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Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces
Journal of Inequalities and Applications volume 2012, Article number: 137 (2012)
Abstract
In this article, we consider the proximal point algorithm for the problem of approximating zeros of maximal monotone mappings. Strong convergence theorems for zero points of maximal monotone mappings are established in the framework of Hilbert spaces.
2000 AMS Subject Classification: 47H05; 47H09; 47J25.
1. Introduction
The theory of maximal monotone operators has emerged as an effective and powerful tool for studying many real world problems arising in various branches of social, physical, engineering, pure and applied sciences in unified and general framework. Recently, much attention has been payed to develop efficient and implementable numerical methods including the projection method and its variant forms, auxiliary problem principle, proximalpoint algorithm and descent framework for solving variational inequalities and related optimization problems (see [1–32] and the references therein). The proximal point algorithm, can be traced back to Martinet [33] in the context of convex minimization and Rockafellar [34] in the general setting of maximal monotone operators, has been extended and generalized in different directions by using novel and innovative techniques and ideas.
In this article, we investigate the problem of approximating a zero of the maximal monotone mapping based on a proximal point algorithm in the framework of Hilbert spaces. Strong convergence of the iterative algorithm is obtained.
2. Preliminaries
Throughout this article, we assume that H is a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ǀǀ · ǀǀ, respectively. Let T be a setvalued mapping.

(a)
The set D(T) defined by
D\left(T\right)=\left\{u\in H:T\left(u\right)\ne \varnothing \right\}
is called the effective domain of T.

(b)
The set R(T) defined by
R\left(T\right)=\bigcup _{u\in H}T\left(u\right)
is called the range of T.

(c)
The set G(T) defined by
G\left(T\right)=\left\{\left(u,v\right)\in H\times H:u\in D\left(T\right),v\in R\left(T\right)\right\}
is said to be the graph of T.
Recall the following definitions.

(c)
T is said to be monotone if
\u3008uv,xy\u3009\ge 0,\phantom{\rule{1em}{0ex}}\forall \left(u,x\right),\left(v,y\right)\in G\left(T\right). 
(d)
T is said to be maximal monotone if it is not properly contained in any other monotone operator.
For a maximal monotone T : D(T) → 2^{H}, we can defined the resolvent of T by
It is well known that J_{ t }: H → D(T) is nonexpansive, and F(J_{ t }) = T^{1}(0), where F(J_{ t }) denotes the set of fixed points of J_{ t }. The Yosida approximation T_{ t }is defined by
It is well known that T_{ t }x ∈ T J_{ t }x, ∀x ∈ H and ǁT_{ t }x ǁ ≤ ǀTx ǀ, where
for all x ∈ D(T).
Let C be a nonempty, closed and convex subset of H. Next, we always assume that T: C → 2^{H}is a maximal monotone mapping with {T}^{1}\left(0\right)\ne \varnothing, where T^{1}(0) denotes the set of zeros of T.
The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone mappings. A classical method to solve the following setvalued equation
is the proximal point algorithm. To be more precise, start with any point x_{0} ∈ H, and update x_{ n }_{+1} iteratively conforming to the following recursion
where {β_{ n }} ⊂ [β, ∞), (β > 0) is a sequence of real numbers. However, as pointed in [15], the ideal form of the method is often impractical since, in many cases, to solve the problem (2.3) exactly is either impossible or the same difficult as the original problem (2.2). Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximately zeroes of T.
In 1976, Rockafellar [35] gave an inexact variant of the method
where {e_{ n }} is regarded as an error sequence. This is an inexact proximal point algorithm. It was shown that, if
then the sequence {x_{ n }} defined by (1.4) converges weakly to a zero of T provided that {T}^{1}\left(0\right)\ne \varnothing. In [16], Güller obtained an example to show that Rockafellar's proximal point algorithm (1.4) does not converge strongly, in general.
Recently, many authors studied the problems of modifying Rockafellar's proximal point algorithm so that strong convergence is guaranteed. Cho et al. [13] proved the following result.
Theorem CKZ. Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T: Ω → 2^{H}a maximal monotone operator with{T}^{1}\left(0\right)\ne \varnothing. Let P_{Ω}be the metric projection of H onto Ω. Suppose that, for any given x_{ n }∈ H, β_{ n }> 0 and e_{ n }∈ H, there exists{\stackrel{\u0304}{x}}_{n}\in \mathrm{\Omega}conforming to the SVME (2.4), where {β_{ n }} ⊂ (0, + ∞) with β_{ n }→ ∞ as n → ∞ and
Let {α_{ n }} be a real sequence in [0, 1] such that

(i)
α _{ n }→ 0 as n → ∞,

(ii)
{\sum}_{n=0}^{\infty}{\alpha}_{n}=\infty .
for any fixed u ∈ Ω, define the sequence {x_{ n }} iteratively as follows:
Then {x_{ n }} converges strongly to a fixed point z of T, where z = lim_{t→ ∞}J_{ t }u.
In this article, motivated by Theorem CKZ, we continue to consider the problem of approximating a zero of the maximal monotone mapping T. Strong convergence theorems are established under mild restrictions imposed on the error sequence {e_{ n }} comparing with the restriction (B). The results which include Cho et al. [13] as a special case also improve the corresponding results announced by many others.
In order to prove our main result, we need the following lemmas.
Lemma 2.1. (Bruck [[35], Lemma 1]). Let H be a Hilbert space and C a nonempty, closed and convex subset H. For all u ∈ C, lim_{t→ ∞}J_{ t }u exists and it is the point of T^{1}(0) nearest u.
Lemma 2.2 (Eckstein [[15], Lemma 2]). For any given x_{ n }∈ C, λ_{ n }> 0, and e_{ n }∈ H, there exists{\stackrel{\u0304}{x}}_{n}\in Cconforming to the following setvalued mapping equation (in short, SVME):
Furthermore, for any p ∈ T^{1}(0), we have
and
Lemma 2.3 (Liu [36]). Assume that {α_{ n }} is a sequence of nonnegative real numbers such that
where {γ_{ n }} is a sequence in (0,1) and {δ_{ n }} is a sequence such that

(i)
{\sum}_{n=1}^{\infty}{\gamma}_{n}=\infty;

(ii)
lim sup_{n→∞} δ _{ n }/γ _{ n }≤ 0 or {\sum}_{n=1}^{\infty}\u01c0{\delta}_{n}\u01c0<\infty.
Then lim_{n→∞}α_{ n }= 0.
3. Main results
Theorem 3.1. Let H be a real Hilbert space, C a nonempty, closed and convex subset of H and T: C → 2^{H}a maximal monotone operator with{T}^{1}\left(0\right)\ne \varnothing. Let P_{ C }be a metric projection from H onto C. For any x_{ n }∈ H and λ_{ n }> 0, find{\stackrel{\u0304}{x}}_{n}\in Cand e_{ n }∈ H conforming to the SVME (2.5), where {λ_{ n }} ⊂ (0, ∞) with λ_{ n }→ ∞ as n → ∞ and
with sup_{ n }_{≥0}η_{ n }= η < 1. Let {α_{ n }} and {β_{ n }} be real sequences in [0, 1] satisfying α_{ n }+ β_{ n }< 1 and the following control conditions:
Let {x_{ n }} be a sequence generated by the following manner:
where u ∈ C is a fixed element. Then the sequence {x_{ n }} generated by (3.1) strongly converges to a zero point z of T, where z = lim_{t→∞}J_{ t }u, if and only if e_{ n }→ 0 as n → ∞.
Proof. First, show that the necessity. Assume that x_{ n }→ z as n → ∞, where z ∈ T^{1}(0). It follows from (2.5) that
This implies that
It follows that {\stackrel{\u0304}{x}}_{n}\to z as n → ∞. Note that
This shows that e_{ n }→ 0 as n → ∞.
Next, we show the sufficiency. The proof is divided into three steps.
Step 1. Show that {x_{ n }} is bounded.
From the assumption (C), we see that
For any p ∈ T^{1} (0), it follows from Lemma 2.2 that
That is,
It follows from (3.2) that
Putting
we show that ǀǀx_{ n }ǀǀ ≤ M for all n ≥ 0. It is easy to see that the result holds for n = 0. Assume that the result holds for some n ≥ 0. That is, ǀǀx_{ n } p ǀǀ ≤ M. Next, we prove that ǀǀx_{ n }_{+1}  p ǀǀ ≤ M. Indeed, we see from (3.3) that
This shows that the sequence {x_{ n }} is bounded.
Step 2. Show that lim sup_{n→∞}〈u  z, x_{ n }_{+1} z〉 ≤ 0, where z = lim_{t→∞}J_{ t }u.
From Lemma 2.1, we see that lim_{t→∞}J_{ t }u exists, which is the point of T^{1}(0) nearest to u. Since T is maximal monotone, T_{ t }u ∈ TJ_{ t }u and T_{ λn }x_{ n }∈ TJ_{ λn }x_{ n }, we see
Since λ_{ n }→ ∞ as n → ∞, for any t > 0, we have
On the other hand, by the nonexpansivity of J_{ λn }, we obtain
From the assumption e_{ n }→ 0 as n → ∞ and (3.4), we arrive at
From (2.5), we see that
That is,
Combining (3.5) with (3.6), we arrive at
On the other hand, we see from the algorithm (3.1) that
It follows from the condition lim_{n→∞}α_{ n }= lim_{n→∞}β_{ n }= 0 that
which combines with (3.7) yields that
From z = lim_{t→∞}J_{ t }u and (3.8), we arrive at
Step 3. Show that x_{ n }→ z as n → ∞.
It follows from (3.2) that
This implies that
Applying Lemma 2.3 to (3.10), we obtain that x_{ n }→ z as n → ∞. This completes the proof.
As a corollary of Theorem 3.1, we have the following.
Corollary 3.2. Let H be a real Hilbert space, C a nonempty, closed and convex subset of H and T: C → 2^{H}a maximal monotone operator with{T}^{1}\left(0\right)\ne \varnothing. Let P_{ C }be a metric projection from H onto C. For any x_{ n }∈ H and λ_{ n }> 0, find{\stackrel{\u0304}{x}}_{n}\in Cand e_{ n }∈ H conforming to the SVME (2.5), where {λ_{ n }} ⊂ (0, ∞) with λ_{ n }→ ∞ as n → ∞ and
with sup_{ n }_{≥0}η_{ n }= η < 1. Let {α_{ n }} be a real sequence in (0,1) satisfying the following control conditions:
Let {x_{ n }} be a sequence generated by the following manner:
where u ∈ C is a fixed element. Then the sequence {x_{ n }} strongly converges to a zero point z of T, where z = lim_{t→ ∞}, J_{ t }u, if and only if e_{ n }→ 0 as n → ∞.
Remark 3.3. Corollary 3.2 improves Theorem CKZ by relaxing the restriction imposed on the sequence {e_{ n }}. In [34], Rockafellar obtained a weak convergence by assuming that {\sum}_{n=0}^{\infty}\u01c1{e}_{n}\u01c1<\infty, see [34] for more details.
Next, as applications of Theorem 3.1, we consider the problem of finding a minimizer of a convex function.
Let H be a Hilbert space, and f: H → (∞, +∞] be a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:
Theorem 3.4. Let H be a real Hilbert space and f: H → (∞, +∞] a proper convex lower semicontinuous function. Let {λ_{ n }} be a sequence in (0, +∞) with λ_{ n }→ ∞ as n → ∞ and {e_{ n }} a sequence in H with e_{ n }→ ∞ as n → ∞. Assume that
with sup_{ n }_{≥0}η_{ n }= η < 1. Let{\stackrel{\u0304}{x}}_{n}be the solution of SVME (2.5) with T replacing by ∂f. That is,
Let {α_{ n }} and {β_{ n }} be real sequences in [0, 1] satisfying α_{ n }+ β_{ n }< 1 and the following control conditions:
Let {x_{ n }} be a sequence generated by the following manner:
where u ∈ H is a fixed element. If\partial f\left(0\right)\ne \varnothing, the sequence {x_{ n }} converges strongly to a minimizer of f nearest to u.
Proof. Since f: H → (∞, +∞] is a proper convex lower semicontinuous function, we have that the subdifferential ∂ f of f is maximal monotone by Theorem 1 of [34]. Notice that
is equivalent to the following
It follows that
By Theorem 3.1, we can obtain the desired conclusion immediately.
As a corollary of Theorem 3.4, we have the following.
Corollary 3.5. Let H be a real Hilbert space and f: H → (∞, +∞] a proper convex lower semicontinuous function. Let {λ_{ n }} be a sequence in (0, +∞) with λ_{ n }→ ∞ as n → ∞ and {e_{ n }} a sequence in H with e_{ n }→ ∞ as n → ∞. Assume that
with sup_{ n }_{≥0}η_{ n }= η < 1. Let{\stackrel{\u0304}{x}}_{n}be the solution of SVME (2.5) with T replacing by ∂f. That is,
Let {α_{ n }} be a real sequence in [0, 1] ssatisfying the following control conditions:
Let {x_{ n }} be a sequence generated by the following manner:
where u ∈ H is a fixed element. If\partial f\left(0\right)\ne \varnothing, the sequence {x_{ n }} converges strongly to a minimizer of f nearest to u.
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Wei, Z., Shi, G. Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces. J Inequal Appl 2012, 137 (2012). https://doi.org/10.1186/1029242X2012137
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DOI: https://doi.org/10.1186/1029242X2012137
Keywords
 fixed point
 nonexpansive mapping
 maximal monotone mapping
 zero