Strong convergence of modified inertial Mann algorithms for nonexpansive mappings
SStrong convergence of modified inertial Mann algorithms for nonexpansive mappings
Bing Tan a , Zheng Zhou a , Songxiao Li a, ∗ a Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
Abstract
In this paper, we introduce two new modified inertial Mann Halpern and viscosity algorithms for solving fixed point prob-lems. We establish strong convergence theorems under some suitable conditions. Finally, our algorithms are applied to splitfeasibility problem, convex feasibility problem and location theory. The algorithms and results presented in this paper cansummarize and improve corresponding results previously known in this area.
Keywords:
Halpern algorithm, Viscosity algorithm, Inertial algorithm, Nonexpansive mapping, Strong convergence
1. Introduction-Preliminaries
Let C be a nonempty closed convex subset of a real Hilbert space H . A mapping T : C → C is said to be nonexpansiveif (cid:107) T x − T y (cid:107) ≤ (cid:107) x − y (cid:107) for all x, y ∈ C . The set of fixed points of a mapping T : C → C is defined by Fix( T ) := { x ∈ C : T x = x } . For any x ∈ H , P C x denotes the metric projection of H onto C , such that P C ( x ):= argmin y ∈ C (cid:107) x − y (cid:107) .In this paper, we consider the following fixed point problem: find x ∗ ∈ C , such that T ( x ∗ ) = x ∗ , where T : C → C is nonexpansive with Fix( T ) (cid:54) = ∅ . Approximation of fixed point problems with nonexpansive mappings has various specificapplications, because many problems can be considered as fixed point problems with nonexpansive mappings. For instance,monotone variational inequalities, convex optimization problems, convex feasibility problems and image restoration problems.It is known that the Picard iteration algorithm may not converge. One way to overcome this difficulty is to use Manns iterationalgorithm that produces a sequence { x n } via the following: x n +1 = ψ n x n + (1 − ψ n ) T x n , n ≥ , (1.1)the iterative sequence { x n } defined by (1.1) converges weakly to a fixed point of T provided that { ψ n } ⊂ (0 , satisfies (cid:80) ∞ n =0 ψ n (1 − ψ n ) = + ∞ .In practical applications, many problems, such as, quantum physics and image reconstruction, are in infinite dimensionalspaces. To investigate these problems, norm convergence is usually preferable to the weak convergence. Therefore, modifyingthe Mann iteration algorithm to obtain strong convergence has been studied by many authors, see [1, 2, 3, 4, 5, 6, 7] and thereferences therein. In 2003, Nakajo and Takahashi [1] established strong convergence of the Mann iteration with the aid of ∗ Corresponding author
Email addresses: [email protected] (Bing Tan), [email protected] (Zheng Zhou), [email protected] (Songxiao Li)
Preprint submitted to Applied Mathematics Letters January 22, 2020 a r X i v : . [ m a t h . O C ] J a n rojections. Indeed, they considered the following algorithm: y n = ψ n x n + (1 − ψ n ) T x n ,C n = { u ∈ C : (cid:107) y n − u (cid:107) ≤ (cid:107) x n − u (cid:107)} ,Q n = { u ∈ C : (cid:104) x n − u, x n − x (cid:105) ≤ } ,x n +1 = P C n ∩ Q n x , n ≥ , (1.2)where { ψ n } ⊂ [0 , , T is a nonexpansive mapping on C and P C n ∩ Q n is the metric projection from C onto C n ∩ Q n . Thismethod is now referred as the CQ algorithm. For further research, see [6, 8, 9, 10, 11]. Recently, Kim and Xu [2] proposed thefollowing modified Mann iteration algorithm based on the Halpern iterative algorithm [12] and the Mann iteration algorithm: y n = ψ n x n + (1 − ψ n ) T x n ,x n +1 = ν n u + (1 − ν n ) y n , n ≥ , (1.3)where u ∈ C is an arbitrary (but fixed) element in C . They obtained a strong convergence theorem of iteration algorithm (1.3)as follows: Theorem 1.1.
Let C be a closed convex subset of a uniformly smooth Banach space X and let T : C → C be a nonexpansivemapping with Fix( T ) (cid:54) = ∅ . Given a point u ∈ C and given sequences { ψ n } and { ν n } in (0 , , the following conditions aresatisfied:(C1) lim n →∞ ψ n = 0 , (cid:80) ∞ n =0 ψ n = ∞ and (cid:80) ∞ n =0 | ψ n +1 − ψ n | < ∞ ;(C2) lim n →∞ ν n = 0 , (cid:80) ∞ n =0 ν n = ∞ and (cid:80) ∞ n =0 | ν n +1 − ν n | < ∞ .Then the sequence { x n } defined by (1.3) converges strongly to a fixed point of T . Inspired by the result of Kim and Xu [2], Yao, Chen and Yao [5] introduced a new modified Mann iteration algorithm bycombines the viscosity approximation algorithm [13] and the modified Mann iteration algorithm [2]. They established strongconvergence in a uniformly smooth Banach space under some fewer restrictions. It should be noted that there is no additionalprojection involved in [2] and [5]. For further research, see [14, 15, 16, 17, 18].In general, the convergence rate of Mann algorithm is slow. Fast convergence of algorithm is required in many practicalapplications. In particular, an inertial type extrapolation was first proposed by Polyak [19] as an acceleration process. Inrecent years, some authors have constructed different fast iterative algorithms by inertial extrapolation techniques, such as,inertial Mann algorithms [20], inertial forward-backward splitting algorithms [21], inertial extragradient algorithms [22, 23]and fast iterative shrinkage-thresholding algorithm (FISTA) [24]. In 2008, Mainge [20] introduced the following inertial Mannalgorithm by unifying the inertial extrapolation and the Mann algorithm: w n = x n + δ n ( x n − x n − ) ,x n +1 = ψ n w n + (1 − ψ n ) T w n , n ≥ . (1.4)Then the iterative sequence { x n } defined by (1.4) converges weakly to a fixed point of T under some mild assumptions.Inspired and motivated by the works of Kim and Xu [2], Yao, Chen and Yao [5] and Mainge [20], we propose modifiedinertial Mann Halpern algorithm and modified inertial Mann viscosity algorithm, respectively. Strong convergence resultsare obtain under some mild conditions. Finally, our algorithms are applied to split feasibility problems, convex feasibility2roblems and location theory. Our algorithms and results generalize and improve some corresponding previously knownresults.Throughout this paper, we denote the strong and weak convergence of a sequence { x n } to a point x ∈ H by x n → x and x n (cid:42) x , respectively. For each x, y ∈ H , we have the following facts.(1) (cid:107) x + y (cid:107) ≤ (cid:107) x (cid:107) + 2 (cid:104) y, x + y (cid:105) ;(2) (cid:107) tx + (1 − t ) y (cid:107) = t (cid:107) x (cid:107) + (1 − t ) (cid:107) y (cid:107) − t (1 − t ) (cid:107) x − y (cid:107) , ∀ t ∈ R ;(3) (cid:104) P C x − x, P C x − y (cid:105) ≤ , ∀ y ∈ C . Lemma 1.1. [25] Let C be a nonempty closed convex subset of a real Hilbert space H , T : C → H be a nonexpansivemapping. Let { x n } be a sequence in C and x ∈ H such that x n (cid:42) x and T x n − x n → as n → + ∞ . Then x ∈ Fix( T ) . Lemma 1.2. [26] Assume { S n } is a sequence of nonnegative real numbers such that S n +1 ≤ (1 − ν n ) S n + ν n σ n , ∀ n ≥ , and S n +1 ≤ S n − η n + π n , ∀ n ≥ , where { ν n } is a sequence in (0 , , { η n } is a sequence of nonnegative real numbers, { σ n } and { π n } are real sequences suchthat (i) (cid:80) ∞ n =0 ν n = ∞ ; (ii) lim n →∞ π n = 0 ; (iii) lim k →∞ η n k = 0 implies lim sup k →∞ σ n k ≤ for any subsequence { η n k } of { η n } . Then lim n →∞ S n = 0 .
2. Modified inertial Mann Halpern and viscosity algorithms
In this section, combining the idea of inertial with the Halpern algorithm and viscosity algorithm, respectively, we intro-duce two modified inertial Mann algorithms and analyzes their convergence.
Theorem 2.1.
Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C → C be a nonexpansivemapping with Fix( T ) (cid:54) = ∅ . Given a point u ∈ C and given two sequences { ψ n } and { ν n } in (0 , , the following conditionsare satisfied:(D1) lim n →∞ ν n = 0 and (cid:80) ∞ n =0 ν n = ∞ ;(D2) lim n →∞ δ n ν n (cid:107) x n − x n − (cid:107) = 0 .Set x − , x ∈ C be arbitarily. Define a sequence { x n } by the following algorithm: w n = x n + δ n ( x n − x n − ) ,y n = ψ n w n + (1 − ψ n ) T w n ,x n +1 = ν n u + (1 − ν n ) y n , n ≥ . (2.1) Then the iterative sequence { x n } defined by (2.1) converges strongly to p = P Fix( T ) u .Proof. First we show that { x n } is bounded. Indeed, taking p ∈ Fix( T ) , we have (cid:107) x n +1 − p (cid:107) ≤ ν n (cid:107) u − p (cid:107) + (1 − ν n ) (cid:107) y n − p (cid:107)≤ ν n (cid:107) u − p (cid:107) + (1 − ν n ) ( ψ n (cid:107) w n − p (cid:107) + (1 − ψ n ) (cid:107) T w n − p (cid:107) ) ≤ (1 − ν n ) (cid:107) x n − p (cid:107) + ν n (cid:107) u − p (cid:107) + (1 − ν n ) δ n (cid:107) x n − x n − (cid:107) . (2.2)3et M := 2 max (cid:110) (cid:107) u − p (cid:107) , sup n ≥ − ν n ) δ n ν n (cid:107) x n − x n − (cid:107) (cid:111) . Then (2.2) reducing to the following: (cid:107) x n +1 − p (cid:107) ≤ (1 − ν n ) (cid:107) x n − p (cid:107) + ν n M ≤ max {(cid:107) x n − p (cid:107) , M } ≤ · · · ≤ max {(cid:107) x − p (cid:107) , M } . (2.3)Combining condition (D2) and (2.3), we obtain that { x n } is bounded. So { w n } and { y n } are also bounded. By the definitionof y n in (2.1), we have (cid:107) y n − p (cid:107) = ψ n (cid:107) w n − p (cid:107) + (1 − ψ n ) (cid:107) T w n − p (cid:13)(cid:13) − ψ n (1 − ψ n ) (cid:13)(cid:13) T w n − w n (cid:107) ≤ (cid:107) w n − p (cid:107) − ψ n (1 − ψ n ) (cid:107) T w n − w n (cid:107) . (2.4)Therefore, from the definition of w n and (2.4), we get (cid:107) x n +1 − p (cid:107) = (cid:107) (1 − ν n )( y n − p ) + ν n ( u − p ) (cid:107)≤ (1 − ν n ) (cid:107) y n − p (cid:107) + 2 ν n (cid:104) u − p, x n +1 − p (cid:105)≤ (1 − ν n ) (cid:107) w n − p (cid:107) − ψ n (1 − ψ n ) (1 − ν n ) (cid:107) T w n − w n (cid:107) + 2 ν n (cid:104) u − p, x n +1 − p (cid:105) = (1 − ν n ) (cid:107) x n − p (cid:107) + δ n (1 − ν n ) (cid:107) x n − x n − (cid:107) + 2 δ n (1 − ν n ) (cid:104) x n − x n − , x n − p (cid:105)− ψ n (1 − ψ n ) (1 − ν n ) (cid:107) T w n − w n (cid:107) + 2 ν n (cid:104) u − p, x n +1 − p (cid:105) . (2.5)For each n ≥ , let S n = (cid:107) x n − p (cid:107) , π n = ν n σ n ,σ n = δ n (1 − ν n ) ν n (cid:107) x n − x n − (cid:107) + 2 δ n (1 − ν n ) ν n (cid:104) x n − x n − , x n − p (cid:105) + 2 (cid:104) u − p, x n +1 − p (cid:105) ,η n = ψ n (1 − ψ n ) (1 − ν n ) (cid:107) T w n − w n (cid:107) . Then (2.5) reduced to the following: S n +1 ≤ (1 − ν n ) S n + ν n σ n , and S n +1 ≤ S n − η n + π n . From conditions (D1) and (D2), we obtain (cid:80) ∞ n =0 ν n = ∞ and lim n →∞ π n = 0 . In order to complete the proof, using Lemma1.2, it remains to show that lim k →∞ η n k = 0 implies lim sup k →∞ σ n k ≤ for any subsequence { η n k } of { η n } . Let { η n k } be a subsequence of { η n } such that lim k →∞ η n k = 0 , which implies that lim k →∞ (cid:107) T w n k − w n k (cid:107) = 0 . From condition (D2),we have (cid:107) w n k − x n k (cid:107) = δ n k (cid:13)(cid:13) x n k − x n k − (cid:13)(cid:13) → . (2.6)Since { x n k } is bounded, there exists a subsequence { x n kj } of { x n k } such that x n kj (cid:42) ¯ x and lim k →∞ sup (cid:104) u − p, x n k − p (cid:105) =lim j →∞ (cid:104) u − p, x n kj − p (cid:105) . By (2.6), we have w n kj (cid:42) ¯ x . Using Lemma 1.1, we get ¯ x ∈ Fix( T ) . Combining the projectionproperty and p = P Fix( T ) u , we obtain lim k →∞ sup (cid:104) u − p, x n k − p (cid:105) = lim j →∞ (cid:104) u − p, x n kj − p (cid:105) = (cid:104) u − p, ¯ x − p (cid:105) ≤ . (2.7)From (2.1), we obtain (cid:107) y n k − w n k (cid:107) = (1 − ψ n k ) (cid:107) T w n k − w n k (cid:107) → . This together with (2.6), we get (cid:107) y n k − x n k (cid:107) ≤(cid:107) y n k − w n k (cid:107) + (cid:107) w n k − x n k (cid:107) → . Further, combining condition (D1), we obtain (cid:13)(cid:13) x n k +1 − x n k (cid:13)(cid:13) ≤ ν n k (cid:107) u − x n k (cid:107) + (1 − ν n k ) (cid:107) y n k − x n k (cid:107) → . (2.8)Combining (2.7) and (2.8), we get that lim sup k →∞ (cid:104) u − p, x n k +1 − p (cid:105) ≤ , this together with condition (D2) implies that lim sup k →∞ σ n k ≤ . From Lemma 1.2 we observe that lim n →∞ S n = 0 and hence x n → p as n → ∞ . This completes theproof. 4 emark 2.1. If f : C → C is a contractive mapping and we replace u by f ( x n ) in (2.1) , we can obtain the following viscosityiteration algorithm, for more details, see [27]. Theorem 2.2.
Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C → C be a nonexpansivemapping with Fix( T ) (cid:54) = ∅ . Let f : C → C be a ρ -contraction with ρ ∈ [0 , , that is (cid:107) f ( x ) − f ( y ) (cid:107) ≤ ρ (cid:107) x − y (cid:107) , ∀ x, y ∈ C .Given two sequences { ψ n } and { ν n } in (0 , , the following conditions are satisfied:(K1) lim n →∞ ν n = 0 and (cid:80) ∞ n =0 ν n = ∞ ;(K2) lim n →∞ δ n ν n (cid:107) x n − x n − (cid:107) = 0 .Set x − , x ∈ C be arbitarily. Define a sequence { x n } by the following algorithm: w n = x n + δ n ( x n − x n − ) ,y n = ψ n w n + (1 − ψ n ) T w n ,x n +1 = ν n f ( x n ) + (1 − ν n ) y n , n ≥ . (2.9) Then the iterative sequence { x n } defined by (2.9) converges strongly to z = P Fix( T ) f ( z ) . Remark 2.2. (i) For special choice, the parameter δ n in the Algorithm (2.1) and the Algorithm (2.9) can be chosen thefollowing: ≤ δ n ≤ ¯ δ n , ¯ δ n = min (cid:110) ξ n (cid:107) x n − x n − (cid:107) , n − n + η − (cid:111) , if x n (cid:54) = x n − , n − n + η − , otherwise , (2.10) for some η ≥ and { ξ n } is a positive sequence such that lim n →∞ ξ n ν n = 0 . This idea derives from the recent inertialextrapolated step introduced in [24, 28].(ii) If δ n = 0 for all n ≥ , in the Algorithm (2.1) and the Algorithm (2.9) , then we obtained the results of proposed by Kimand Xu [2] and Yao, Chen and Yao [5], respectively.
3. Numerical experiments
In this section, we provide some numerical examples to illustrate the computational performance of the proposed algo-rithms. All the programs are performed in MATLAB2018a on a PC Desktop Intel(R) Core(TM) i5-8250U CPU @ 1.60GHz1.800 GHz, RAM 8.00 GB.
Example 3.1.
Let H and H be real Hilbert spaces and T : H → H a bounded linear operator. Let C and Q be nonemptyclosed and convex subsets of H and H , respectively. We consider the following split feasibility problem (in short, SFP):find x ∗ ∈ C such that T x ∗ ∈ Q. (3.1)For any f, g ∈ L ([0 , π ]) , we consider H = H = L ([0 , π ]) with the inner product (cid:104) f, g (cid:105) := (cid:82) π f ( t ) g ( t ) dt and theinduced norm (cid:107) f (cid:107) := (cid:16)(cid:82) π | f ( t ) | dt (cid:17) . Consider the half-space C = (cid:26) x ∈ L ([0 , π ]) | (cid:90) π x ( t ) dt ≤ (cid:27) , and Q = (cid:26) x ∈ L ([0 , π ]) | (cid:90) π | x ( t ) − sin( t ) | dt ≤ (cid:27) . C and Q are nonempty closed and convex subsets of L ([0 , π ]) . Assume that T : L ([0 , π ]) → L ([0 , π ]) is abounded linear operator with its adjoint T , it is defined by ( T x )( t ) := x ( t ) . Then ( T ∗ x ) ( t ) = x ( t ) and (cid:107) T (cid:107) = 1 . Therefore,(3.1) is actually a convex feasibility problem: find x ∗ ∈ C ∩ Q . Moreover, observe that the solution set of (3.1) is nonemptysince x ( t ) = 0 is a solution. For solving the (3.1), Byrne [29] proposed the following algorithm: x n +1 = P C ( x n − λT ∗ ( I − P Q ) T x n ) , where < λ < L with Lipschitz constant L = 1 / (cid:107) T (cid:107) . For the purpose of our numerical computation, we use the followingformula for the projections onto C and Q , respectively, see [25]. P C ( x ) = − a π + x, a > ,x, a ≤ . and P Q ( x ) = sin( · ) + x − sin( · )) √ b , b > ,x, b ≤ , where a = (cid:82) π x ( t ) dt and b = (cid:82) π | x ( t ) − sin( t ) | dt . We consider different initial points x − = x and use the stoppingcriterion E n = 12 (cid:107) P C x n − x n (cid:107) + 12 (cid:107) P Q T x n − T x n (cid:107) < (cid:15). We use the modified Mann Halpern algorithm (MMHA, i.e., MIMHA with δ n = 0 ) [2], the modified inertial Mann Halpernalgorithm (2.1) (MIMHA), the modified Mann viscosity algorithm (MMVA, i.e., MIMVA with δ n = 0 ) [5] and the modifiedinertial Mann viscosity algorithm (2.9) (MIMVA) to solve Example 3.1. In all algorithms, set (cid:15) = 10 − , ψ n = n +1) , ν n = n +1 , λ = 0 . . In MIMHA algorithm and MIMVA algorithm, update δ n by (2.10) with ξ n = n +1) and η = 4 .Set u = 0 . x in the MIMHA algorithm and f ( x ) = 0 . x n in the MIMVA algorithm, respectively. Numerical results arereported in Table 1 and Fig. 1. In Table 1, “Iter.” and “Time(s)” denote the number of iterations and the cpu time in seconds,respectively. Table 1: Computation results for Example 3.1
MMHA MIMHA MMVA MIMVACases Initial points Iter. Time(s) Iter. Time(s) Iter. Time(s) Iter. Time(s)I x = t
58 15.62 56 16.83 14 3.64 8 2.35II x = e t
195 53.20 194 60.40 16 4.34 10 2.90III x = t
47 12.85 43 13.03 13 3.36 8 2.35IV x = 3 sin(2 t )
98 26.61 95 28.54 8 2.08 5 1.47
Example 3.2.
We consider the convex feasibility problem, for any nonempty closed convex set C i ⊂ R N ( i = 0 , , . . . , m ),find x ∗ ∈ C := (cid:84) mi =0 C i , where one assumes that C (cid:54) = ∅ . Define a mapping T : R N → R N by T := P (cid:0) m (cid:80) mi =1 P i (cid:1) , where P i = P C i stands for the metric projection onto C i . Since P i is nonexpansive and hence the mapping T is also nonexpansive.Moreover, we find that Fix( T ) = Fix ( P ) (cid:84) mi =1 Fix ( P i ) = C (cid:84) mi =1 C i = C . In this experiment, we set C i as a closed ballwith center c i ∈ R N and radius r i > . Thus P i can be computed with P i ( x ) := c i + r i (cid:107) c i − x (cid:107) ( x − c i ) if (cid:107) c i − x (cid:107) > r i ,x if (cid:107) c i − x (cid:107) ≤ r i .
10 20 30 40 50 60
Number of iterations -3 -2 -1 (a) Case I Number of iterations -4 -3 -2 -1 (b) Case II Number of iterations -4 -3 -2 -1 (c) Case III Number of iterations -5 -4 -3 -2 -1 (d) Case IVFigure 1: Convergence behavior of iteration error { E n } for Example 3.1 Choose r i = 1( i = 0 , , . . . , m ) , c = [0 , , . . . , , c = [1 , , . . . , , and c = [ − , , . . . , . c i ∈ ( − / √ N , / √ N ) N ( i =3 , . . . , m ) are randomly chosen. From the choice of c , c and r , r , we get that Fix( T ) = { } . Denote E n = (cid:107) x n (cid:107) ∞ theiteration error of the algorithms.We use th CQ algorithm (1.2) (CQ) [1], the inertial Mann algorithm (1.4) (iMann) [20], the modified inertial Mannalgorithm (MIMA) [30], the modified Mann viscosity algorithm (MMVA, i.e., MIMVA with δ n = 0 ) [5] and the modifiedinertial Mann viscosity algorithm (2.9) (MIMVA) to solve Example 3.2. In all algorithms, set N = 30 , m = 30 . Set ψ n = n +1 in the CQ algorithm and δ n = 0 . , ψ n = n +1 in the iMann algorithm, respectively. Set α n = 0 . , λ = 1 , β n = n +1) , µ = 1 and γ n = 0 . in the MIMA algorithm and ξ n = n +1) , η = 4 , ψ n = n +1) , ν n = n +1 and f ( x ) = 0 . x n inthe MIMVA algorithm, respectively. Take maximum iteration of 1000 as a common stopping criterion. The initial values arerandomly generated by the MATLAB function × rand. Numerical results are reported in Fig. 2. Number of iterations -8 -6 -4 -2 (a) Convergence behavior of iteration error { E n } Number of iterations -14 -12 -10 -8 -6 -4 -2 (b) MIMVA algorithm with different η Figure 2: Numerical results for Example 3.2
Example 3.3.
The FermatWeber problem is a famous model in location theory, which described as follows: find x ∈ R n thatsolves min f ( x ) := m (cid:88) i =1 ω i (cid:107) x − a i (cid:107) , (3.2)where ω i > are given weights and a i ∈ R n are anchor points. We know that the objective function f in (3.2) is convexand coercive and hence the problem has a nonempty solution set. It should be noted that f is not differentiable at the anchorpoints. The most famous method to solve the problem (3.2) is Weiszfeld algorithm, see [31] for more discussion. Weiszfeld7roposed the following fixed point algorithm: x n +1 = T ( x n ) , n ∈ N . The mapping T : R n \ A (cid:55)−→ R n is defined by T ( x ) := (cid:80) mi =1 ωi (cid:107) x − ai (cid:107) (cid:80) mi =1 ω i a i (cid:107) x − a i (cid:107) , where A = { a , a , . . . , a m } . We consider a small example with n = 3 , m = 8 anchorpoints, A = , and ω i = 1 for all i . From the special selection of anchor points a i ( i = 1 , , , · · · , , we know that the optimal value of(3.2) is x ∗ = (5 , , (cid:62) . Fig. 3 shows a schematic diagram of the anchor points and the optimal solution. Figure 3: Schematic diagram of anchor points and optimal solution for Example 3.3
We use the modified inertial Mann Halpern algorithm (2.1) (MIMHA) and the modified inertial Mann viscosity algorithm(2.9) (MIMVA) to solve Example 3.3. In MIMHA algorithm and MIMVA algorithm, set ξ n = n +1) , η = 4 , ψ n = n +1) , ν n = n +1 . Set u = 0 . x in the MIMHA algorithm and f ( x ) = 0 . x n in the MIMVA algorithm, respectively. Take E n = (cid:107) x n − x ∗ (cid:107) as iteration error of the algorithms and maximum iteration of 1000 as a common stopping criterion. Theinitial values are randomly generated by the MATLAB function × rand. Numerical results are reported in Fig. 4. (a) Convergence process of iterative sequence { x n } Number of iterations -3 -2 -1 (b) Convergence behavior of iteration error { E n } Figure 4: Convergence behavior of { x n } and { E n } for Example 3.3 Remark 3.1. (i) From Example 3.1–Example 3.3, we observe that Algorithm (2.9) is efficient, easy to implement, and most mportantly very fast. In addition, the inertial parameter (2.10) can significantly improve the convergence speed, seeFig. 2(b).(ii) The Algorithm (2.9) proposed in this paper can improve some known results in the field, see Fig. 2(a). It should be notedthat the choice of initial values does not affect the calculation performance of the algorithm, see Table 1.
4. Conclusions
This paper discussed the modified inertial Mann Halpern and viscosity algorithms. Strong convergence results are obtainedunder some suitable conditions. Finally, our proposed algorithms are applied to split feasibility problem, convex feasibilityproblem and location theory. Note that the algorithms and results presented in this paper can summarize and improve someknown results in the area.
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