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Strong convergence of a proximal-type algorithm for an occasionally pseudomonotone operator in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 190 (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:
Find such that
A wide variety of the problems, such as optimization problems and related fields, min-max problems, complementarity problems, variational inequalities, equilibrium problems and fixed point problems, fall within this general framework. For example, if T is the subdifferential ∂f of a proper lower semicontinuous convex function , then T is a maximal monotone operator and the equation is reduced to . One method of solving is the proximal point algorithm. Let I denote the identity operator on H. Rockafellar’s proximal point algorithm generates, for any starting point , a sequence in H by the rule
where is a sequence of positive real numbers. Note that (1.2) is equivalent to
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.
On the other hand, Kamimura and Takahashi [14, 15], Solodov and Svaiter  one decade ago modified the proximal point algorithm to generate a strongly convergent sequence. In 1999, Solodov and Svaiter  introduced the following algorithm :
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 .
In 2003, Kamimura and Takahashi  extended Solodov and Svaiter’s result to more general Banach spaces like the spaces () by further modifying the proximal-point algorithm (1.2) in the following form in a smooth Banach space E:
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.
Example 2.1 Let and be a multi-valued operator defined by
Then for any , in , if and , then we have
Thus, if , then T is monotone. However, if , then T is neither monotone nor pseudomonotone. Indeed, for , then we have , and , but .
Further, we see that T is occasionally pseudomonotone. To effect this, for any and in , if , , then we have
Example 2.2 The rotation operator on given by
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.
Let . A Banach space E is said to be smooth if the limit
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.
For any , the mapping defined by
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.
Proposition 2.1 Let E be a real Banach space. For , the duality mapping has the following properties:
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 ;
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 ).
Proposition 2.2 Let E be a real Banach space, and let , , be the duality mapping. Then for any ,
Proof It is a straightforward consequence of the assertion (5) of Proposition 2.1 applied for x and . Alternatively, from Proposition 2.1(9), it follows that (subdifferential of the functional ), where . Also, it follows from the definition of the subdifferential of ψ that
Now, substituting by , we have
This completes the proof. □
Remark 2.1 If E is a uniformly smooth Banach space, it follows from Proposition 2.1(10) that () is a single-valued mapping. We now define the functions by
and ϕ is the support function satisfying the following condition:
It is obvious from the definition of Ψ and Proposition 2.1(5) that
Also, we see that
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 .
Proof It follows from that
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. □
Proposition 2.4 Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let . Then there exists a unique element such that
Proof Since E is reflexive and implies , there exists such that . Since E is strictly convex, is a strictly convex function, that is,
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.
Proposition 2.5 Let E be a smooth Banach space, let C be a convex subset of E, let and . Then
if and only if
Proof First, we show that (2.6) ⇒ (2.7). Let and . It follows from that
Taking , since is norm-to-weak∗ continuous, we obtain
which shows (2.7).
Next, we show that (2.7) ⇒ (2.6). For any , we have
which proves (2.6). This completes the proof. □
Proposition 2.6 Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E, let and with
Then we have
Proof It follows from Proposition 2.5 that
This completes the proof. □
3 Main results
Throughout this section, unless otherwise stated, we assume that is a occasionally pseudomonotone maximal monotone operator. In this section, we study the following algorithm in a smooth Banach space E, which is an extension of (1.2):
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.
Proof From the definition of the sequence , it is obvious that both and are closed convex sets. Let . From Theorem 3.2, there exists such that
Since T is occasionally pseudomonotone and , from , it follows that
for some . It follows that . On the other hand, it is clear that . Then and so is well defined. Suppose that is well defined for some . Again, by Theorem 3.2, we obtain such that
Then since T is occasionally pseudomonotone and , from , it follows that
for some , and so . It follows from Proposition 2.5 that
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. □
Remark 3.1 From the above proof, we obtain
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 .
Proof It follows from the definition of and Proposition 2.5 that . Further, from and Proposition 2.6, we have
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 .
Now, we show that . It follows from (3.2) that . On the other hand, we have
Since and , it follows that
and so . Further, since , we have
which yields that
Then it follows from that . Consequently, by Proposition 2.3, we have , which implies . Moreover, since J is uniformly norm-to-norm continuous on bounded subsets and , we obtain
It follows from with and that
Then, since T is occasionally pseudomonotone, it follows that for some . Therefore, from the maximality of T, we obtain . Let . Now, from and , we have
Then we have
Thus, from Proposition 2.5, we have
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
Let be a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined by
Using Theorem 3.4, we consider the problem of finding a minimizer of the function f.
Theorem 4.1 Let E be reflexive, strictly convex and uniformly smooth Banach space, and let be a proper convex lower semicontinuous function. Assume that satisfies and is the sequence generated by
If , then the sequence generated by (4.1) converges strongly to the minimizer of f.
Proof Since is a proper convex lower semicontinuous function, by Rockafelllar , the subdifferential ∂f of f is a maximal monotone operator and so it is also an occasionally pseudomonotone maximal operator. We also know that
is equivalent to the following:
This implies that
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.
Martinet B: Régularisation d’inéquations variationnelles par approximatios successives. Rev. Fr. Autom. Inform. Rech. Opér. 1970, 4: 154–159.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
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.2005.01.012
Brezis H, Lions PL: Produits infinis de resolvantes. Isr. J. Math. 1978, 29: 329–345. 10.1007/BF02761171
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 598191
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 7
Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403–419. 10.1137/0329022
Lions PL: Une méthode itérative de résolution d’une inéquation variationnelle. Isr. J. Math. 1978, 31: 204–208. 10.1007/BF02760552
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)90234-8
Qin X, Cho YJ, Kang SM: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s10898-009-9410-6
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.2010.01.124
Solodov MV, Svaiter BF: A hybrid projection proximal point algorithm. J. Convex Anal. 1999, 6: 59–70.
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 168468
Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493
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:1026592623460
Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 2000, 87: 189–202.
Minty GJ: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 1962, 29: 341–346. 10.1215/S0012-7094-62-02933-2
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2003, 13: 938–945.
Karamardian S: Complementarity problems over cones with monotone or pseudomonotone maps. J. Optim. Theory Appl. 1976, 18: 445–454. 10.1007/BF00932654
Asplund E: Positivity of duality mappings. Bull. Am. Math. Soc. 1967, 73: 200–203. 10.1090/S0002-9904-1967-11678-1
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-O
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. 18. Proc. Sympos. Pure Math. 1976.
Rockafellar RT: Characterization of the subdifferentials of convex functions. Pac. J. Math. 1966, 17: 497–510. 10.2140/pjm.1966.17.497
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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Pathak, H.K., Cho, Y.J. Strong convergence of a proximal-type algorithm for an occasionally pseudomonotone operator in Banach spaces. Fixed Point Theory Appl 2012, 190 (2012). https://doi.org/10.1186/1687-1812-2012-190
- proximal point algorithm
- monotone operator
- maximal monotone operator
- pseudomonotone operator
- occasionally pseudomonotone operator
- maximal pseudomonotone operator
- maximal occasionally pseudomonotone operator
- Banach space
- strong convergence