Inexact Halpern-type proximal point algorithm

J Glob Optim (2011) 51:11–26 DOI 10.1007/s10898-010-9616-7 Inexact Halpern-type proximal point algorithm O. A. Boikanyo · G. Moroşanu Received: 29 January 2010 / Accepted: 17 September 2010 / Published online: 30 September 2010 © Springer Science+Business Media, LLC. 2010 Abstract We present several strong convergence results for the modified, Halpern-type, proximal point algorithm xn+1 = αn u + (1 − αn )Jβn xn + en (n = 0, 1, . . .; u, x0 ∈ H given, and Jβn = (I + βn A)−1 , for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on αn ∈ (0, 1) and βn ∈ (0, ∞). These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional. Keywords Proximal point algorithm · prox-Tikhonov algorithm · Monotone operator · Control conditions · Strong convergence · Convex function · Minimizer · Minimum value Mathematics Subject Classification (2000) 47J25 · 47H05 · 47H09 1 Introduction Let H be a real Hilbert space with inner product ·, · and norm  · . A map T : H → H is said to be nonexpansive if for every x, y ∈ H the inequality T x − T y ≤ x − y holds. In the case when T x − T y ≤ ax − y holds for some a ∈ (0, 1), then T is said to be a contraction with Lipschitz constant a. We recall that a mapping A : D(A) ⊂ H → 2 H is said to be a monotone operator if x − x ′ , y − y ′  ≥ 0, ∀ (x, y), (x ′ , y ′ ) ∈ A. O. A. Boikanyo (B) · G. Moroşanu Department of Mathematics and its Applications, Central European University, Nador u. 9, 1051 Budapest, Hungary e-mail: [email protected] G. Moroşanu e-mail: [email protected] 123 12 J Glob Optim (2011) 51:11–26 In other words, the graph of A, G(A) = {(x, y) ∈ H × H : x ∈ D(A), y ∈ Ax} is a monotone subset of H × H . The operator A is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. It is well known that if A is maximal monotone and β > 0, then the resolvent of A, the operator Jβ : H → H defined by Jβ (x) = (I + β A)−1 (x), is single-valued and nonexpansive (see, e.g. [13]). For a fixed u ∈ H and t ∈ (0, 1), let z t denote the fixed point of the contraction Tt given by the rule x → tu + (1 − t)T x, i.e., z t = tu + (1 − t)T z t . (1) The strong convergence of z t to a fixed point of T was proved in 1967 by Browder [3]. This result of Browder has been widely used in the theory of fixed points and extended in different directions by several authors. Motivated by Browder’s (implicit) convergence result, Halpern [7] considered the (explicit) iteration xn+1 = αn u + (1 − αn )T xn , for any u, x0 ∈ H with αn ∈ (0, 1) and all n ≥ 0, (2) in a Hilbert space and proved that under certain assumptions on αn , the sequence {xn } given by the iterative process (2) is strongly convergent, and the limit is the point of F(T ) = {x ∈ H | T x = x} which is nearest to u. Later, Lions [12] proved the strong convergence of (2) still in a Hilbert space under the control conditions (C1) lim αn = 0, (C2) n→∞ ∞  n=0 αn = ∞ and (C3) lim n→∞ (αn+1 − αn ) = 0. 2 αn+1 Unfortunately, Lions’ result excludes the natural choice αn = n −1 . This was overcome in 1992 by Wittmann [17] who showed strong convergence of {x n } under the control conditions (C1), (C2), and (C4) ∞  n=0 |αn+1 − αn | < ∞. In 2002, Xu [18] studied algorithm 2 extensively. First, he showed that in a Banach space setting, {xn } still maintains its strong convergence on removing the square in the denominator of (C3), thereby improving Lions’ result twofold. The conditions used were (C1), (C2), and (C5) lim n→∞ (αn+1 − αn ) = 0, or equivalently, αn+1 lim n→∞ αn = 1. αn+1 He then showed that the conditions (C3) and (C4) are not comparable, and did the same for (C4) and (C5). Xu then observed that Halpern actually showed that the conditions (C1) and (C2) are necessary to have strong convergence to the metric projection of u on F(T ). This provided a partial answer to Reich’s question: Concerning {αn }, what are the necessary and sufficient conditions for {xn } to converge strongly? To the best of our knowledge, the other part of the question concerning sufficiency remains open. However, in a recent paper of Suzuki [16], it is shown that if the nonexpansive mapping T in (2) is of the form T := λS + (1 − λ)I (with λ ∈ (0, 1), S a nonexpansive mapping and I the identity operator), then the conditions (C1) and (C2) are not only necessary for {x n } to converge strongly, but they are also sufficient. In fact, Suzuki showed strong convergence of the iterative process xn+1 = αn u + (1 − αn )(λSxn + (1 − λ)xn ), for any u, x0 ∈ H and all n ≥ 0, (3) in Banach spaces. The same result was obtained by Chidume and Chidume [4] independently. Very recently, He et. al. [8] showed also in Banach spaces that if the nonexpansive map S 123 J Glob Optim (2011) 51:11–26 13 above is replaced by the resolvent, Jβn , of an m-accretive operator, then strong convergence is still guaranteed under (C1), (C2), and the condition (C6) lim (βn+1 − βn ) = 0, n→∞ with βn bounded from below away from zero. Notice that it is possible to prove strong convergence results if one replaces the nonexpansive map T in algorithm 2 by a sequence of nonexpansive mappings. For instance, one may consider the iterative process, known as the (modified) proximal point algorithm of Halpern-type, defined by xn+1 = αn u + (1 − αn )Jβn xn , for any u, x0 ∈ H and all n ≥ 0, (4) where {βn } ⊂ (0, ∞). Under additional assumptions on βn , the strong convergence of {xn } defined by (4) can be obtained. In 2000, Kamimura and Takahashi [10] showed that {xn } is strongly convergent to the point of the set F(Jc ) = {x ∈ H : Jc x = x} = A−1 (0) (for all c > 0) nearest to u if one assumes (C1), (C2) and βn → ∞. In fact, they considered the following algorithm which is the inexact form of algorithm 4:  yn ≈ Jβn xn , for all n ≥ 0, (5) xn+1 = αn u + (1 − αn )yn , for any u, x0 ∈ H , where the criterion for the approximate computation of yn is given by yn − Jβn xn  ≤ δn with ∞  n=0 δn < ∞. It is worth mentioning that Xu [18] also obtained the same result independently, and in [1] (see also [2]), we extended this result to include non-summable errors, en . It is unclear if the same conclusion can also be derived for bounded βn and the general condition that the error sequence tend to zero in norm. We refer the interested reader to the paper of Rockafellar [14] to see what happens in the case when αn = 0 for all n ≥ 0. The so called prox-Tikhonov regularization method have also been under investigation from several researchers. In 2006, Xu [19] extended the result of Lehdili and Moudafi [11] by considering the iterative process xn+1 = Jβn (αn u + (1 − αn )xn + en ), for any u, x0 ∈ H and all n ≥ 0, (6) were {en } is a sequence of errors, and proved strong convergence of {x n } defined by (6) to the metric projection of u into the fixed point set A−1 (0) under the control conditions which appear as a combination of αn and βn . More precisely, his conditions were    ∞    αn βn+1  1 αn βn+1  lim (C7) 1− = 0. 1 − α β  < ∞ or, (C8) n→∞ αn αn+1 βn n+1 n n=0 Note that for βn → ∞, the natural choices of αn = n −1 and βn = n, fails under both conditions. In fact, for any choice of αn and βn , condition (C7) is impossible to achieve as shall be shown in this paper, (see Remark 4). In another result of Xu, Theorem 3.3 [19], it is shown that for summable errors, strong convergence is still maintained under the conditions (C1), (C2), (C4), and βn bounded (from above and from below away from zero) with (C9) (as defined below) being satisfied. Song and Yang [15] established strong convergence of the prox-Tikhonov algorithm 6 when the errors are summable, (C1), (C2), (C4) being satisfied, 123 14 J Glob Optim (2011) 51:11–26 and the following condition on βn imposed: βn is bounded from below away from zero with either (C9) ∞  n=0 |βn+1 − βn | < ∞, or (C9)′ ∞  |βn+1 − βn | n=0 βn+1 < ∞. They remarked that their result (Theorem 2) contains Theorem 3.3 [19] as a special case. Although this seems to be the case at first glance, it turns out that the two theorems are equivalent. In fact, the condition (C9)′ on βn is equivalent to (C9) and βn bounded from below away from zero. Obviously, from this equivalence follows the equivalence of the two theorems. This equivalence is not so obvious and it is discussed in Lemma 4 below. The main purpose of this paper is to prove strong convergence of {x n } conforming to the iterative process xn+1 = αn u + (1 − αn )Jβn xn + en , for any u, x0 ∈ H and all n ≥ 0, (7) under new conditions on αn and βn . The conditions we are about to introduce will allow choices such as αn = n −1 and βn = n, and they are weaker than those previously studied, so our results can be viewed as significant improvements and refinements of previously known results. Theorem 5 deals with the conditions  ∞    αn−1 αn  1  (αn−1 βn+1 − αn βn ) = 0, lim either (C10)  < ∞ or, (C11) n→∞  β −β αn βn2 n n+1 n=1 and Theorem 6 is concerned with the conditions   1 1 − (C6)∗ lim = 0, and either n→∞ βn βn−1 ∞  |αn − αn−1 | (αn − αn−1 ) = 0 or, (C13) < ∞. (C12) lim n→∞ αn−1 βn βn n=1 In particular, our results provide an answer to the question we asked in [2]: Can one design a proximal point algorithm by choosing appropriate regularization parameters αn such that strong convergence of {xn } is preserved, for en  → 0 and βn bounded? Of course, for constant βn , (C10) reduces to (C4) and (C11) reduces to (C5). If A is the subdifferential of a proper, convex, lower semicontinuous function ϕ : H → (−∞, ∞], then our convergence results provide sequences which converge strongly to the minimum point of ϕ nearest to u. In addition, we give convergence rate estimates for a sequence converging to inf ϕ (see Theorem 7 of Sect. 4). The reader interested in theoretical and practical aspects of convex and non-convex optimization theory is referred to the recent excellent six-volume resource [6]. See also [5,9]. 2 Preliminaries In the sequel, H is a real Hilbert space, F denotes the set A−1 (0) = {x ∈ H : Jc x = x} = F(Jc ) for all c > 0, and given any sequence {x n }, its weak ω-limit set will be denoted by ωw ({xn }), that is,   ωw ({xn }) := x ∈ H | xn k ⇀ x for some subsequence {xn k } of {xn } . 123 J Glob Optim (2011) 51:11–26 15 Here “⇀” denotes weak convergence. Setting vn := xn − αn−1 u − en−1 , 1 − αn−1 (8) we see that (7) can be reformulated as vn+1 = Jβn (αn−1 u + (1 − αn−1 )vn + en−1 ), for n ≥ 1. (9) It is worth pointing out that, for αn → 0 and en → 0, the algorithms (7) and (9) are equivalent, that is, {vn } converges if and only if {xn } does. We shall therefore always use either form of the algorithm at our convenience. Obviously, (9) has the form (6), with αn−1 , βn , and en−1 instead of αn , βn , and en . If we consider (9) instead of (6), then conditions (C7) and (C8) take the form    ∞     αn−1 βn+1 1 − αn−1 βn+1  < ∞ or, (C8)′ lim 1 (C7)′ 1 − = 0.  n→∞ αn−1 αn βn  αn βn n=1 Theorem 4 and Remark 4 are concerned with these conditions. Let us now recall some Lemmas which will be useful in proving our main results. The first Lemma can be proved easily. Lemma 1 For all x, y ∈ H , we have x + y2 ≤ y2 + 2x, x + y. Lemma 2 (Resolvent Identity). For any β, γ > 0, and x ∈ H , the identity     γ γ Jβ x = Jγ x + 1− Jβ x β β holds true. Proof The proof of this Lemma is well known, but we provide it for the sake of completeness. Let β, γ > 0, and x ∈ H be arbitrary but fixed. Set y := Jβ x. Then using the definition of the resolvent, we have   γ γ y = Jβ x ⇔ y + β Ay ∋ x ⇔ y + γ Ay ∋ x + 1 − y ⇔ y β β     γ γ x + 1− y . = Jγ β β This completes the proof of the resolvent identity. ⊔ ⊓ Lemma 3 [18]. Let {sn } be a sequence of non-negative real numbers satisfying sn+1 ≤ (1 − an )sn + an bn + cn , n ≥ 0, where {an }, {bn }, {cn } satisfy the conditions: (i) {an } ⊂ [0, 1], with ∞ n=0 an = ∞, (ii) lim supn→∞ bn ≤ 0, and (iii) cn ≥ 0 for all n ≥ 0 with ∞ c < ∞. Then limn→∞ sn = 0. n=0 n We next show that any sequence of positive real numbers satisfying the condition of (C9)′ is bounded (with the lower bound being strictly positive). Lemma 4 For any sequence {bn } of positive real numbers, the following conditions are equivalent: (i) ∞ n=0 |bn+1 − bn | < ∞ and 0 < lim inf n→∞ bn (=lim n→∞ bn ), |bn+1 −bn | |bn+1 −bn | (ii) ∞ < ∞, and (iii) ∞ < ∞. n=0 n=0 bn bn+1 123 16 J Glob Optim (2011) 51:11–26 Proof First, it is easily seen that (i) ⇒ (ii), and (i) ⇒ (iii). Now let us prove that (ii) ⇒ (i). For this, it suffices to show that there exist constants m, M > 0 such that m ≤ bn ≤ M for all n = 0, 1, . . . From (ii), there exists a sequence {an } ⊂ R, such that ∞ n=0 |an | < ∞, and bn+1 bn+1 − bn = an ⇔ = 1 + an , n = 0, 1, . . . bn bn Note that in particular, limn→∞ an = 0. Therefore, we may assume without any loss of generality that |an | < 1 for all n. Then by simple induction, we have bn = b0 n−1 k=0 (1 + ak ). (10) Since 1 + x ≤ exp(x) for all x ≥ 0, it follows from (10) that bn = b0 n−1 n−1 k=0 (1 + ak ) ≤ k=0 (1 + |ak |) ≤ exp n−1  k=0 |ak | ≤ exp ∞  k=0 |ak | =: M0 < ∞. (11) On the other hand, ∞  k=0 |ak | < ∞ ⇔ ∞ k=0 (1 − |ak |) > 0, and again from (10) we obtain bn = b0 n−1 n−1 k=0 (1 + ak ) ≥ k=0 (1 − |ak |) ≥ ∞ k=0 (1 − |ak |) =: m 0 > 0. (12) The conclusion then follows from (11) and (12). Replacing bn by bn−1 in (ii), one readily gets (iii), showing that (iii) ⇒ (i) as desired. ⊔ ⊓ Let the mapping h : H → H be defined by x → tu + (1 − t)Jc x + e(t) for c > 0, u ∈ H and t ∈ (0, 1), where e = e(t) is a given function defined on (0, 1) with values in H . For any fixed t (and c, u), one can easily check that the map h is a contraction with Lipschitz constant 1 − t. The Banach contraction principle asserts that h has a unique fixed point, say, z t . That is, z t = tu + (1 − t)Jc z t + e(t) for c > 0 and u ∈ H. (13) In fact z t depends on u and c as well. Theorem 1 Take any c > 0 and u ∈ H , and assume t −1 e(t) → 0 as t → 0+ . (14) 0+ If F  = ∅, then {z t } defined in (13) converges strongly as t → to the point of F nearest to u, denoted by PF u. Moreover, this limit is attained uniformly with respect to c ≥ δ for every δ > 0. Proof For every p ∈ F, we have from Lemma 1 z t − p2 ≤ (1 − t)2 z t − p2 + 2tu − p + t −1 e(t), z t − p. 123 J Glob Optim (2011) 51:11–26 17 In other words, (2 − t)z t − p2 ≤ 2u − p + t −1 e(t), z t − p. This shows that {z t } is bounded as t → 0+ . (15) Now setting vt := (1 − t)−1 (z t − tu − e(t)) = Jc z t , we see that {vt } is also bounded as t → 0+ and the weak ω-limit sets of {z t } and {vt } (as t → 0+ ) coincide, that is, ωw ({z t }) = ωw ({vt }). Since Avt ∋ 1 (z t − vt ) → 0 as t → 0+ , c we have ωw ({z t }) ⊂ F. By (14) and (15) with p = PF u we get lim sup z t − PF u2 ≤ 0, t→0+ which shows that lim z t − PF u = 0. t→0+ Obviously, the above limit is attained uniformly with respect to c ≥ δ for every δ > 0. ⊔ ⊓ Remark 1 Theorem 1 is an extension of Theorem 3.1 in [19], since vt converges strongly to PF u (as t → 0+ ) if and only if z t does. We note that Theorem 3.1 in [19] contains a mistake, since the strong limit of vt (as t → 0+ ) is not attained uniformly for c > 0 (but for c ≥ δ for every δ > 0). 3 Main results We devote this section to demonstrate the strong convergence of algorithm 7 under different sets of assumptions on the parameters αn and βn . We begin by proving a strong convergence result satisfying similar conditions to those of Lions. One of the conditions (C3)′ lim n→∞ |αn+1 − αn | = 0, αn2 is weaker than Lions’ condition (C3) in the case when αn is decreasing. Theorem 2 Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let {xn } be the sequence generated by algorithm (7) with the conditions: (i) αn ∈ (0, 1), (C1), (C2) and (C3)′ , (ii) either ∞ n=0 en  < ∞ or en /αn → 0, and (iii) βn ∈ (0, ∞) with (C6)′ limn→∞ βn = β for some β > 0, being satisfied. Then {xn } converges strongly to PF u, the projection of u on F. Proof Note that it was shown in [19] that {xn } is bounded if ∞ n=0 en  < ∞. Also it was shown in [2] that {xn } is bounded if {en /αn } is bounded. For each n, let z n be the unique fixed point of the contraction x → αn u + (1 − αn )Jβ x. According to Theorem 1, z n → PF u as n → ∞. Therefore it is enough to show that xn − z n  → 0 as n → ∞. For this purpose, we estimate xn+1 − z n+1  as follows xn+1 − z n+1  ≤ xn+1 − z n  + z n − z n+1 . (16) 123 18 J Glob Optim (2011) 51:11–26 Noting that z n = αn u + (1 − αn )Jβ z n and the fact that Jβ is nonexpansive for all β > 0, we get xn+1 − z n  ≤ (1 − αn )Jβn xn − Jβ z n  + en  ≤ (1 − αn )Jβn xn − Jβn z n  + Jβn z n − Jβ z n  + en  |β − βn | ≤ (1 − αn )xn − z n  + z n − Jβ z n  + en  β |β − βn | ≤ (1 − αn )xn − z n  + αn u − Jβ z n  + en , β (17) where the third inequality follows from the application of the resolvent identity. On the other hand, we compare z n and z n+1 as follows z n − z n+1  = (αn − αn+1 )(u − Jβ z n+1 ) + (1 − αn )(Jβ z n − Jβ z n+1 ) ≤ |αn − αn+1 |u − Jβ z n+1  + (1 − αn )z n − z n+1 , which gives z n − z n+1  ≤ |αn − αn+1 | K, αn (18) where K is a positive constant such that u − Jβ z n  ≤ K for all n. Combining (16), (17) and (18) we get xn+1 − z n+1  ≤ (1 − αn )xn − z n  + αn bn + cn , where bn = K  ∞  |β − βn | |αn − αn+1 | → 0 and c en  < ∞, = e  with + n n β αn2 n=0 or xn+1 − z n+1  ≤ (1 − αn )xn − z n  + αn bn′ , where bn′  |β − βn | |αn − αn+1 | en  =K →0 + + β αn2 αn for the case en /αn → 0. In either case Lemma 3 gives the required conclusion. ⊔ ⊓ Remark 2 For β > 0 and βn = β + (−1)n /(n + 1), the condition (C6)′ is satisfied, whereas (C9) is not, showing that our condition on βn is weaker than the one used in the following theorem due to Xu [19]. On the other hand, the sequences αn = n −3/4 and αn = 1/ ln n satisfy condition (i) of Theorem 2. Since (C3) and (C3)′ are not comparable to (C4) (see Remark 3.1 [18]), Theorem 2 is new. Theorem 3 [19]. Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let {xn } be the sequence generated by algorithm (7) with the conditions: (i) αn ∈ (0, 1), (C1), (C2) and (C4), (ii) βn ∈ (0, ∞) with ∞ n=0 |βn+1 − βn | < ∞ and 0 < lim inf n→∞ βn (= lim n→∞ βn ), being satisfied. If ∞ n=0 en  < ∞, then {x n } converges strongly to PF u, the projection of u on F. 123 J Glob Optim (2011) 51:11–26 19 Remark 3 Although it appears from Lemma 3 and inequality (18) that ∞  |αn − αn+1 | <∞ αn n=0 can be a possible assumption on αn , there is no sequence {αn } ⊂ (0, 1) satisfying (C1) and this condition. Indeed, if this condition is satisfied, then Lemma 4 implies that αn is bounded below away from zero, contradicting (C1). We next give a result similar to Theorem 3.2 of Xu [19]. In the next result, if we consider algorithm 6 instead of algorithm 7, then we can prove the same result with (C8)′ being replaced by (C8). In that case, the result extends Theorem 3.2 [19] to a larger class of errors which include those that are non-summable and still converge to zero in norm. Moreover, we can show that Theorem 3.2 [19] fails to hold under the condition (C7). Theorem 4 Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let {xn } be the sequence generated by algorithm (7), where (i) αn ∈ (0, 1), with (C1), and (C2), (ii) either ∞ n=0 en  < ∞ or en /αn → 0, and (iii) βn ∈ (0, ∞) with lim inf n→∞ βn > 0, βn+1 ≥ αn βn and (C8)′ . Then {xn } converges strongly to PF u, the projection of u on F. Proof For each fixed n, let yn be the unique fixed point of the contraction x → αn−1 u + (1 − αn−1 )Jβn x. Then according to Theorem 1, yn → PF u as n → ∞. Set vn := xn − αn−1 u − en−1 yn − αn−1 u and wn := . 1 − αn−1 1 − αn−1 (19) As a consequence of the boundedness of {xn } and {yn } (see [2] and [19]), the sequences {vn } and {wn } are bounded. Also by virtue of (19), wn → PF u as n → ∞. It follows from (7) and the definition of yn that vn+1 = Jβn ((1 − αn−1 )vn + αn−1 u + en−1 ) and wn = Jβn ((1 − αn−1 )wn + αn−1 u). As before, using the nonexpansivity of the resolvent, we estimate vn+1 − wn+1  as follows vn+1 − wn+1  ≤ vn+1 − wn  + wn+1 − wn  ≤ (1 − αn−1 )vn − wn  + wn+1 − wn  + en−1 . (20) Now using the resolvent identity and the nonexpansivity of the resolvent, we can estimate wn+1 − wn  as follows       βn βn J ((1 − α )w + α u) + 1 − w wn+1 − wn  =  n n+1 n n+1 β n  βn+1 βn+1   −Jβn ((1 − αn−1 )wn + αn−1 u)       αn βn αn βn  ≤ 1− K, wn+1 − wn  + αn−1 − βn+1 βn+1  which gives    αn−1 βn+1  wn+1 − wn  ≤ 1 − K, αn βn  (21) 123 20 J Glob Optim (2011) 51:11–26 for some positive constant K . Combining (20) and (21) we get    αn−1 βn+1   K + en−1 . vn+1 − wn+1  ≤ (1 − αn−1 )vn − wn  + 1 − αn βn  (22) Hence from Lemma 3, we see that vn − wn  → 0, and the proof is complete. ⊔ ⊓ Remark 4 In view of Lemma 3 and (22), it is tempting to infer that the theorem is still valid under the condition (C7)′ . However we show that this condition is impossible to attain for any sequences {βn } and {αn } satisfying the conditions of the above theorem. To this end, we assume that (C7)′ holds true. Denote bn := αn−1 /βn . Then  ∞  ∞     |bn+1 − bn | 1 − αn−1 βn+1  < ∞ ⇔ < ∞.  αn βn  bn+1 n=1 n=1 Therefore, it follows from Lemma 4 that αn−1 lim inf = lim inf bn > 0, n→∞ n→∞ βn which implies that βn → 0 (since αn → 0). This is a contradiction as βn is bounded below away from zero. However, if we allow βn → 0, then Theorem 1 is no longer applicable. Indeed, from wn = Jβn ((1 − αn−1 )wn + αn−1 u), we have αn−1 (23) (u − wn ) ∈ Awn . βn From the above inclusion relation, we can not derive ωw ({wn }) ⊂ F := A−1 (0), even if wn is strongly convergent (since by (21), ∞ n=1 wn+1 − wn  < ∞) because αn−1 /βn may not necessarily converge to zero. Therefore, in this case {x n } is still strongly convergent (according to (22)) but we can not derive that its limit is in F. In fact, its limit need not be in F. We give an example to that effect. Example 1 Let βn = 1/n and αn = 1/(n + 2) for n ≥ 1. Then we have 1− 1 βn+1 αn−1 βn+1 = =: an , for all n ≥ 1, and → 1 as n → ∞. αn βn (n + 1)2 αn Clearly the condition βn+1 ≥ αn βn for all n ≥ 1 is fulfilled. Let H = R, and let the sequence {en } ⊂ R satisfy either the condition ∞ n=0 |en | < ∞ or |en |/αn → 0, (for example, |en | = (n + 2)−2 or |en | = 1/(n ln n) for n ≥ 2 with ∞ n=2 |en | = ∞, respectively), and let A : D(A) = [0, ∞) ⊂ R → R be defined by ⎧ if x > 0, ⎨ ax, Ax = (−∞, 0], if x = 0, ⎩ ∅, if x < 0, for some a > 0. Then if u > 0, we have for sufficiently large n, αn−1 u + en−1 > 0 and 0 < wn = Jβn ((1 − αn−1 )wn + αn−1 u + en−1 ) 1 = ((1 − αn−1 )wn + αn−1 u + en−1 ), 1 + βn a which implies that wn → w∞ := conclusion is true if u < 0. 123 1 1+a u ∈ / F = {0}. Hence xn → w∞ ∈ / F. The same J Glob Optim (2011) 51:11–26 21 The argument given above shows that if βn is bounded away from zero in Theorem 3.2 of [19], then the condition (C7) is impossible to achieve. Also the above example shows that the result may not hold if βn → 0. We now give an example to show the applicability of Theorem 4. Example 2 Choose βn = β0 > 0 for all n, αn = (n + 1)−1/2 and en  = 1/(n + 1) for all n ≥ 0. Theorem 5 Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let {xn } be the sequence generated by algorithm (7) where conditions (i) and (ii) of Theorem 4 are fulfilled. If βn ∈ (0, ∞) is increasing and either (C10) or (C11) is satisfied, then {xn } converges strongly to PF u, the projection of u on F. Proof We know that {xn } (and hence {vn }) is bounded, see Theorem 2. Claim: lim supn→∞ u − PF u, xn − PF u ≤ 0. Let {xn k } be a subsequence of {xn } converging weakly to some x∞ , such that lim supu − PF u, xn − PF u = lim u − PF u, xn k − PF u = u − PF u, x∞ − PF u. n→∞ k→∞ To prove the claim, we only need to show that x ∞ ∈ F, or more generally ωw ({xn }) ⊂ F. If βn is unbounded, then the conclusion follows from the inclusion relation 1 αn−1 vn+1 − vn + A(vn+1 ) ∋ en−1 . (u − vn ) + βn βn βn (24) Otherwise, from Eq. 9, the boundedness of {en /αn } and {vn }, the nonexpansivity of Jβn and taking advantage of the resolvent identity, we can compare vn+2 and vn+1 as follows       n n vn+2 ((1 − αn )vn+1 + αn u + en ) + 1 − ββn+1 vn+2 − vn+1  = Jβn ββn+1   −Jβn ((1 − αn−1 )vn + αn−1 u + en−1 )      βn n (vn+2 − vn+1 ) + 1 − αβnn+1 (vn+1 − vn ) ≤  1 − ββn+1     en−1  βn αn βn en + αn−1 − αβnn+1 (vn − u − αen−1 ) + − βn+1 αn αn−1  n−1     βn n vn+2 − vn+1  + 1 − αβnn+1 vn+1 − vn  ≤ 1 − ββn+1      αn βn   βn   en − en−1  , + αn−1 − αβnn+1 K +   β α α n+1 n n−1 which implies that     vn+2 − vn+1  αn αn βn vn+1 − vn  αn−1 αn  K+ ≤ 1− + − βn+1 βn+1 βn βn βn+1 βn+1     αn β0 vn+1 − vn  αn−1 αn  αn ≤ 1− + − K+ βn+1 βn βn βn+1 βn+1    en   − en−1  α  α n n−1    en en−1   . − α αn−1  n 123 22 J Glob Optim (2011) 51:11–26 ∞ n=0 en  < ∞, we have      βn n (vn+2 − vn+1 ) + 1 − αβnn+1 (vn+1 − vn ) vn+2 − vn+1  ≤  1 − ββn+1      βn n + αn−1 − αβnn+1 (vn − u) + ββn+1 en − en−1      βn n ≤ 1 − ββn+1 vn+2 − vn+1  + 1 − αβnn+1 vn+1 − vn       βn   βn  ′ en − en−1  , + αn−1 − αβnn+1  K +  βn+1 Similarly, for the case which implies that     vn+2 − vn+1  αn β0 vn+1 − vn   αn−1 αn  ′ 1 ≤ 1− + − K + (en  + en−1 ) . βn+1 βn+1 βn βn βn+1  β0 Denote an := αn β0 /βn+1 . Since {αn } satisfy αn ∈ (0, 1), αn → 0 and do {an }. Therefore, from Lemma 3, we have (in both cases) ∞ n=0 αn = ∞, so vn+1 − vn  → 0 ⇔ vn+1 − vn  → 0. βn Moreover, (24) implies that ωw ({vn }) ⊂ F, and from (8), we derive ωw ({vn }) = ωw ({xn }), hence the claim. Finally we show that {xn } converges strongly to PF u. We have from Lemma 1   en , xn+1 − PF u . (25) xn+1 − PF u2 ≤ (1 − αn )xn − PF u2 + 2αn u − PF u + αn In the case when en /αn → 0, inequality (25) implies by Lemma 3 that xn → PF u. If ∞ n=0 en  < ∞, then we derive from inequality (25) xn+1 − PF u2 ≤ (1 − αn )xn − PF u2 + 2αn u − PF u, xn+1 − PF u + K en , for some K > 0, and Lemma 3 again implies that xn → PF u as desired. ⊔ ⊓ Remark 5 The condition (C10) is weaker than the conditions (C4) and (C9) if βn ≥ δ for all n and for some δ > 0. Indeed,     1  αn−1 1 αn  1    ≤ − |α − α | + α − n−1 n n  β βn+1  βn βn βn+1  n  |βn+1 − βn | 1 ≤ |αn−1 − αn | + . δ δ Note that if βn = n 2 for n ≥ 1, then (C10) holds true for any choice of αn ∈ (0, 1). Remark 6 Observe that (C11) is satisfied for βn = n and αn = (n + 1)−1 , whereas the condition (C8)′ of Theorem 3 fails. Moreover, (C11) works if βn is constant and αn taken as before but (C8)′ fails. Although the condition (C10) is weaker than (C4) and (C9) if lim inf n→∞ βn > 0, our result is restricted only to those βn ’s which are increasing. The next result is designed to cater for those βn ’s who does not satisfy this restrictive condition. It is actually an extension and improvement of Theorem 3 above. Our proof differs from those given in [15] and [19], and it relies on the equivalence of the algorithms 6 and 7. Note that it was observed in [15] that a gap exists in the proof of Theorem 3. We remark here that our method of transforming 123 J Glob Optim (2011) 51:11–26 23 Eq. 9 into Eq. 7 is an alternative way of solving this gap as can be seen from the proof of Theorem 6 below. Theorem 6 Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let the sequence {xn } be generated by algorithm (7) with the following conditions being satisfied: (i) αn ∈ (0, 1), (C1), (C2), (ii) either ∞ n=0 en  < ∞ or en /αn → 0, (iii) lim inf n→∞ βn > 0, and (C6)*. If either (C12) or (C13) hold, then {xn } (and hence {vn }) converges strongly to PF u, the projection of u on F. Proof We know from [2] and [19] that {xn } is bounded. For en /αn → 0, we have, (by the resolvent identity and the nonexpansivity of the resolvent),        n n Jβn−1 xn−1  xn−1 + 1 − ββn−1 xn+1 − xn  ≤ (1 − αn−1 ) Jβn xn − Jβn ββn−1 +|αn − αn−1 | · u − Jβn xn + en /αn  + αn−1 en /αn − en−1 /αn−1        n n (xn − Jβn−1 xn−1 ) ≤ (1 − αn−1 )  ββn−1 (xn − xn−1 ) + 1 − ββn−1 +|αn − αn−1 | · u − Jβn xn + en /αn  + αn−1 en /αn − en−1 /αn−1  so that n xn − xn−1  + |αn − αn−1 | · u − Jβn xn + en /αn  ≤ (1 − αn−1 ) ββn−1         en−1   n   + αn−1  en − en−1  , +αn−1 1 − ββn−1  u − Jβn−1 xn−1 +   αn−1 αn αn−1    1 xn+1 − xn  xn − xn−1  1  αn−1  ≤ (1 − αn−1 ) + αn−1  − K+ βn βn−1 βn βn−1  βn |αn − αn−1 | +K , βn    en en−1    α − α  n n−1 (26) for some positive constant K . From Lemma 3 and inequality (26), we have xn+1 − xn  → 0, βn which is equivalent to vn+1 − vn  → 0. βn Hence we can derive (see (24) above), ωw ({xn }) = ωw ({vn }) ⊂ F. Consequently, we have lim supu − PF u, xn − PF u ≤ 0. (27) n→∞ −1 Note that for some positive constant C, |βn+1 − βn−1 | ≤ C (since lim inf n→∞ βn > 0) and xn+1 − xn = (αn − αn−1 )(u − Jβn xn ) + (en − en−1 ) + (1 − αn−1 )(Jβn xn − Jβn−1 xn−1 ), so that in the case when ∞ n=1 en  < ∞, we again get inequality (27) on applying similar arguments as above. As in the proof of Theorem 5, we derive strong convergence of {xn } to PF u. ⊔ ⊓ Remark 7 Clearly (C6)∗ is weaker than the conditions    ∞   1 1 1  1 1  − − < ∞ and (C15) lim (C14) = 0, β n→∞ αn βn−1  βn βn−1 n n=1 123 24 J Glob Optim (2011) 51:11–26 both of which hold true if αn = n −1 and βn = n while (C6) fails for this choice of βn . However, both (C6) and (C6)∗ hold if βn = ln n. We point out that the first inequality in Remark 5 suggest that for lim inf n→∞ βn > 0, the condition (C10) is weaker than (C4) and (C14). Also, the condition (C14) is weaker than (C9) whenever lim inf n→∞ βn > 0 holds. But the condition that βn is increasing is stronger than the assumption lim inf n→∞ βn > 0, so there are cases in which the following corollary is applicable and Theorem 5 is not. We remark that both (C4) and (C5) are not satisfied by  1/n, if n is odd, αn = 1/(2n), if n is even. This choice of αn however fulfills the assumptions of (C13) for the case when βn = n and (C12) for any βn → ∞. Corollary 1 Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let the sequence {xn } be generated by algorithm (7), where αn ∈ (0, 1) and βn ∈ (0, ∞), with the conditions (i) and (ii) taken as in Theorem 6, and lim inf n→∞ βn > 0 with either (C14) or (C15). If either (C12) or (C13) hold, then {xn } (and hence {vn }) converges strongly to PF u. The following corollary is an extension of Theorem 3. Corollary 2 Assume that A : D(A) ⊂ H → 2 H is a maximal monotone operator and F := A−1 (0)  = ∅. For any fixed u, x0 ∈ H , let the sequence {xn } be generated by algorithm 7, where αn ∈ (0, 1) and βn ∈ (0, ∞), with the conditions (i) and (ii)taken as in  Theβn 1 orem 6, and (iii) lim inf n→∞ βn > 0 and either (C9) or (C16) limn→∞ αn 1 − βn+1 = 0. If either (C12) or (C13) hold, then {xn } (hence {vn }) converges strongly to PF u. We give an example to show that the conditions of (iii) are different. Example 3 Let αn = (n + 2)−1/4 and βn = 2(n + 1)(n + 2)−1 for all n ≥ 0. Then αn and βn satisfy both conditions of (iii) while βn = (n + 1) and αn as above satisfy only (C16). Remark 8 Let us observe that if en /αn → 0 and ∞ n=0 en  = ∞, then automatically ∞ n=0 αn = ∞. Also the trend that has been followed by many authors in order to obtain strong convergence of the PPA was to use the criterion which restricts the error sequence to be summable. We have deviated from this tradition by allowing any sequence of errors converging strongly to zero and still derived strong convergence of the PPA. Indeed, if ∞ n=0 en  = ∞ and en  → 0, then we can construct (or choose) a sequence {αn } of parameters √ depending on {en } such that the condition en /αn → 0 holds (for example αn = en  if en  = 0 and all n big enough). Otherwise, (i.e., if ∞ n=0 en  < ∞), we can choose freely (independent of en ) αn ∈ (0, 1) such that the conditions αn → 0 and ∞ n=0 αn = ∞ are satisfied. 4 The case when A is a subdifferential Recall that the subdifferential of a proper and convex function ϕ : H → (−∞, +∞] is the operator (possibly multivalued) ∂ϕ : H → H defined by ∂ϕ(x) = {w ∈ H | ϕ(x) − ϕ(v) ≤ w, x − v, ∀ v ∈ H }. 123 J Glob Optim (2011) 51:11–26 25 If in addition, ϕ is lower semicontinuous, then its subdifferential is a maximal monotone operator and a point p ∈ H minimizes ϕ if and only if 0 ∈ ∂ϕ( p). In other words, A−1 (0) for A = ∂ϕ is the set of minimum points of ϕ. Note that for A = ∂ϕ, where ϕ is a proper, convex and lower semicontinuous function, algorithm (9) is equivalent to vn+1 = arg minx∈H ϕn (x), where ϕn (x) = ϕ(x) + 1 x − αn−1 u − (1 − αn−1 )vn − en−1 2 . 2βn Obviously, ϕn is a coercive function having a unique minimizer vn+1 due to the quadratic term added to ϕ(x). Under the assumptions of the previously proved results, {vn } (equivalently, {xn }) converges strongly to the minimizer of ϕ nearest to u. We now give two convergence rate estimates for the residual ϕ(wk+1 ) − ϕ(z) where ϕ is a proper, convex and lower semicontinuous function and z is an arbitrary point of H , and wn = σn−1 n  k=1 βk vk+1 where σn = n  βk . (28) k=1 In general, if a sequence {vn } converges strongly (resp. weakly) to a point, say p, then the sequence of its weighted means with positive weights {βk } defined by (28) also converges strongly (resp. weakly) to the same limit p, provided σn → ∞. Theorem 7 Let A = ∂ϕ and A−1 (0)  = ∅ where ϕ : H → (−∞, +∞] is a proper, convex and lower semicontinuous function. For any fixed u, v1 ∈ H , let {vn } be the sequence generated by algorithm (9) and {wn } be as in (28). • If ∞ k=1 ek−1  < ∞, then for some K > 0, the following estimate holds  n  n v1 − z2 + K k=1 αk−1 + k=1 ek−1  ϕ(wn ) − ϕ(z) ≤ , for all z ∈ H. (29) 2σn • If {en /αn } is bounded, then for some M > 0, we have ϕ(wn ) − ϕ(z) ≤ If in addition, σn−1 n k=1 αk−1 v1 − z2 + M 2σn n k=1 αk−1 , for all z ∈ H. (30) → 0 as n → ∞, then ϕ(wn ) → inf y∈H ϕ(y). Proof Let us prove estimate (30). Note that for A = ∂ϕ, we have from (9), αk−1 (u − vk ) + ek−1 + (vk − vk+1 ) ∈ βk ∂ϕ(vk+1 ), and for all z ∈ H , we have from the boundedness of {ek /αk } and {vk } (see the proof of Theorem 1 [2]), 2βk (ϕ(vk+1 ) − ϕ(z)) ≤ 2vk − vk+1 , vk+1 − z + 2αk−1 u − vk + ek−1 /αk−1 , vk+1 − z   ≤ vk − z2 − vk+1 − vk 2 − vk+1 − z2 + Mαk−1 , (31) for some M > 0. Summing (31) from k = 1, . . . , n and rearranging terms, we get  n  n v1 − z2 + M nk=1 αk−1 βk ϕ(vk+1 ) k=1 βk vk+1 2ϕ(z) + ≥2 k=1 ≥2ϕ . (32) σn σn σn Therefore (30) follows from (32). The proof of the other estimate is similar. The final assertion of the theorem is obvious. ⊔ ⊓ 123 26 Acknowledgments J Glob Optim (2011) 51:11–26 The authors thank Alexandru Kristály for his insightful comments on this paper. References 1. Boikanyo, O.A., Moroşanu, G.: Modified Rockafellar’s algorithms. Math. Sci. Res. J. 13(5), 101– 122 (2009) 2. Boikanyo, O.A., Moroşanu, G.: A proximal point algorithm converging strongly for general errors. Optim. 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