Strong convergence of a proximal-type algorithm for an occasionally pseudomonotone operator in Banach spaces
© Pathak and Cho; licensee Springer 2012
Received: 9 May 2012
Accepted: 24 September 2012
Published: 24 October 2012
It is known that the proximal point algorithm converges weakly to a zero of a maximal monotone operator, but it fails to converge strongly. Then, in (Math. Program. 87:189-202, 2000), Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for the algorithm in Hilbert spaces. Further, Kamimura and Takahashi (SIAM J. Optim. 13:938-945, 2003) extended Solodov and Svaiter’s result to more general Banach spaces and obtained strong convergence of a proximal-type algorithm in Banach spaces. In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi.
Let H be a real Hilbert space with inner product , and let be a maximal monotone operator (or a multifunction) on H. We consider the classical problem:
This algorithm was first introduced by Martinet  and generally studied by Rockafellar  in the framework of Hilbert spaces. Later, many authors studied the convergence of (1.2) in Hilbert spaces (see Agarwal et al. , Brezis and Lions , Cho et al. , Cholamjiak et al. , Güler , Lions , Passty , Qin et al. , Song et al. , Solodov and Svaiter , Wei and Cho  and the references therein). Rockafellar  proved that, if and , then the sequence generated by (1.2) converges weakly to an element of . Further, Rockafellar  posed an open question of whether the sequence generated by (1.2) converges strongly or not. This question was solved by Güler , who introduced an example for which the sequence generated by (1.2) converges weakly, but not strongly.
To explain how the sequence is generated, we formally state the above algorithm as follows.
Choose any and . At iteration n, having , choose and find , an inexact solution of , with tolerance σ. Define and as in (1.3). Take . Note that at each iteration, there are two subproblems to be solved: find an inexact solution of the proximal point subproblem and find the projection of onto , the intersection of two half-spaces. By a classical result of Minty , the proximal subproblem always has an exact solution, which is unique. Notice that computing an approximate solution makes things easier. Hence, this part of the method is well defined. Regarding the projection step, it is easy to prove that is never empty, even when the solution set is empty. Therefore, the whole algorithm is well defined in the sense that it generates an infinite sequence and an associated sequence of pairs .
to generate a strongly convergent sequence. They proved that if and , then the sequence generated by (1.3) converges strongly to a point .
In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi.
2 Preliminaries and definitions
Let E be a real Banach space with norm , and let denote the dual space of E. Let denote the value of at . Let be a sequence in E. We denote the strong convergence of to by and the weak convergence by , respectively.
Definition 2.1 A multivalued operator with domain and range is said to be monotone if for any and , . A monotone operator T is said to be maximal if its graph is not properly contained in the graph of any other monotone operator.
Definition 2.2 A multivalued operator with domain and range is said to be pseudomonotone (see also Karamardian ) if implies for any and , .
It is obvious that each monotone operator is pseudomonotone, but the converse is not true.
We now introduce the concept of occasionally pseudomonotone as follows.
Definition 2.3 A multivalued operator is said to be occasionally pseudomonotone if, for any , there exist , , such that implies .
It is clear that every monotone operator is pseudomonotone and every pseudomonotone operator is occasionally pseudomonotone, but the converse implications need not be true. To this end, we observe the following examples.
Thus, if , then T is monotone. However, if , then T is neither monotone nor pseudomonotone. Indeed, for , then we have , and , but .
is monotone and hence it is pseudomonotone. Thus, it follows that A is also occasionally pseudomonotone.
Maximality of pseudomonotone and occasionally pseudomonotone operators are defined as similar to maximality of a monotone operator. We denote by the ray passing through , .
A Banach space E is said to be strictly convex if for all with and . It is also said to be uniformly convex if for any two sequences , in E such that and .
It is known that a uniformly convex Banach space is reflexive and strictly convex. The spaces and are neither reflexive nor strictly convex. Note also that a reflexive Banach space is not necessarily uniformly convex. For example, consider a finite dimensional Banach space in which the surface of the unit ball has a ‘flat’ part. We note that such a Banach space is reflexive because of finite dimension. But the ‘flat’ portion in the surface of the ball makes it nonuniformly convex. It is also well known that a Banach space E is reflexive if and only if every bounded sequence of elements of E contains a weakly convergent sequence.
exists for all . It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for .
It is well known that the spaces , and (Sobolev space) (, m is a positive integer) are uniformly convex and uniformly smooth Banach spaces.
is called the duality mapping with the gauge function . In particular, for , the duality mapping with gauge function is called the normalized duality mapping.
The following proposition gives some basic properties of the duality mapping.
for all and , where denotes the domain of ;
for all with ;
for all ;
for all and ;
If E is smooth, then is norm-to-weak∗ continuous;
If E is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of E;
is bounded, i.e., for any bounded subset , is a bounded subset in ;
- (9)can be equivalently defined as the subdifferential of the functional (Asplund ), i.e.,
E is a uniformly smooth Banach space (equivalently, is a uniformly convex Banach space) if and only if is single-valued and uniformly continuous on any bounded subset of E (see, for instance, Xu and Roach , Browder ).
This completes the proof. □
In particular, for , we have .
Further, we can show the following two propositions.
Proposition 2.3 Let E be a smooth Banach space, and let , be two sequences in E. If , then .
because of (2.2) and (2.3). Therefore, if is bounded, then (and also if is bounded, then ) is also bounded and . This completes the proof. □
for all with , and . Then the function is also strictly convex. Therefore, is unique. This completes the proof. □
For each nonempty closed convex subset C of a reflexive, convex and smooth Banach space E, we define the mapping of E onto C by , where is defined by (2.5). For the case , it is easy to see that the mapping is coincident with the metric projection in the setting of Hilbert spaces. In our discussion, instead of the metric projection, we make use of the mapping . Finally, we prove two results concerning Proposition 2.4 and the mapping . The first one is the usual analogue of a characterization of the metric projection in a Hilbert space.
which shows (2.7).
which proves (2.6). This completes the proof. □
This completes the proof. □
3 Main results
where is a sequence of positive real numbers.
First, we investigate the condition under which the algorithm (3.1) is well defined. Rockafellar  proved the following theorem.
Theorem 3.1 Let E be a reflexive, strictly convex and smooth Banach space, and let be a monotone operator. Then T is maximal if and only if for all .
By the appropriate modification of arguments in Theorem 3.1, we can prove the following.
Theorem 3.2 Let E be a reflexive, strictly convex and smooth Banach space, and let be an occasionally pseudomonotone operator. Then T is maximal if and only if for all .
Using Theorem 3.2, we can show the following result.
Proposition 3.3 Let E be a reflexive, strictly convex and smooth Banach space. If , then the sequence generated by (3.1) is well defined.
which implies . Therefore, and hence is well defined. Then by induction, the sequence generated by (3.1) is well defined for each . This completes the proof. □
Now, we are ready to prove our main theorem.
Theorem 3.4 Let E be a reflexive, strictly convex and uniformly smooth Banach space. If , ϕ satisfies the condition (2.2) and satisfies , then the sequence generated by (3.1) converges strongly to .
Since the sequence is monotone decreasing and bounded below by 0, it follows that exists and, in particular, is bounded. Then by (2.3), is also bounded. This implies that there exists a subsequence of such that for some .
Then we obtain , and hence . It follows from Proposition 2.3 that . This means that the whole sequence generated by (3.1) converges weakly to and each weakly convergent subsequence of converges strongly to . Therefore, converges strongly to . This completes the proof. □
4 An application
Using Theorem 3.4, we consider the problem of finding a minimizer of the function f.
If , then the sequence generated by (4.1) converges strongly to the minimizer of f.
Thus, we have such that . Therefore, using Theorem 3.4, we get the conclusion. This completes the proof. □
5 Concluding remarks
We presented a modified proximal-type algorithm with the varied degree of rate of the convergence depending upon the choice of p () for an occasionally pseudomonotone operator, which is a generalization of a monotone operator, to extend Kamimura and Takahashi’s result to more general Banach spaces which are not necessarily uniformly convex like locally uniformly Banach spaces. As an application, we consider the problem of finding a minimizer of a convex function in a more general setting of Banach spaces than what Kamimura and Takahashi have considered.
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2012-0008170).
- Martinet B: Régularisation d’inéquations variationnelles par approximatios successives. Rev. Fr. Autom. Inform. Rech. Opér. 1970, 4: 154–159.MathSciNetGoogle Scholar
- Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056MathSciNetView ArticleGoogle Scholar
- Agarwal RP, Zhou HY, Cho YJ, Kang SM: Zeros and mappings theorems for perturbations of m -accretive operators in Banach spaces. Comput. Math. Appl. 2005, 49: 147–155. 10.1016/j.camwa.2005.01.012MathSciNetView ArticleGoogle Scholar
- Brezis H, Lions PL: Produits infinis de resolvantes. Isr. J. Math. 1978, 29: 329–345. 10.1007/BF02761171MathSciNetView ArticleGoogle Scholar
- Cho YJ, Kang SM, Zhou H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J. Inequal. Appl. 2008., 2008: Article ID 598191Google Scholar
- Cholamjiak P, Cho YJ, Suantai S: Composite iterative schemes for maximal monotone operators in reflexive Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 7Google Scholar
- Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403–419. 10.1137/0329022MathSciNetView ArticleGoogle Scholar
- Lions PL: Une méthode itérative de résolution d’une inéquation variationnelle. Isr. J. Math. 1978, 31: 204–208. 10.1007/BF02760552View ArticleGoogle Scholar
- Passty GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 1979, 72: 383–390. 10.1016/0022-247X(79)90234-8MathSciNetView ArticleGoogle Scholar
- Qin X, Cho YJ, Kang SM: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s10898-009-9410-6MathSciNetView ArticleGoogle Scholar
- Song Y, Kang JI, Cho YJ: On iterations methods for zeros of accretive operators in Banach spaces. Appl. Math. Comput. 2010, 216: 1007–1017. 10.1016/j.amc.2010.01.124MathSciNetView ArticleGoogle Scholar
- Solodov MV, Svaiter BF: A hybrid projection proximal point algorithm. J. Convex Anal. 1999, 6: 59–70.MathSciNetGoogle Scholar
- Wei L, Cho YJ: Iterative schemes for zero points of maximal monotone operators and fixed points of nonexpansive mappings and their applications. Fixed Point Theory Appl. 2008., 2008: Article ID 168468Google Scholar
- Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493MathSciNetView ArticleGoogle Scholar
- Kamimura S, Takahashi W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 2000, 8: 361–374. 10.1023/A:1026592623460MathSciNetView ArticleGoogle Scholar
- Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 2000, 87: 189–202.MathSciNetGoogle Scholar
- Minty GJ: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 1962, 29: 341–346. 10.1215/S0012-7094-62-02933-2MathSciNetView ArticleGoogle Scholar
- Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2003, 13: 938–945.MathSciNetView ArticleGoogle Scholar
- Karamardian S: Complementarity problems over cones with monotone or pseudomonotone maps. J. Optim. Theory Appl. 1976, 18: 445–454. 10.1007/BF00932654MathSciNetView ArticleGoogle Scholar
- Asplund E: Positivity of duality mappings. Bull. Am. Math. Soc. 1967, 73: 200–203. 10.1090/S0002-9904-1967-11678-1MathSciNetView ArticleGoogle Scholar
- Xu ZB, Roach GF: Characteristic inequalities in uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 1991, 157: 189–210. 10.1016/0022-247X(91)90144-OMathSciNetView ArticleGoogle Scholar
- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. 18. Proc. Sympos. Pure Math. 1976.Google Scholar
- Rockafellar RT: Characterization of the subdifferentials of convex functions. Pac. J. Math. 1966, 17: 497–510. 10.2140/pjm.1966.17.497MathSciNetView ArticleGoogle Scholar
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