∈Γ, then it solves the fixed point equation system
(18)x=PΩ1x−λ∑i=1tαix−PCi,kx+ATAx−By,y=PΩ2y−β∑j=1rβjy−PQj,ky−BTAx−By.Proof.
To solve the problem (16), we consider the minimization problem
(19)minpkx,y∣x∈Ω1,y∈Ω2.
(19) leads to the following unconstrained optimization problem:
(20)minx∈Ω1,y∈Ω2δΩ1x+δΩ2y+pkx,y,where δΩi is a indicator function of Ωi for i=1,2 defined by
(21)δΩix=0,x∈Ωi,+∞,otherwise.Note that∂δΩ1x=NΩ1x and ∂δΩ2y=NΩ2y, where NΩ1 and NΩ2 are the normal cone of the convex sets Ω1 and Ω1, respectively. From the optimality conditions of (20), it yields
(22)0∈∑i=1tαix−PCi,kx+ATAx−By+∂δΩ1x,0∈∑j=1rβjy−PQj,ky−BTAx−By+∂δΩ2y,which means that, for λ>0,β>0,
(23)x−λ∑i=1tαix−PCi,kx+ATAx−By=x+λ∂δΩ1x,y−β∑j=1rβjy−PQj,ky−BTAx−By=y+β∂δΩ2y,that is,
(24)x=I+λNΩ1−1x−λ∑i=1tαix−PCi,kx+ATAx−By,y=I+βNΩ2−1y−β∑j=1rβjy−PQj,ky−BTAx−By.SinceI+λNΩ1−1=PΩ1 and I+βNΩ2−1=PΩ2, we obtain
(25)x=PΩ1x−λ∑i=1tαix−PCi,kx+ATAx−By,y=PΩ2y−β∑j=1rβjy−PQj,ky−BTAx−By.Thus, the desired result can be obtained.The following lemma reveals that ESEP (16) is equivalent to the fixed point equation system (18).Lemma 5.
Assume that the problem (16) is consistent. x∗,y∗∈Γ solves ESEP (2) if and only if x∗,y∗ solves the fixed point equation system (18).Proof.
From Lemma4, we reveal that x∗,y∗ can solve (16); it also can solve (18). Next, we will prove that x∗,y∗ can solve (18), it also can solve (16). Obviously, one has x∗∈Ω1, and y∗∈Ω2. It follows from the proposition of projection that
(26)x∗−λ∑i=1tαix∗−PCi,kx∗+ATAx∗−By∗−x∗,u−x∗≤0,u∈Γ,y∗−β∑j=1rβjy∗−PQj,ky∗−BTAx∗−By∗−y∗,v−y∗≤0,v∈Γ.which means
(27)−λ∑i=1tαix∗−PCi,kx∗+ATAx∗−By∗,u−x∗≤0,u∈Γ,−β∑j=1rβjy∗−PQj,ky∗−BTAx∗−By∗,v−y∗≤0,v∈Γ.Hence, from Lemma3, we add two inequalities to obtain
(28)∑i=1tαix∗−PCi,kx∗2+∑j=1rβjy∗−PQj,ky∗2+Ax∗−By∗,Bv−Au+Ax∗−By∗≤0.Furthermore, fromAu=Bv, we deduce
(29)∥x∗−PCi,kx∗∥=0,fori=1,2,⋯,t,y∗−PQj,ky∗2=0,forj=1,2,⋯,r,Ax∗−By∗2=0.Thus,x∗,y∗ solves ESEP (16). This completes the proof.Based on (18), we can introduce a relaxed self-adaptive projection algorithm to solve (16), with σk∈0,1.Alggorithm 6.
Letx0∈H1,y0∈H2 be arbitrary. We calculate the k+1th iterate via the following formula
(30)uk=PΩ1xk−λk∑i=1tαixk−PCi,kxk+ATAxk−Byk,xk+1=γkxk+1−γkuk,vk=PΩ2yk−λk∑j=1rβjyk−PQj,kyk−BTAxk−Byk,yk+1=γkyk+1−γkvk,where the stepsize λk is chosen by
(31)λk=2σk∑i=1tαi∥xk−PCi,kxk∥2+∑j=1rβj∥yk−PQj,kyk∥2+∥Axk−Byk∥2∥∑i=1tαixk−PCi,kxk+ATAxk−Byk∥2+∥∑j=1rβjyk−PQj,kyk−BTAxk−Byk∥2=4σkpkxk,yk∥∇pkxk,yk∥2, with σk∈(1, 0).Next, we will focus on the convergence analysis of Algorithm6.Theorem 7.
Assume limk→∞γk=0, ∑k=1∞γk=∞ and σk∈M1,M2⊂0,1, then the sequence xk,yk generated by Algorithm 6 converges to a solution of (1).Proof.
Takingx∗,y∗∈Γ, one has
(32)Ax∗=By∗.From (30) and the fact that the projection is nonexpansive, we have
(33)uk−x∗2=PΩ1xk−λk∑i=1tαixk−PCi,kxk+ATAxk−Byk−x∗2≤xk−λk∑i=1tαixk−PCi,kxk+ATAxk−Byk−x∗2=xk−x∗∥2+λk2∥∑i=1tαixk−PCi,kxk+ATAxk−Byk2−2λk∑i=1tαixk−PCi,kxk+ATAxk−Byk,xk−x∗.Since(34)−2λk∑i=1tαixk−PCi,kxk+ATAxk−Byk,xk−x∗=−2λk∑i=1tαixk−PCi,kxk,xk−x∗−2λkATAxk−Byk,xk−x∗=−2λk∑i=1tαixk−PCi,kxk,xk−x∗−2λkAxk−Byk,Axk−Ax∗≤−2λk∑i=1tαi∥xk−PCi,kxk∥2−λk∥Axk−Byk∥2−λk∥Axk−Ax∗∥2+λk∥Byk−Ax∗∥2,together with (33), we deduce
(35)∥uk−x∗∥2≤∥xk−λk∑i=1tαixk−PCi,kxk+ATAxk−Byk−x∗∥2=∥xk−x∗∥2+λk2∥∑i=1tαixk−PCi,kxk+ATAxk−Byk∥2−2λk∑i=1tαi∥xk−PCi,kxk∥2−λk∥Axk−Byk∥2−λk∥Axk−Ax∗∥2+λk∥Byk−Ax∗∥2.Similarly, we have(36)∥vk−y∗∥2=∥PΩ2yk−λk∑j=1rβjyk−PQj,kyk−BTAxk−Byk−y∗∥2≤∥yk−λk∑j=1rβjyk−PQj,kyk−BTAxk−Byk−y∗∥2≤∥yk−y∗∥2+λk2∥∑j=1rβjyk−PQj,kyk−BTAxk−Byk∥2−2λk∑i=1tβj∥yk−PQj,kyk∥2−λk∥Byk−By∗∥2−λk∥Axk−Byk∥2+λk∥Axk−By∗∥2.From (35) and (36), it follows
(37)∥uk−x∗∥2+∥vk−y∗∥2≤∥xk−x∗∥2+∥yk−y∗∥2−λk2∑i=1tαi∥xk−PCi,kxk∥2+∑i=1tβj∥yk−PQj,kyk∥2+∥Axk−Byk∥2−λk∥∑i=1tαixk−PCi,kxk+ATAxk−Byk∥2+∥∑j=1rβjyk−PQj,kyk−BTAxk−Byk∥2,which together with (31) means
(38)∥uk−x∗∥2+∥vk−y∗∥2≤∥xk−x∗∥2+∥yk−y∗∥2.Furthermore, it follows from (31) and (38) that
(39)∥xk+1−x∗∥2+∥yk+1−y∗∥2=∥γkxk+1−γkuk−x∗∥2+∥γkyk+1−γkvk−y∗∥2≤γk∥xk−x∗∥2+1−γk∥uk−x∗∥2+γk∥yk−y∗∥2+1−λk∥vk−y∗∥2≤γk∥xk−x∗∥2+∥yk−y∗∥2+1−γk∥uk−x∗∥2+∥vk−y∗∥2≤∥xk−x∗∥2+∥yk−y∗∥2,By induction, one has(40)∥xk+1−x∗∥2+∥yk+1−y∗∥2≤∥x0−x∗∥2+∥y0−y∗∥2.Hence,xn and yn are bounded. Following (31), (36), and (39), we have
(41)∥xk+1−x∗∥2+∥yk+1−y∗∥2≤γk∥xk−x∗∥2+∥yk−y∗∥2+1−γk∥uk−x∗∥2+∥vk−y∗∥2≤∥xk−x∗∥2+∥yk−y∗∥2−1−γkλk2∑i=1tαi∥xk−PCi,kxk∥2+∑i=1tβj∥yk−PQj,kyk∥2+∥Axk−Byk∥2−λk∥∑i=1tαixk−PCi,kxk+ATAxk−Byk∥2+∥∑j=1rβjyk−PQj,kyk−BTAxk−Byk∥2.Without loss of generality, we can assume that there isσ>0 such that 41−γkσk1−σk>σ for all k. Setting sk=∥xk−x∗∥2+∥yk−y∗∥2, together with (41), we have the following inequality
(42)σpkxk,yk2∥∇pkxk,yk∥2+sk+1−sk≤0.Sincesk is eventually decreasing, we obtain sk as convergent. From (42), we have limk→∞pkxk,yk=0. Furthermore,
(43)limk→∞∥xk−PCi,kxk∥2=0,fori=1,2,⋯,t,(44)limk→∞∥yk−PQj,kyk∥2=0,forj=1,2,⋯,r,(45)limk→∞∥Axk−Byk∥2=0.Furthermore,(46)∥xk+1−xk∥=∥γkxk+1−γkuk−xk∥=1−γk∥uk−xk∥≤1−γkλk∑i=1tαi∥xk−PCi,kxk∥+∥ATAxk−Byk∥,which with (41), (45), and the assumption on γk means
(47)limk→∞∥xk+1−xk∥2=0.Note that(48)∥xk+1−uk∥=∥γkxk+1−γkuk−uk∥=γk∥xk−uk∥,we have
(49)limk→∞∥xk+1−uk∥2=0.(47) and (49) imply
(50)limk→∞∥xk−uk∥2=0.Similarly, we have(51)limk→∞∥yk+1−yk∥2=0,limk→∞∥yk+1−vk∥2=0,limk→∞∥yk−vk∥2=0.Thus,xk and yk are asymptotically regular. Notice that
(52)∑i=1tαixk−PCi,kxk+ATAxk−Byk∥2+∥∑j=1rβjyk−PQj,kyk−BTAxk−Byk2≤2∑i=1tαi∥xk−PCi,kxk∥2+∥A∥2∥Axk−Byk∥2+∑j=1rβj∥yk−PQj,kyk∥2+∥B∥2∥Axk−Byk∥2≤2max1,∥A∥2+∥B∥2∑i=1tαi∥xk−PCi,kxk∥2+∥Axk−Byk∥2+∑j=1rβj∥yk−PQj,kyk∥2,which implies that
(53)λk≥σk1max1,∥A∥2+∥B∥2.Moreover, it follows from (22) that
(54)∥xk+1−xkλk∥=1−γk1λk∥uk−xk∥≤1−γk∑i=1tαi∥xk−PCi,kxk∥+∥ATAxk−Byk∥,which with (43), (45), and the assumption on γk yields
(55)limk→∞∥xk+1−xkλk∥=0.Similarly, one has(56)limk→∞∥yk+1−ykλk∥=0.Letx¯ and y¯ be, respectively, weak cluster points of the sequences xk and yk, then there exist two subsequences of xk and yk (again labeled xk and yk which converge weakly to x¯ and y¯). Next, we will show that x¯,y¯∈Γ. It follows from (30) that
(57)xk+1−xkλk1−γk−λk∑i=1tαixk−PCi,kxk+ATAxk−Byk∈NΩ1xk+1−γkxk1−γk,yk+1−ykλk1−γk−λk∑j=1rβjyk−PQj,kyk−BTAxk−Byk∈NΩ2yk+1−γkyk1−γk.From the graphs of the maximal monotone operators,NCandNQ are weakly-strongly closed, and by passing to the limit in the last inclusions, we obtain that x¯∈Ω1 and y¯∈Ω2.On the other hand, from Lemma1 and the definition of Ci,k, one has
(58)cixk≤ξi,k,xk−PCi,kxk≤∥ξi,k∥∥xk−PCi,kxk∥≤M1∥xk−PCi,kxk∥,where M satisfies ∥ξi,k∥≤M1 for all k. The lower semicontinuity of function cix and (41) assert that
(59)cix¯≤liminfk→∞cixk≤0.Thus,x¯∈Ci for i=1,2,⋯,t. Likewise, we can obtain
(60)qjxk≤ηj,k,yk−PQj,kyk≤∥ηj,k∥∥yk−PQj,kyk∥≤M2∥yk−PQj,kyk∥,where M2 satisfies ∥ηj,k∥≤M2 for all k. The lower semicontinuity of function qjx and (42) lead to
(61)qiy¯≤liminfk→∞qiyk≤0.Thus,y¯∈Qj for j=1,2,⋯,r. Moreover, the weak convergence of Axk−Byk to Ax¯−By¯ and the lower semicontinuity of the squared norm imply
(62)∥Ax¯−By¯∥≤liminfk→∞∥Axk−Byk∥=0,hence, x¯,y¯∈Γ. This completes the proof.


## 4. Numerical Examples

We are in a position to show numerical examples to demonstrate the performance and convergence of Algorithm6. The whole programs are written in MATLAB 7.0. All the numerical results are carried out on a personal Lenovo computer with Intel®Core™ i7-7500 U CPU 2.70 GHz and RAM 4.00 GB. We denote the vector with all elements 1 by e in what follows.Example 8.
Let
(63)A=3−12210303,B=4−4213−1435104
C1=x∈R3∣x1+5x22+4x3≤0,C2=x∈R3∣3x1+10x3≤0,Q1=y∈R4∣2y1−3y2−2y3+4y4≤0,andQ2=y∈R4∣2y12−y2+4y3−3y4≤0. Find x∈C=C1∩C2,y∈Q=Q1∩Q2 such that Ax=By.Example 9.
Let
(64)A=0.26200.02680.25890.56970.50040.04580.35950.82700.2464,B=0.66070.01300.03350.92130.32940.71800.40600.98400.65940.39110.71630.9834
C1=x∈R3∣x14+x22−2x32−1≤0,C2=x∈R3∣2x12+x23−3x32−1≤0,Q1=y∈R4∣2y13−y22+2y33+6y4−2≤0,andQ2=y∈R4∣2y12+3y32+2y42−2≤0. Find x∈C=C1∩C2,y∈Q=Q1∩Q2 such that Ax=By.Example 10.
Let A=aijJ×N and B=bijJ×M. C1=x1∈RN∣∥x1∥≤2,C2=x2∈RN∣−e≤x2≤3e.Q1=y1∈RM∣−2e≤y1≤6e, and Q2=y2∈RM∣∥y2∥≤4, where aij,bij∈0,1 are all generated randomly; J, N and M are positive integers. Find x∈C=C1∩C2,y∈Q=Q1∩Q2 such that Ax=By.In this example, we considerJ=10,N=10,andM=20;J=20,N=30,andM=40;andJ=40,N=50,andM=60 and three initial values:
(i)
Case 1x=onesN,1,y=onesM,1;
(ii)
Case 2x=10∗onesN,1,y=10∗onesM,1;
(iii)
Case 3x=−10∗onesN,1,y=−10∗onesM,1.We takeΩ1=C1,n, Ω2=Q1,n when the algorithm iterates to step n, γk=1/20k,σk=1/4+1/2k,α1=α2=β1=β2=1/4 in Algorithm 6. In the following tables and figures, we denote Algorithm 6 and the algorithm in reference [45] by QSPA and RTPPM, respectively. And we set "n","s" and "x∗," and "y∗" to express the number of iteration, CPU time in seconds, and the final solution, respectively. Init. denote the initial points, and pkx,y≤ε=10−4 is used as the stop criterion. The numerical results can be seen from Tables 1–3 and Figures 1–4. For Figures 3 and 4, take J=20,N=30,andM=40 in Example 10.Table 1
The numerical results of Example8.
Init.
QSPR
RTPPM
x1=0,0,0T
n=14,s=0.000671
n=7684,s=0.198442
y1=1,0,−1,1T
x∗=0.0196,−0.0222,−0.0055T
x∗=0.0342,−0.0734,−0.0153T
y∗=0.0345,0.0119,0.0071,−0.0357T
y∗=0.0509,0.0012,−0.0023,−0.0495T
x1=0,1,1T
n=15,s=0.000636
n=49,s=0.001585
y1=0,0,0,0T
x∗=−0.0750,0.0232,0.0181T
x∗=−0.2420,−0.0354,0.0589T
y∗=−0.0279,0.0205,0.0014,−0.0109T
y∗=−0.0721,0.0426,−0.0254,−0.0566T
x1=1,1,1T
n=258,s=0.008365
n=34587,s=0.816461
y1=1,1,−1,−1T
x∗=−0.1554,−0.0125,−0.0255T
x∗=0.0836,−0.1013,−0.0566T
y∗=−0.0951,0.0345,0.0180,−0.0257T
y∗=0.0496,−0.0180,0.0025,−0.0371TTable 2
The numerical results of Example9.
Init.
QSPR
RTPPM
x1=1,1,1T
n=16,s=0.001268
n=478,s=0.029797
y1=1,1,1,1T
x∗=0.1710,0.1525,0.1824T
x∗=1.2294,0.7761,0.9712T
y∗=0.1110,0.1218,0.1283,0.0100T
y∗=0.6171,0.7022,0.6533,0.1692T
x1=101,1,1T
n=37,s=0.001877
n=1687,s=0.079442
y1=101,1,1,1T
x∗=0.2973,0.3380,0.7617T
x∗=1.5257,0.4196,1.5156T
y∗=0.6560,0.3316,0.2638,−0.1878T
y∗=0.9622,0.8416,0.2119,0.1513T
x1=1001,1,1T
n=63,s=0.003101
n=2651,s=0.110352
y1=1001,1,1,1T
x∗=0.5363,−0.1663,2.0146T
x∗=1.5276,0.3916,1.5165T
y∗=0.9380,0.0704,−0.1862,0.0354T
y∗=0.9713,0.8464,0.1785,0.1454TTable 3
The numerical results of Example10.
QSPR withλn
QSPR with 0.5λn
RTPPM withλn
J
N
M
n
s
n
s
n
s
Case 1
10
10
20
30
0.003122
48
0.004302
808
0.050933
20
30
40
37
0.005304
94
0.011647
1994
0.258627
40
50
60
91
0.013088
188
0.028659
4014
1.286561
Case 2
10
10
20
56
0.011269
125
0.031055
3236
0.201379
20
30
40
107
0.038957
171
0.039266
1762
0.233102
40
50
60
295
0.041663
351
0.100628
5084
1.673274
Case 3
10
10
20
67
0.008644
98
0.016412
812
0.051558
20
30
40
118
0.015576
178
0.028107
1953
0.270206
40
50
60
233
0.058625
302
0.108850
4176
1.360695Figure 1
The iteration number of QSPA and RTPPM.Figure 2
The iteration number of QSPA and RTPPM.Figure 3
The iteration number of QSPA and RTPPM.Figure 4
The iteration number of QSPA and RTPPM.From Tables1–3, we can see that the iterative number and CPU time of Algorithm 6 is less algorithm RTPPM. Figures 1–4 indicate that Algorithm 6 is more stable than RTPPM.Furthermore, for testing the stationary property of iterative number, we carry 500 experiments for the initial point which are presented randomly, such as(65)x1=rand3,1,y1=rand4,1,in Example 9, the results can be found in Figure 1.On the other initial point, such as(66)x1=rand3,1∗10,y1=rand4,1∗10,in Example 9, the results can be found in Figure 2.Similarly, we carry 500 experiments for the initial point which are presented randomly, such as(67)x1=randN,1,y1=randM,1,in Example 10, the results can be found in Figure 3.On the other initial point, such as(68)x1=randN,1∗10,y1=randM,1∗10,in Example 10, the results can be found in Figure 4.



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