Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces

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.

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, proximal-point 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.

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 set-valued mapping.

(a) The set D(T) defined by

is called the effective domain of T.

(b) The set R(T) defined by

is called the range of T.

(c) The set G(T) defined by

is said to be the graph of T.

Recall the following definitions.

(c) T is said to be monotone if

(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_tx ∈ T J_tx, ∀x ∈ H and ǁT_txǁ ≤ ǀTxǀ, 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 , 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 set-valued equation

is the proximal point algorithm. To be more precise, start with any point x₀ ∈ 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 . 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 . 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 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)

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_tu.

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_tu 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 conforming to the following set-valued 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) ;

(ii) lim sup_n→∞ δ_n/γ_n ≤ 0 or .

Then lim_n→∞ α_n = 0.

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 . Let P_C be a metric projection from H onto C. For any x_n ∈ H and λ_n > 0, find and 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_tu, 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 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_nǀǀ ≤ 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 - pǀǀ ≤ M. Next, we prove that ǀǀx_n+1 - pǀǀ ≤ 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_tu.

From Lemma 2.1, we see that lim_t→∞ J_tu exists, which is the point of T^-1(0) nearest to u. Since T is maximal monotone, T_tu ∈ TJ_tu 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_tu 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 . Let P_C be a metric projection from H onto C. For any x_n ∈ H and λ_n > 0, find and 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_tu, 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 , 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 semi-continuous 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 semi-continuous 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 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 , the sequence {x_n} converges strongly to a minimizer of f nearest to u.

Proof. Since f: H → (-∞, +∞] is a proper convex lower semi-continuous 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 semi-continuous 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 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 , the sequence {x_n} converges strongly to a minimizer of f nearest to u.

The authors declare that they have no competing interests.

All authors contributed equally to this paper. All authors read and approved the final manuscript.

The authors are grateful to the referees for their valuable comments and suggestions which improve the contents of the article.

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