# A Modified Proximal Point Algorithm and Some Convergence Results

Authors: Shengquan Weng; Dingping Wu; Zengfu Chao
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, the convergence to minimizers of a convex function of a modified proximal point algorithm involving a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in CAT(0) spaces is studied and a numerical example is given to support our main results.


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## Body

## 1. Introduction

In daily life, no matter what we do, there are always many options available and many possible outcomes. When we do these things, we always consciously or unconsciously choose an optimal solution in order to achieve the optimal result. The discipline of seeking the best solution to achieve the best result is optimization. The way to find the optimal solution is the optimization method.Given a real numberκ (curvature), let Mκ2 denote the following space: if κ<0, then Mκ2 is a real hyperbolic space H2 with the distance function scaled by a factor of 1/−κ; if κ=0, then Mκ2 is the Euclidean plane; if κ>0, then Mκ2 is the 2-sphere S2 with the metric scaled by a factor 1/κ. Let Dκ denote the diameter of Mκ2. Let Δ be a geodesic triangle in X with a perimeter less than 2Dκ. Let Δ¯∈Mκ2 be a comparison triangle for Δ¯. Then, Δ is said to satisfy the CAT(κ) inequality if for x,y∈Δ and all comparison points x¯,y¯∈Δ¯,(1)dx,y≤dx¯,y¯.Ifκ≤0, then X is called a CAT(κ) space if X is a geodesic space all of whose geodesic triangles satisfy the CAT(κ) inequality. If κ>0, then X is called a CAT(κ) space if X is Dκ-geodesic and all geodesic triangles in X of perimeter less than 2Dκ satisfy the CAT(κ) inequality. Thus, it can be seen that a CAT(0) space is a special case of CAT(κ) spaces when the curvature κ=0. Furthermore, it is possible that the metric on CAT(κ) spaces (κ>0) may take infinite values.A metric space is said to be a geodesic metric space if every two points ofX are joined by a geodesic in this metric space; a geodesic metric space X,d is said to be a CAT(0) space if each geodesic triangle of geodesic metric space X,d is at least as “thin” as its comparison triangle in R2. In addition, a CAT(0) space is said to be a Hadamard space if it is complete; see for more details in [1–8]. Let X be a geodesic metric space and D be a nonempty subset of X. One of the major problems for optimization is to find a point x∈X such that(2)fx=miny∈Xfy,where f is a proper convex lower semicontinuous function and the set of all minimizers of f on X is denoted by argminy∈Xfy. There are many ways to study this problem.For anyλ>0,∀x∈X, the Moreau–Yosida resolvent of f is defined in CAT(0) spaces X as(3)Jλx=argminy∈Xfy+12λd2y,x.In fact, heref is a proper convex and lower semicontinuous function. The concept of Moreau−Yosida resolvent of f appeared in [9]. Furthermore, the fact that the set FJλ of fixed points of the resolvent associated with f coincides with the set argminy∈Xfy of minimizers of f is shown in [10, 11]. Also, for any λ>0, the resolvent Jλ of f is nonexpansive [12].In 1970, Martinet [13] published an article and a new algorithm was proposed to solve this optimization problem; the new algorithm is called as the proximal point algorithm. In 1976, in a Hilbert space, Rockafellar [14] studied the convergence to a solution of the convex minimization problem by the proximal point algorithm and proved and obtained a main conclusion that the sequence xn converges weakly to a minimizer of a convex function f such that ∑n=1∞λn=∞. In 2013, the proximal point algorithm was introduced by Bac̆ák [15] into CAT(0) spaces X,d as follows: x1∈X, for each n∈N,(4)xn+1=argminy∈Xfy+12λnd2y,xn.Here,λn>0, ∀n∈N, and this shows that, if f has a minimizer, ∑n=1∞λn=∞, then the sequence xnΔ-converges to its minimizer. In fact, the proximal point algorithm has been combined with many iterative methods, and a new construction algorithm is further proposed to find approximating fixed points of nonlinear mappings and a proper convex lower semicontinuous function f. In 1953, a known iteration method was proposed by Mann, and it was named as Mann iteration [16]; the Mann iteration process is defined as follows: x1∈D and(5)xn+1=αnxn+1−αnTxn.Here,αn,n∈N, is a real sequence in (0,1). In 1974, another well-known iteration method was proposed by Ishikawa, and this iteration method was named as Ishikawa iteration [17]; the Ishikawa iteration process concretely is expressed as follows: x1∈D and(6)yn=1−αnTxn+αnxn,xn+1=1−βnTyn+βnxn,for each n∈N, where αn,βn is a real sequence in (0,1).In 2017, Suthep Suantai and Withum Phuengtattana [18] proposed a proximal point algorithm for a hybrid pair of nonexpansive single-valued and multivalued mappings in geodesic metric spaces as follows:(7)zn=argminy∈Xfy+12λnd2y,xn,yn=βnzn⊕1−βnwn,wn∈Syn,xn+1=αnxn⊕1−αnTzn,∀n∈N,where αn,βn∈0,1, T is a single-valued nonexpansive mapping, and S is a multi-valued nonexpansive mapping.Motivated and inspired by the above research work, Ishikawa iteration process, multivalued mapping, nonexpansive mapping, and convex function are considered as some elements for a new idea; then, the plan is to make use of these elements to reconstruct an algorithm; thus in this article, a modified proximal point algorithm involving a convex function and two nonexpansive mappings will be proposed. Under some suitable conditions, the convergence of the proposed algorithm is studied and its convergence analysis in the end is given.


## 2. Preliminaries

Ifz,x,andy are three points in CAT0 spaces and if x⊕y/2 is the midpoint of a geodesic segment x,y, then the CAT0 inequality implies(8)d2z,x⊕y2≤12d2z,x+12d2z,y−14d2x,y,which is the CN inequality (see [19]).LetX be a CAT0 space; a subset D∈X is convex, if for any x,y∈D, x,y∈D, where x,y=λx⊕1−λy:0≤λ≤1 and x,y is the unique geodesic joining x and y. Indeed, a geodesic space X,d is said to be a CAT0 space, if and only if the inequality (CN∗ inequality [20]),(9)d21−λx⊕λy,z≤1−λd2x,z+λd2y,z−λ1−λd2x,y,is satisfied for all x,y,z∈X and λ∈0,1. Moreover, if x,y,andz are points in a CAT0 space X,d and λ∈0,1, then(10)d1−λx⊕λy,z≤1−λdx,z+λdy,z.LetCBD and KCD denote the families of nonempty closed bounded subsets and compact convex subsets of D, respectively. The Pompeiu−Hausdorff distance [21] on CBD is defined by(11)HA,B=maxsupx∈Adistx,B,supy∈Bdisty,A,for A,B∈CBD, where distx,D=infdx,y:y∈D is the distance from a point x to a subset D. Let S:D⟶CBD be a multivalued mapping. An element x∈D is said to be a fixed point of mapping S, if an element x∈Sx. The set of fixed points of mapping S is denoted by FS; in brief, FS=x∈D:x∈Sx. More references for a multivalued mapping can be seen in [18, 22, 23].Definition 1 (see [18]).
A single-valued mappingT:D⟶D is nonexpansive if ∀x,y∈D, dTx,Ty≤dx,y.Definition 2 (see [18]).
A multivalued mappingS:D⟶CBD is nonexpansive if ∀x,y∈D, HSx,Sy≤dx,y.Definition 3 (see [18]).
Letxn be a bounded sequence in a CAT0 space X. For any x∈X, we define a mapping r⋅,xn:X⟶0,∞ by rx,xn=limn⟶∞supdx,xn: (i) the asymptotic radius of xn is given by rxn=infrx,xn:x∈X; (ii) the asymptotic center of xn is the set Axn=x∈X:rx,xn=rxn.The asymptotic centerAxn in a complete CAT0 space consists of exactly one point [24].Definition 4 (see [18]).
A sequencexn in a CAT0 space X is said to Δ-converge to x∈X if x is the unique asymptotic center of every subsequence of xn. In this case, we write Δ−limn⟶∞xn=x and call x as the Δ-limit of xn.In fact, givenxn in X such that xnΔ-converges to x and given y∈X with x≠y, limn⟶∞infdxn,x<limn⟶∞infdxn,y. So, the Opial condition in Banach space is also satisfied in every CAT0 space X.Lemma 1 (see [25]).
Every bounded sequence in aCAT0 space has a Δ-convergent subsequence.Lemma 2 (see [26]).
LetD be a nonempty closed convex subset of a CAT0 space X. If xn is a bounded sequence in D, then the asymptotic center of xn is in D.Lemma 3 (see [20]).
Ifxn is a bounded sequence in a complete CAT0 space with Axn=x, un is a subsequence of xn with Aun=u, and the sequence dxn,u converges, then x=u.Lemma 4 (see [20]).
LetD be a nonempty closed convex subset of a complete CAT0 space X and T:D⟶D be a nonexpansive mapping. If xn is a bounded sequence in D such that limn⟶∞dxn,Txn=0 and Δ−limn⟶∞xn=x, then x=Tx.Lemma 5 (see [12]).
LetX,d be a complete CAT0 space and f:X⟶−∞,∞ be a proper convex and lower semicontinuous function. Then, the following identity holds:(12)Jλx=Jμλ−μλJλx⊕μλx,∀x∈X,λ>μ>0,where Jλ is the Moreau−Yosida resolvent of f.Lemma 6 (see [27]).
LetX,d be a complete CAT0 space and f:X⟶−∞,∞ be a proper convex and lower semicontinuous function. Then, for all x,y∈X and λ>0, the following inequality holds:(13)12λd2Jλx,y−12λd2x,y+12λd2x,Jλx+fJλx≤fy,where Jλ is the Moreau−Yosida resolvent of f.Lemma 7 (see [28, 29]).
LetX be a CAT0 space and D be a nonempty closed and convex subset of X. Let xii=1n be any finite subset of D and αi∈0,1,i=1,2,…,n such that ∑i=1nαi=1. Then, the following inequalities hold:(i)
d⊕i=1nαixi,z≤∑i=1nαidxi,z,∀z∈D(ii)
d2⊕i=1nαixi,z≤∑i=1nαid2xi,z−∑i,j=1,i≠jnd2xi,xj,∀z∈D


## 3. Main Results

Theorem 1.
Let (X, d) denote a CAT(0) space and be complete, assuming that the subsetD⊆X is nonempty, closed, and convex. Suppose that S:D⟶CBD is a multivalued nonexpansive mapping, T is a single-valued nonexpansive mapping, and f:D⟶−∞,∞ is a proper convex and lower semicontinuous function. Suppose that the set(14)Ω=FT∩FS∩argminy∈Cfy≠∅and Sp=p for p∈Ω. For x1∈D, let the sequence xn be defined by(15)wn=argminy∈Cfy+12λnd2y,xn,xn+1=αnxn⊕βnpn⊕γnTpn,pn∈Swn,the sequences αn,βn,γn⊆0,1, and 0<a≤αn,βn,γn≤b<1,αn+βn+γn=1n∈N, and λn is a sequence such that λn≥λ>0 for all n∈N and some λ. Then, the following statements hold:(i)
limn⟶∞dxn,p exists for all p∈Ω(ii)
limn⟶∞dxn,wn=0(iii)
limn⟶∞dxn,Txn=0(iv)
limn⟶∞distxn,Sxn=0(v)
limn⟶∞dxn,Jλxn=0Proof.
Letp∈Ω; then, we get that p=Tp∈Sp and fp≤fy for all y∈D. Thus, it shows that(16)fp+12λnd2p,p≤fy+12λnd2y,p,∀y∈D,and hence, p=Jλp for each n∈N.(i)
First of all, the first step is to prove the fact that for allp∈Ω, limn⟶∞dxn,p exists. Since wn=Jλnxn, with the nonexpansiveness of Jλn, then(17)dwn,p=dJλnxn,Jλnp≤dxn,p.Forp∈Ω, by p∈Sp, (17), and Lemma 7, we have(18)dxn+1,p=dαnxn⊕βnpn⊕γnTpn,p≤αndxn,p+βndpn,p+γndTpn,p≤αndxn,p+βnHSwn,Sp+γndTpn,p≤αndxn,p+βndwn,p+γndpn,p≤αndxn,p+βndwn,p+γndwn,p≤αndxn,p+1−αndwn,p≤dxn,p.This shows that the sequencedxn,p is decreasing and bounded. So, the limit limn⟶∞dxn,p exists for all p∈Ω.(ii)
Next, prove thatlimn⟶∞dxn,wn=0. Now, we let(19)limn⟶∞dxn,p=c,wherec is a constant in 0,∞ and p∈Ω. In fact, by the inequality of Lemma 6, it implies that(20)12λnd2wn,p−12λnd2xn,p+12λnd2wn,xn≤fp−fwn.Because offp≤fwn for all n≥1, it follows that(21)d2wn,xn≤d2xn,p−d2wn,p.Sincelimn⟶∞dxn,p=c, in order to show the fact that limn⟶∞dxn,wn=0, it is sufficient to show that(22)limn⟶∞dwn,p=c.From (18), we have(23)dwn,p≥11−αndxn+1,p−αndxn,p.Then, this shows that(24)limn⟶∞infdwn,p≥limn⟶∞inf11−αndxn+1,p−αndxn,p=c.At the same time, from (17), we get that(25)limn⟶∞supdwn,p≤limn⟶∞supdxn,p=c.Thus, from (24) and (25), it is implied that(26)limn⟶∞dwn,p=c.Then, from (19), (21), and (26), it can be shown that(27)limn⟶∞dxn,wn=0.(iii)
Next, prove thatlimn⟶∞dxn,Txn=0. By Lemma 7, we have(28)d2xn+1,p=d2αnxn⊕βnpn⊕γnTpn,p≤αnd2xn,p+βnd2pn,p+γnd2Tpn,p−αnβnd2xn,pn−αnγnd2xn,Tpn−βnγnd2pn,Tpn≤αnd2xn,p+βnH2Swn,Sp+γnd2Tpn,Tp−αnβnd2xn,p−αnγndxn,Tpn−βnγnd2pn,Tpn≤αnd2xn,p+βnd2wn,p+γnd2pn,p−αnβnd2xn,p−αnγndxn,Tpn−βnγnd2pn,Tpn≤d2xn,p−αnβnd2xn,pn−αnγndxn,Tpn−βnγnd2pn,Tpn,that is,(29)αnβnd2xn,pn+αnγndxn,Tpn+βnγnd2pn,Tpn≤d2xn,p−d2xn+1,p.Then from (19) and (29), we get(30)limn⟶∞dxn,pn=0,limn⟶∞dxn,Tpn=0,limn⟶∞dpn,Tpn=0.Therefore, from (30), we can obtain(31)dxn,Txn≤dxn,Tpn+dTpn,Txn≤dxn,Tpn+dpn,xn.From (30) and (31), it can be shown that(32)limn⟶∞dxn,Txn=0.(iv)
Thus,(33)distxn,Sxn≤dxn,pn+distpn,Sxn≤dxn,pn+HSwn,Sxn≤dxn,pn+dwn,xn.From (27), (30), and (33), it is implied that(34)distxn,Sxn=0.(v)
Sinceλn≥λ>0, making use of wn=Jλnxn, the nonexpansiveness of Jλ, and Lemma 5, it can be seen that(35)dxn,Jλxn≤dxn,wn+dwn,Jλxn≤dxn,wn+dJλnxn,Jλxn=dxn,wn+dJλλn−λλnJλnxn⊕λλnxn,Jλxn≤dxn,wn+λn−λλndJλnxn,xn+λλndxn,xn=2−λλndxn,wn.
So, this implies the fact that the limit(36)limn⟶∞dxn,Jλxn=0.
This completes the proof.Theorem 2.
LetD be a nonempty closed convex subset of a complete CAT(0) space (X, d). Let function f:D⟶−∞,∞ be a proper convex and lower semicontinuous function, T be a nonexpansive single-valued mapping, and S:D⟶KCD be a multivalued nonexpansive mapping. Suppose that(37)Ω=FT∩FS∩argminy∈Dfy≠∅and Sp=p for p∈Ω. For x1∈D, let the sequence xn be defined by (15), the sequences αn,βn,γn∈0,1, and 0<a≤αn,βn,γn≤b<1,αn+βn+γn=1n∈N. In addition, λn is a sequence such that λn≥λ>0 for all n∈N and some λ. Then, the sequence xnΔ-converges to a point in Ω.Proof.
LetωΔ=∪Aun, where the union is taken over all subsequences un of xn. Let p∈ωΔxn. Then, there exists a subsequence un of xn such that Aun=p. D∈X is a nonempty closed convex subset, and it is easy to know from Theorem 1 that the sequence xn is bounded. Then, by Lemmas 1 and 2, it can be shown that there exists a subsequence vn of un such that(38)Δ−limn⟶∞vn=v∈D.
From Theorem1 (iii) and (v), we have limn⟶∞dvn,Tvn=0 and limn⟶∞dvn,Jλvn=0. Then, by the nonexpansiveness of T and Jλ, through Lemma 4, it can be shown that the fact that v=Tv=Jλv. Thus, we get(39)v∈FT∩FJλ=FT∩argminu∈Dfu.
Since the mappingS is compact valued, for each n∈N, then there exist rn∈Svn and γn∈Sv such that dvn,rn=distvn,Svn and drn,γn=distrn,Sv. By Theorem 1 (iv), this implies the fact that(40)limn⟶∞dvn,rn=0.
Owing to the factSv is compact, there exists a subsequence γni of γn such that limi⟶∞γni=γ∈Sv. Thus, this shows that(41)limi⟶∞supdvni,γ≤limi⟶∞supdvni,rni+drni,γni+dγni,γ≤limi⟶∞supdvni,rni+distrni,Sv+dγni,γ≤limi⟶∞supdvni,rni+HSrni,Sv+dγni,γ≤limi⟶∞supdvni,rni+drni,v+dγni,γ=limi⟶∞supdvni,v.
Through (38) and the uniqueness of asymptotic centers, we can obtain that v=γ∈Sv. Therefore, by (39), we can show that(42)v∈FT∩FS∩argminu∈Dfu=Ω.
It follows by Lemma3 and Theorem 1 (i) that p=v, and hence, ωΔxn⊆Ω.
In order to show thatxnΔ-converges to a point in Ω, it suffices to show that ωΔxn consists of exactly one point. Suppose that un is a subsequence of xn with Aun=u∗ and Axn=x. Since u∗∈ωΔxn⊆Ω and dxn,u∗ converge, it implies by Lemma 3 that x=u∗.
This completes the proof. □Remark 1.
(i)
The results of Shuntai and Phuaengrattana [18] and Cholamjiak [30] are extended and improved by Theorem 2. In fact, a new proximal point algorithm can be used for solving the constrained convex minimization problem as well as the fixed-point problem of a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in a CAT(0) space.(ii)
Since every real Hilbert space H is a complete CAT(0) space, the above result can also be obtained in Hilbert spaces, so a convergence weakly theorem can be obtained in a real Hilbert space as follows.Corollary 1.
LetD be a nonempty closed convex subset of a real Hilbert space H. Let f:D⟶−∞,∞ be a proper convex and lower semicontinuous function, T be a single-valued nonexpansive mapping, and S:D⟶CBD be a multivalued nonexpansive mapping. Suppose that(43)Ω=FT∩FS∩argminy∈Cfy≠∅and Sp=p,p∈Ω. For x1∈D, let the sequence xn be defined by(44)wn=argminy∈Cfy+12λnd2y,xn,xn+1=αnxn+βnpn+γnTpn,pn∈Swn,the sequences αn,βn,γn⊆0,1, and 0<a≤αn,βn,γn≤b<1,αn+βn+γn=1n∈N, and λn is a sequence such that λn≥λ>0,∀n∈N and some λ. Then, the sequence xn converges weakly to an element in Ω.Remark 2.
In fact, the construction of our proposed algorithm is rather peculiar and it is different from references [31–33]. Convex combination of the sequences xn,pn, and Tpn are given here, especially pn∈Swn, and Tpn is the nonexpansive mapping T that operates on the sequences pn.


## 4. Numerical Experiments

In this section, a numerical example is given to illustrate reckoning the convergence of modified proximal point algorithm with iteration (15) by numerical experiment for supporting Theorems 1 and 2.LetX=ℝ2 with Euclidean norm and D=x=x1,x2∈ℝ2:0≤x1,x2≤1. For each x=x1,x2∈D, the concrete definition of nonexpansive mappings TandS is shown as(45)Tx=2x1+14,x2+78,Sx=x1×3x2+14,1.For eachx∈D, assume that x1=∑i=1nxi, and x2=∑i=1nxi2. The function f:D⟶−∞,∞ is defined in the following manner:(46)fx=x1+12x22+−1.5,−2x+7.From the fact thatTandS are nonexpansive and f is a proper convex lower semicontinuous function easy to prove, we skip their proofs here. Furthermore, by making use of the soft thresholding operator [34] and the proximity operator [35], let λn=1; we have(47)J0x=argminv∈Dfv+12v−x2=proxfx=prox⋅1/2x−−1.5,−22=maxx1+1.5−12,0sgnx1+1.5,maxx2+2−12,0sgnx2+2,where sgnξ is a signum function, that is,(48)sgnξ=1,ξ>0,sgnξ=0,ξ=0,sgnξ=−1,ξ<0.Further simplification of the proposed iterative algorithm is in the following expression:(49)wn=J0xn,xn+1=αnxn+βnpn+γnTpn,pn∈Swn,where xn=xn1,xn2 and wn=wn1,wn2 are points in ℝ2. In addition, we choose some points and the sequences of parameters as follows:(50)pn=wn1,3wn2+14∈Swn,αn=50n−3100n,βn=35n−4100n,γn=15n+7100n.Next, we use Algorithm (49) with an initial point x1=0.1,0.2 and obtain numerical results in Table 1.Table 1
Numerical results of Algorithm (49).
nxn=xn1,xn2xn−xn−12fxn1(0.100000, 0.200000)0.3472056.5258753(0.310620, 0.705335)0.1242696.4007535(0.406401, 0.884487)0.0508386.3798307(0.453275, 0.954060)0.0217176.3759689(0.476569, 0.981610)0.0095876.37520511(0.488219, 0.992611)0.0043596.37504513(0.494066, 0.997023)0.0020346.37501015(0.497008, 0.998799)0.0009706.37500217(0.498490, 0.999514)0.0004716.37500019(0.499237, 0.999803)0.0002316.37500021(0.499614, 0.999920)0.0001156.37500023(0.499805, 0.999967)0.0000576.37500025(0.499901, 0.999986)0.0000286.37500027(0.499950, 0.999994)0.0000146.37500029(0.499974, 0.999997)0.0000076.37500031(0.499987, 0.999999)0.0000036.37500033(0.499993, 0.999999)0.0000016.37500035(0.499996, 0.999999)0.0000006.375000Figure 1
Number of iterations and errors.Remark 3.
(i)
From Figure1, with the increase in the number of iterations, the errors between xn and xn+1 decrease. When the iteration is 17 times, the minimum of f is obtained.(ii)
From Table1 and Figure 1, it is observed that the sequence xn converges to a point (0.5, 1).(iii)
The point (0.5, 1) is a solution of the constrained convex minimization problems (46) and also a solution of the fixed-point problems of a pair of a nonexpansive single-valued mapping T and a nonexpansive multivalued mapping S.



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