A Modified Proximal Point Algorithm and Some Convergence Results

Weng, Shengquan; Wu, Dingping; Chao, Zengfu

doi:https://doi.org/10.1155/2021/6951062

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6951062 | https://doi.org/10.1155/2021/6951062

A Modified Proximal Point Algorithm and Some Convergence Results

Shengquan Weng,¹Dingping Wu,²and Zengfu Chao¹

Academic Editor: Jen-Chih Yao

Received20 Sept 2021

Revised17 Oct 2021

Accepted21 Oct 2021

Published15 Nov 2021

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.

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 number (curvature), let denote the following space: if , then is a real hyperbolic space with the distance function scaled by a factor of ; if , then is the Euclidean plane; if , then is the 2-sphere with the metric scaled by a factor . Let denote the diameter of . Let be a geodesic triangle in with a perimeter less than . Let be a comparison triangle for . Then, is said to satisfy the CAT() inequality if for and all comparison points ,

If , then is called a CAT() space if is a geodesic space all of whose geodesic triangles satisfy the CAT() inequality. If , then is called a CAT() space if is -geodesic and all geodesic triangles in of perimeter less than satisfy the CAT() inequality. Thus, it can be seen that a CAT(0) space is a special case of CAT() spaces when the curvature . Furthermore, it is possible that the metric on CAT() spaces () may take infinite values.

A metric space is said to be a geodesic metric space if every two points of are joined by a geodesic in this metric space; a geodesic metric space is said to be a CAT(0) space if each geodesic triangle of geodesic metric space is at least as “thin” as its comparison triangle in . 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 be a geodesic metric space and be a nonempty subset of . One of the major problems for optimization is to find a point such thatwhere is a proper convex lower semicontinuous function and the set of all minimizers of on is denoted by . There are many ways to study this problem.

For any , the Moreau–Yosida resolvent of is defined in CAT(0) spaces as

In fact, here is a proper convex and lower semicontinuous function. The concept of resolvent of appeared in [9]. Furthermore, the fact that the set of fixed points of the resolvent associated with coincides with the set of minimizers of is shown in [10, 11]. Also, for any , the resolvent of 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 converges weakly to a minimizer of a convex function such that . In 2013, the proximal point algorithm was introduced by Bac̆ák [15] into CAT(0) spaces as follows: , for each ,

Here, , , and this shows that, if has a minimizer, , then the sequence -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 . 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: and

Here, , 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: andfor each , where 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:where , is a single-valued nonexpansive mapping, and 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

If are three points in spaces and if is the midpoint of a geodesic segment , then the inequality implieswhich is the inequality (see [19]).

Let be a space; a subset is convex, if for any , , where and is the unique geodesic joining and . Indeed, a geodesic space is said to be a space, if and only if the inequality ( inequality [20]),is satisfied for all and . Moreover, if are points in a space and , then

Let and denote the families of nonempty closed bounded subsets and compact convex subsets of , respectively. The distance [21] on is defined byfor , where is the distance from a point to a subset . Let be a multivalued mapping. An element is said to be a fixed point of mapping , if an element . The set of fixed points of mapping is denoted by ; in brief, . More references for a multivalued mapping can be seen in [18, 22, 23].

Definition 1 (see [18]). A single-valued mapping is if , .

Definition 2 (see [18]). A multivalued mapping is if , .

Definition 3 (see [18]). Let be a bounded sequence in a space . For any , we define a mapping by : (i) the of is given by ; (ii) the of is the set .

The asymptotic center in a complete space consists of exactly one point [24].

Definition 4 (see [18]). A sequence in a space is said to -converge to if is the unique asymptotic center of every subsequence of . In this case, we write and call as the -limit of .

In fact, given in such that -converges to and given with , . So, the condition in Banach space is also satisfied in every space .

Lemma 1 (see [25]). Every bounded sequence in a space has a -convergent subsequence.

Lemma 2 (see [26]). Let be a nonempty closed convex subset of a space . If is a bounded sequence in , then the asymptotic center of is in .

Lemma 3 (see [20]). If is a bounded sequence in a complete space with , is a subsequence of with , and the sequence converges, then .

Lemma 4 (see [20]). Let be a nonempty closed convex subset of a complete space and be a nonexpansive mapping. If is a bounded sequence in such that and , then .

Lemma 5 (see [12]). Let be a complete space and be a proper convex and lower semicontinuous function. Then, the following identity holds:where is the resolvent of .

Lemma 6 (see [27]). Let be a complete space and be a proper convex and lower semicontinuous function. Then, for all and , the following inequality holds:where is the resolvent of .

Lemma 7 (see [28, 29]). Let be a space and be a nonempty closed and convex subset of . Let be any finite subset of and such that . Then, the following inequalities hold:(i)(ii)

3. Main Results

Theorem 1. Let (X, d) denote a CAT(0) space and be complete, assuming that the subset is nonempty, closed, and convex. Suppose that is a multivalued nonexpansive mapping, is a single-valued nonexpansive mapping, and is a proper convex and lower semicontinuous function. Suppose that the setand for . For , let the sequence be defined bythe sequences , and , and is a sequence such that for all and some . Then, the following statements hold:(i) exists for all (ii)(iii)(iv)(v)

Proof. Let ; then, we get that and for all . Thus, it shows thatand hence, for each .(i)First of all, the first step is to prove the fact that for all , exists. Since , with the nonexpansiveness of , then For , by , (17), and Lemma 7, we have This shows that the sequence is decreasing and bounded. So, the limit exists for all .(ii)Next, prove that . Now, we let where is a constant in and . In fact, by the inequality of Lemma 6, it implies that Because of for all , it follows that Since , in order to show the fact that , it is sufficient to show that From (18), we have Then, this shows that At the same time, from (17), we get that Thus, from (24) and (25), it is implied that Then, from (19), (21), and (26), it can be shown that(iii)Next, prove that . By Lemma 7, we have that is, Then from (19) and (29), we get Therefore, from (30), we can obtain From (30) and (31), it can be shown that(iv)Thus, From (27), (30), and (33), it is implied that(v)Since , making use of , the nonexpansiveness of , and Lemma 5, it can be seen thatSo, this implies the fact that the limitThis completes the proof.

Theorem 2. Let D be a nonempty closed convex subset of a complete CAT(0) space (X, d). Let function be a proper convex and lower semicontinuous function, be a nonexpansive single-valued mapping, and be a multivalued nonexpansive mapping. Suppose thatand for . For , let the sequence be defined by (15), the sequences , and . In addition, is a sequence such that for all and some . Then, the sequence -converges to a point in .

Proof. Let , where the union is taken over all subsequences of . Let . Then, there exists a subsequence of such that . is a nonempty closed convex subset, and it is easy to know from Theorem 1 that the sequence is bounded. Then, by Lemmas 1 and 2, it can be shown that there exists a subsequence of such thatFrom Theorem 1 (iii) and (v), we have and . Then, by the nonexpansiveness of and , through Lemma 4, it can be shown that the fact that . Thus, we getSince the mapping is compact valued, for each , then there exist and such that and . By Theorem 1 (iv), this implies the fact thatOwing to the fact is compact, there exists a subsequence of such that . Thus, this shows thatThrough (38) and the uniqueness of asymptotic centers, we can obtain that . Therefore, by (39), we can show thatIt follows by Lemma 3 and Theorem 1 (i) that , and hence, .
In order to show that -converges to a point in , it suffices to show that consists of exactly one point. Suppose that is a subsequence of with and . Since and converge, it implies by Lemma 3 that .
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. Let D be a nonempty closed convex subset of a real Hilbert space H. Let be a proper convex and lower semicontinuous function, be a single-valued nonexpansive mapping, and be a multivalued nonexpansive mapping. Suppose thatand . For , let the sequence be defined bythe sequences , and , and is a sequence such that and some . Then, the sequence converges weakly to an element in .

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 , and are given here, especially , and is the nonexpansive mapping that operates on the sequences .

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.

Let with Euclidean norm and . For each , the concrete definition of nonexpansive mappings is shown as

For each , assume that , and . The function is defined in the following manner:

From the fact that are nonexpansive and 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 ; we havewhere is a signum function, that is,

Further simplification of the proposed iterative algorithm is in the following expression:where and are points in . In addition, we choose some points and the sequences of parameters as follows:

Next, we use Algorithm (49) with an initial point and obtain numerical results in Table 1.

Remark 3. (i)From Figure 1, with the increase in the number of iterations, the errors between and decrease. When the iteration is 17 times, the minimum of is obtained.(ii)From Table 1 and Figure 1, it is observed that the sequence 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 and a nonexpansive multivalued mapping .

Data Availability

Some data obtained by the authors themselves were used to support this study.

Conflicts of Interest

All authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this study.

Acknowledgments

This work was jointly supported by the High-Level Talent Sailing Project of Yibin University (2021QH07).

References

G. C. Bento, O. P. Ferreira, and P. R. Oliveira, “Proximal point method for a special class of nonconvex functions on Hadamard manifolds,” Optimization, vol. 64, no. 2, pp. 289–319, 2015.
View at: Publisher Site | Google Scholar
B. Ahmadi Kakavandi and M. Amini, “Duality and subdifferential for convex functions on complete metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 10, pp. 3450–3455, 2010.
View at: Publisher Site | Google Scholar
K. Sokhuma, “Δ-convergence theorems for a pair of single-valued and multivalued nonexpansive mappings in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 4, no. 2, pp. 23–31, 2013.
View at: Google Scholar
U. F. Mayer, “Gradient flows on nonpositively curved metric spaces and harmonic maps,” Communications in Analysis and Geometry, vol. 6, no. 2, pp. 199–253, 1998.
View at: Publisher Site | Google Scholar
S. H. Khan and M. Abbas, “Strong and △-convergence of some iterative schemes in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 61, no. 1, pp. 109–116, 2011.
View at: Publisher Site | Google Scholar
O. P. Ferreira and P. R. Oliveira, “Proximal point algorithm on Riemannian manifolds,” Optimization, vol. 51, no. 2, pp. 257–270, 2002.
View at: Publisher Site | Google Scholar
M. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer, Berlin, Germany, 1999.
G. C. Bento, O. P. Ferreira, and P. R. Oliveira, “Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 2, pp. 564–572, 2010.
View at: Publisher Site | Google Scholar
O. Güler, “On the convergence of the proximal point algorithm for convex minimization,” SIAM Journal on Control and Optimization, vol. 29, no. 2, pp. 403–419, 1991.
View at: Publisher Site | Google Scholar
D. Ariza-Ruiz, L. Leuştean, and G. López-Acedo, “Firmly nonexpansive mappings in classes of geodesic spaces,” Transactions of the American Mathematical Society, vol. 366, no. 8, pp. 4299–4322, 2014.
View at: Publisher Site | Google Scholar
R. P. Agarwal, D. ORegan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, pp. 61–79, 2007.
View at: Google Scholar
J. Jost, “Convex functionals and generalized harmonic maps into spaces of non positive curvature,” Commentarii Mathematici Helvetici, vol. 70, no. 1, pp. 659–673, 1995.
View at: Publisher Site | Google Scholar
B. Martinet, “Brève communication. Régularisation d'inéquations variationnelles par approximations successives,” Revue française d'informatique et de recherche opérationnelle. Série rouge, vol. 4, no. R3, pp. 154–158, 1970.
View at: Publisher Site | Google Scholar
R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
View at: Publisher Site | Google Scholar
M. Bacak, “The proximal point algorithm in metric spaces,” Israel Journal of Mathematics, vol. 194, pp. 689–701, 2013.
View at: Google Scholar
W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, p. 506, 1953.
View at: Publisher Site | Google Scholar
S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, p. 147, 1974.
View at: Publisher Site | Google Scholar
S. Suantai and W. Phuengrattana, “Proximal point Algorithms for a hybrid pair of nonexpansive single-valued and multi-valued mappings in geodesic metric spaces,” Mediterranean Journal of Mathematics, vol. 14, no. 2, p. 62, 2017.
View at: Publisher Site | Google Scholar
F. Bruhat and J. Tits, “Groupes réductifs sur un corps local,” Publications Mathématiques de L'IHÉS, vol. 41, no. 1, pp. 5–251, 1972.
View at: Publisher Site | Google Scholar
S. Dhompongsa and B. Panyanak, “On △-convergence theorems in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2572–2579, 2008.
View at: Publisher Site | Google Scholar
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, Germany, 2005.
G. Durmaz and I. AltunOn, “Nonlinear set-valued -contractions,” Bulletin of the Malaysian Mathematical Sciences Society, 2018.
View at: Publisher Site | Google Scholar
I. Altun and G. Minak, “On fixed point theorems for multivalued mappings of feng-liu type,” Bulletin of the Korean Mathematical Society, vol. 52, no. 6, pp. 1901–1910, 2015.
View at: Publisher Site | Google Scholar
S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly Lipschitzian mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 4, pp. 762–772, 2006.
View at: Publisher Site | Google Scholar
W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 12, pp. 3689–3696, 2008.
View at: Publisher Site | Google Scholar
S. Dhompongsa, W. A. Kirk, and B. Panyanak, “Nonexpansive set-valued mappings in metric and Banach spaces,” J. Nonlinear Convex Anal., vol. 8, pp. 35–45, 2007.
View at: Google Scholar
L. Ambrosio, N. Gigli, and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zrich, Birkhuser, Basel, Switzerland, 2nd edition, 2008.
J. F. Tang, S. S. Chang, and J. Dong, “Strong convergence theorems of Cesaro-type means for nonexpansive mapping in CAT(0) spaces,” Fixed Point Theory and Applications, vol. 100, 2015.
View at: Google Scholar
C. E. Chidume, A. U. Bello, and P. Ndambomve, “Strong and -convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces,” Abstract and Applied Analysis, 2014.
View at: Google Scholar
P. Cholamjiak, “The modified proximal point algorithm in CAT(0) spaces,” Optimization Letters, vol. 9, no. 7, pp. 1401–1410, 2015.
View at: Publisher Site | Google Scholar
Y. Shehu, “Weak and linear convergence of proximal point algorithm with reflections,” Journal of Nonlinear and Convex Analysis, vol. 22, pp. 299–307, 2021.
View at: Google Scholar
S. Y. Cho, “A convergence theorem for generalized mixed equilibrium problems and multivalued asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 21, pp. 1017–1026, 2020.
View at: Google Scholar
J. Fan, X. Qin, and B. Tan, “Tseng's extragradient algorithm for pseudomonotone variational inequalities on Hadamard manifolds,” Applicable Analysis, pp. 1–14, 2021.
View at: Publisher Site | Google Scholar
E. T. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing,” Tech.rep.,CAAM, 2007.
View at: Google Scholar
P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Eds., pp. 185–212, Springer, New York, 2011.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Shengquan Weng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies