Regularity and Stability for a Convex Feasibility Problem

https://doi.org/10.1007/s11228-021-00602-3

Regularity and Stability for a Convex Feasibility Problem

Carlo Alberto De Bernardi¹ ·Enrico Miglierina¹

Received: 23 July 2020 / Accepted: 31 July 2021 /

©The Author(s) 2021

Abstract

Let us consider two sequences of closed convex sets{A_n}and{B_n}converging with respect to the Attouch-Wets convergence toAandB, respectively. Given a starting pointa₀, we consider the sequences of points obtained by projecting onto the “perturbed” sets, i.e., the sequences{an}and{bn}defined inductively bybn =PB_n(an−1)andan =PA_n(bn). Sup- pose thatA∩Bis bounded, we prove that if the couple(A, B)is (boundedly) regular then the couple(A, B)isd-stable, i.e., for each{a_n}and{b_n}as above we have dist(a_n, A∩B)→0 and dist(bn, A∩B)→0. Similar results are obtained also in the caseA∩B= ∅, considering the set of best approximation pairs instead ofA∩B.

Keywords Convex feasibility problem·Stability·Regularity·Set-convergence· Alternating projections method

Mathematics Subject Classification (2010) Primary: 47J25; secondary: 90C25·90C48

1 Introduction

LetAandBbe two closed convex nonempty sets in a Hilbert spaceX. The (2-set) convex feasibility problem asks to find a point in the intersection ofAandB(or, whenA∩B= ∅, a pair of points, one inA and the other inB, that realizes the distance between A and B). The relevance of this problem is due to the fact that many mathematical and concrete problems in applications can be formulated as a convex feasibility problem. As typical examples, we mention solution of convex inequalities, partial differential equations, mini- mization of convex nonsmooth functions, medical imaging, computerized tomography and image reconstruction.

Carlo Alberto De Bernardi

carloalberto.debernardi@unicatt.it; carloalberto.debernardi@gmail.com Enrico Miglierina

enrico.miglierina@unicatt.it

1 Dipartimento di Matematica per le Scienze economiche, finanziarie ed attuariali, Universit`a Cattolica del Sacro Cuore, Via Necchi 9, 20123 Milano, Italy

The method of alternating projections is the simplest iterative procedure for finding a solution of the convex feasibility problem and it goes back to von Neumann [15]: let us denote byPA andPB the projections onto the setsA andB, respectively, and, given a starting pointc0∈X, consider thealternating projections sequences{c_n}and{d_n}given by

dn=PB(cn−1) and cn=PA(dn) (n∈N).

If the sequences {cn}and{dn} converge in norm, we say that the method of alternating projections converges. Originally, von Neumann proved that the method of alternating pro- jection converges whenAandBare closed subspaces. Then, for two generic convex sets, the weak convergence of the alternating projection sequences was proved by Bregman in 1965 [5]. Nevertheless, the problem of whether the alternating projections algorithm con- verges in norm for each couple of convex sets remained open till the example given by Hundal in 2004 [12]. This example shows that the alternating projections do not converge in norm whenAis a suitable convex cone andBis a hyperplane touching the vertex ofA.

Moreover, this example emphasizes the importance of finding sufficient conditions ensuring the norm convergence of the alternating projections algorithm. In the literature, conditions of this type were studied (see, e.g., [1,3]), even before the example by Hundal. Here, we focus on those conditions based on the notions of regularity, introduced in [1]. Indeed, in the present paper, we investigate the relationships between regularity of the couple(A, B) (see Definition 3.1 below) and “stability” properties of the alternating projections method in the following sense. Let us suppose that{A_n}and{B_n}are two sequences of closed con- vex sets such thatA_n → AandB_n → B for the Attouch-Wets variational convergence (see Definition 2.2) and let us introduce the definition ofperturbed alternating projections sequences.

Definition 1.1 Givena₀ ∈ X, theperturbed alternating projections sequences{a_n}and {bn}, w.r.t.{An}and{Bn}and with starting pointa0, are defined inductively by

b_n=P_Bn(a_n−1) and a_n=P_An(b_n) (n∈N).

Our aim is to find some conditions on the limit setsAandBsuch that, for each choice of the sequences{A_n}and{B_n}and for each choice of the starting pointa₀, the correspond- ing perturbed alternating projections sequences{a_n}and{b_n}satisfy dist(a_n, A∩B)→0 and dist(b_n, A∩B)→0. If this is the case, we say that the couple(A, B)isd-stable. In particular, we show that the regularity of the couple(A, B)implies not only the norm con- vergence of the alternating projections sequences for the couple(A, B)(as already known from [1]), but also that the couple(A, B)isd-stable. This result might be interesting also in applications since real data are often affected by some uncertainties. Hence stability of the convex feasibility problem with respect to data perturbations is a desirable property, also in view of computational developments.

Let us conclude the introduction by a brief description of the structure of the paper. In Section2, we list some notations and definitions, and we recall some well-known facts about the alternating projections method. Section3is devoted to various notions of regularity and their relationships. It is worth pointing out that in this section we provide a new and alternative proof of the convergence of the alternating projections algorithm under regularity assumptions. This proof well illustrates the main geometrical idea behind the proof of our main result Theorem 4.9, stated and proved in Section4. This result shows thata regular couple(A, B)isd-stable wheneverA∩B(or a suitable substitute ifA∩B= ∅) is bounded.

Corollaries 4.16, 4.18, and 4.19 simplify and generalize some of the results obtained in [9],

since there we considered only the case whereA∩B= ∅whereas, in the present paper, we encompass also the situation where the intersection ofAandBis empty. We conclude the paper with Section5, where we discuss the necessity of the assumptions of our main result and we state a natural open problem: suppose thatA∩Bis bounded, is regularity equivalent tod-stability?

2 Notation and Preliminaries

Throughout all this paper,Xdenotes a nontrivial real normed space with the topological dualX^∗. We denote byBXandSXthe closed unit ball and the unit sphere ofX, respectively.

Ifα >0,x∈X, andA, B⊂X, we denote as usual

x+A:= {x+a;a∈A}, αA:= {αa;a∈A}, A+B:= {a+b;a∈A, b∈B}. Forx, y∈X,[x, y]denotes the closed segment inXwith endpointsxandy, and(x, y)= [x, y] \ {x, y}is the corresponding “open” segment. For a subsetA ofX, we denote by int(A), conv(A)and conv(A)the interior, the convex hull and the closed convex hull ofA, respectively. Let us recall that a body is a closed convex set inXwith nonempty interior.

We denote by

diam(A):=sup_x,y∈A x−y , the (possibly infinite) diameter ofA. Forx∈X, let

dist(x, A):= inf

a∈A a−x .

Moreover, givenA, Bnonempty subsets ofX, we denote by dist(A, B)the usual “distance”

betweenAandB, that is,

dist(A, B):= inf

a∈Adist(a, B).

Now, we recall two notions of convergence for sequences of sets (for a wide overview about this topic see, e.g., [2]). By c(X)we denote the family of all nonempty closed subsets ofX. Let us introduce the (extended) Hausdorff metrichon c(X). ForA, B ∈c(X), we define the excess ofAoverBas

e(A, B):=sup

a∈A

dist(a, B).

Moreover, ifA= ∅andB= ∅we pute(A, B)= ∞, ifA= ∅we pute(A, B)=0. Then, we define

h(A, B):=max

e(A, B), e(B, A) .

Definition 2.1 A sequence{Aj}in c(X)is said to Hausdorff converge toA∈c(X)if lim_jh(A_j, A)=0.

As the second notion of convergence, we consider the so called Attouch-Wets con- vergence (see, e.g., [14, Definition 8.2.13]), which can be seen as a localization of the Hausdorff convergence. IfN∈NandA, B∈c(X), define

eN(A, B) := e(A∩N BX, B)∈ [0,∞), h_N(A, B) := max{e_N(A, B), e_N(B, A)}.

Definition 2.2 A sequence{A_j}in c(X)is said to Attouch-Wets converge toA∈c(X)if, for eachN∈N,

limjhN(Aj, A)=0.

In the last section of our paper we shall need the following elementary fact, related to localized Hausdorff distance between two sets.

Fact 2.3 LetD:X→Xbe a bounded linear operator. Letε, δ∈(0,1)and suppose that D−I ≤εandb∈δB_X. IfA⊂XandN∈Nthen

hN

A, b+D(A)

≤δ+N+δ 1−εε.

Proof Ifx∈A∩N BXthen

d(x, b+D(A)) = inf

y∈D(A) b+y−x

≤ b+D(x)−x

≤ b + D−I x ≤δ+N ε.

Therefore, we conclude that

eN(A, b+D(A))=sup_{x∈A∩N B}

Xd(x, b+D(A))≤δ+N ε≤δ+^N+δ_1−εε.

Now, suppose thatx∈Aandy=b+D(x)∈ [b+D(A)] ∩N B_X, then x ≤ x−D(x) + y + b ≤ε x +N+δ, and hence x ≤^N_1−ε^+δ. Proceeding as above, we have

eN(b+D(A), A)≤δ+N+δ 1−ε ε.

The notions of distance between two convex sets and of projection of a point onto a convex set of a Hilbert space play a fundamental role in our paper. Unless otherwise stated, from now on,Xwill denote an Hilbert spaceendowed with the inner product·,·. The projection onto a closed convex nonempty subsetCsends any pointx₀ ∈Xto its nearest point inC, denoted byP_C(x₀). We shall frequently use in the paper the following result, usually calledvariational characterization of the projection ontoC. Letc₀∈Candx₀∈X, thenc0=PC(x0)if and only if

x₀−c₀, c−c₀ ≤0, wheneverc∈C. (1) We recall that theangleang(u, v) between two nonnull vectorsu, v ∈ X is defined by means of the equality

ang(u, v):=arccos _u,v

u v

.

In the sequel of the paper, we denote thecosineof the angle between two nonnull vectors u, v∈Xas

cos(u, v):=cos(ang(u, v))= _{u v} ^u,v . It is clear that, ifx0∈/C, (1) is equivalent to the following condition:

π

2 ≤ang(x₀−c₀, c−c₀)≤π, wheneverc∈C\ {c₀}.

Finally, we recall that the projectionP_C is a nonexpansive map fromXtoC, i.e., it holds PC(x)−PC(y) ≤ x−y (see, e.g., [14, Proposition 10.4.8]). Now, let us consider two closed convex nonempty subsetsAandBofX, we denote by

E := {a∈A;d(a, B)=d(A, B)}, F := {b∈B;d(b, B)=d(A, B)}.

We say thatv :=P_B−A(0)is thedisplacement vectorfor the couple(A, B). It is clear that ifA∩B= ∅thenE=F =A∩Band the displacement vector for the couple(A, B) is null. We recall the following fact, where, given a mapT :X→X, Fix(T )denotes the set of all fixed points ofT.

Fact 2.4([1, Fact 1.1]) Suppose thatXis a Hilbert space and thatA, Bare closed convex nonempty subsets ofX. Then we have:

(i) v =dist(A, B)andE+v=F;

(ii) E=Fix(PAPB)=A∩(B−v)andF =Fix(PBPA)=B∩(A+v);

(iii) PBe=PFe=e+v(e∈E) andPAf =PEf =f−v(f ∈F).

We conclude this section by proving a relationship between the Attouch-Wets conver- gence of a sequence{An}of closed convex sets and the convergence of the sequence{PAn} of projections ontoAn(see Lemma 2.6). This results is probably known but we were not able to find any reference in the literature, hence we provide a detailed proof for the sake of completeness. In order to prove this result we need to prove a preliminary lemma based on a geometrical property of the unit ball that holds in every Hilbert space. This property, called uniform rotundity, can be seen as a strengthening of the convexity of the unit ball and it is widely studied in the framework of the geometry of Banach space (see, e.g., [11]) Let us recall that, given a normed spaceZ, themodulus of convexity ofZis the function δ_Z: [0,2] → [0,1]defined by

δ_Z(η)=inf

1− x+y

2

:x, y∈B_X, x−y ≥η

.

It is clear thatδZ(η1) ≤ δZ(η2), whenever 0 ≤ η1 ≤ η2 ≤ 2. Moreover, ifr > 0 and η∈ [0,2], by recalling the positive homogeneity of the norm, we have

rδZ_η

r

=inf

r− x+y

2

:x, y∈rBX, x−y ≥η

. In particular, ifr, M >0 andx, y∈rB_Xare such that x−y ≥Mthen we have

x+y 2

≤r

1−δ_ZM

r

. (2) We say thatZisuniformly rotundifδ_Z(η) >0, wheneverη∈(0,2]. It is well known (see, e.g., [11]) that Hilbert spaces are uniformly rotund, but there are uniformly rotund spaces that are not Hilbert spaces. Therefore, it is worth to state and prove the following lemma in this general framework. Moreover, this result, roughly speaking, says that if a convex set in a uniformly rotund space is contained in a sufficiently tight annulus between two spheres, then its diameter is as small as we want.

Lemma 2.5 LetZbe a uniformly rotund normed space. LetH, K, M >0, then there exists ε∈(0, H )such that, ifρ∈ [0, K]and ifCis a convex set such thatρ−ε≤ c ≤ρ+ε, wheneverc∈C, thendiam(C)≤M.

Proof Suppose without any loss of generality thatM ≤2 andH ≤1. We claim that any ε∈(0, H )such thatε

2−δ_Z

K+1M

< ^M₄δ_Z

K+1M

works. Letρ∈ [0, K]and letC be a convex set such thatρ−ε≤ c ≤ρ+ε, wheneverc∈C. First, observe that, since δ_Zassumes values in[0,1], we haveε< ^M₄. Hence, ifρ < ^M₄, we have

diam(C)≤2(ρ+ε)≤M.

Now, suppose thatρ ≥ ^M₄ and let us prove that diam(C) ≤ M. Suppose on the contrary that there existc₁, c₂ ∈Csatisfying c₁−c₂ > M. Putr := ρ+ε. By (2) and since

c1+c2

2 ∈C, we have ρ−ε≤

c1+c2

2

≤r

1−δ_ZM

r

≤r 1−δ_Z

M K+1

. Therefore, we have ε

2−δZ

M K+1

≥ ρδZ

M K+1

≥ ^M₄δZ

M K+1

, against the definition ofε.

We are now in position to state and prove the result that links convergence of setsA_n with that of projections ontoAn.

Lemma 2.6 LetXbe a Hilbert space. Suppose that a sequence{A_n}inc(X)Attouch-Wets converges toA ∈ c(X). Then the corresponding sequence of projections{P_An}uniformly converges on bounded set toPA.

Proof Without any loss of generality we can suppose that 0∈A. Let us prove that, for each K, M >0, there existsn₀∈Nsuch that

x∈KBsupX

PA_nx−PAx ≤M,

whenevern≥n0. By Lemma 2.5 (where we takeH=K), there existsε∈(0, K)such that, ifρ∈ [0, K]and ifCis a convex set such thatρ−ε≤ c ≤ρ+ε, wheneverc∈C, then diam(C)≤M. Since{A_n}Attouch-Wets converges toA, there existsn₀∈Nsuch that, for n≥n₀, we have

(i) A_n∩3KB_X⊂A+εB_X; (ii) A∩3KB_X⊂A_n+εB_X;

Letx∈KBX,y=PAx,n≥n0andyn=PA_nx. Putρ= x−y and observe thatρ≤K since 0∈A. By (ii),

x−yn ≤ x−y + y−yn ≤ρ+ε

and hence y_n ≤ x + x−y_n ≤3K. Therefore, by (i),y_nbelongs to the convex set C:=(A+εB_X)∩ [x+(ρ+ε)B_X].

Moreover, since dist(x, A)=ρ, we have dist(x, C) ≥ρ−ε. It follows that everyc∈C satisfies

ρ−ε≤ c−x ≤ρ+ε.

Hence, the assumptions of Lemma 2.5 hold for the setC−xand we obtain that diam(C− x)=diamC≤M. It follows that y_n−y = P_Anx−P_Ax ≤M. By the arbitrariness of x∈KB_X, the proof is concluded.

3 Notions of Regularity for a Couple of Convex Sets

In this section we introduce some notions of regularity for a couple of nonempty closed convex setsAandB. This class of notions was originally introduced in [1], in order to obtain some conditions ensuring the norm convergence of the alternating projections algorithm (see, also, [4]). Here we list three different type of regularity: (i) and (ii) are exactly as they appeared in [1], whereas (iii) is new. See [1] for concrete examples of couple of sets satisfying or not properties (i) and (ii). In particular, observe that, by [1, Theorem 3.9], bounded regularity always holds whenXis finite-dimensional.

Definition 3.1 LetXbe a Hilbert space andA, Bclosed convex nonempty subsets ofX.

Suppose thatE, F are nonempty. We say that the couple(A, B)is:

(i) regularif for eachε >0 there existsδ >0 such that dist(x, E)≤ε, wheneverx∈X satisfies

max{dist(x, A),dist(x, B−v)} ≤δ;

(ii) boundedly regularif for each bounded setS ⊂ Xand for eachε > 0 there exists δ >0 such that dist(x, E)≤ε, wheneverx∈Ssatisfies

max{dist(x, A),dist(x, B−v)} ≤δ;

(iii) linearly regular for points bounded away fromEif for eachε >0 there existsK >0 such that

dist(x, E)≤Kmax{dist(x, A),dist(x, B−v)}, whenever dist(x, E)≥ε.

The following proposition shows that (i) and (iii) in the definition above are equivalent.

The latter part of the proposition is a generalization of [1, Theorem 3.15].

Proposition 3.2 LetXbe a Hilbert space andA, Bclosed convex nonempty subsets ofX.

Suppose thatE, F are nonempty. Let us consider the following conditions.

(i) The couple(A, B)is regular.

(ii) The couple(A, B)is boundedly regular.

(iii) The couple(A, B)is linearly regular for points bounded away fromE.

Then(iii)⇔(i)⇒(ii). Moreover, ifEis bounded, then(ii)⇒(i).

Proof The implication(i)⇒ (ii)is trivial. The implication(iii) ⇒(i)follows directly from the definition. Indeed, by contradiction let us suppose that(i)does not hold, i.e., there existε >¯ 0 and a sequence{xn} ⊂Xsuch that dist(x_n, E) >ε¯and

max{dist(x, A),dist(x, B−v)} →0.

By(iii)we have that dist(xn, E)→0, a contradiction.

Now, let us prove that(i)⇒(iii). Suppose on the contrary that there existε >0 and a sequence{x_n} ⊂Xsuch that dist(x_n, E) > ε(n∈N) and

max{dist(xn,A),dist(xn,B−v)}

dist(x_n,E) →0.

For eachn∈ N, lete_n ∈E,a_n ∈ A, andb_n ∈ Bbe such that e_n−x_n = dist(x_n, E), a_n−x_n =dist(x_n, A), and b_n−v−x_n =dist(x_n, B−v). Putλ_n= _en−x^ε _n ∈(0,1)

and definez_n=λ_nx_n+(1−λ_n)e_n,a_n =λ_na_n+(1−λ_n)e_n∈A, andb_n=λ_nb_n+(1− λn)(en+v)∈B. By our construction, it is clear that

dist(zn,A)

ε ≤ ^zn−a_ε ⁿ = ^x _eⁿ_n−x^−a_nⁿ and ^dist(zn_ε^,B−v)≤ ^bn−v−z_ε ⁿ = ^b _e^n−v−x_n−xnⁿ . Hence, dist(z_n, E)=εand max{dist(z_n, A),dist(z_n, B−v)} →0. This contradicts (i).

Now, suppose thatE is bounded, and let us prove that (ii) ⇒ (i). Suppose on the contrary that there existε >0 and a sequence{x_n} ⊂Xsuch that dist(x_n, E) > ε(n∈N) and

max{dist(x_n, A),dist(x_n, B−v)} →0.

For eachn ∈N, leten ∈ E,an ∈A, andbn ∈Bbe such that en−xn =dist(xn, E), an−xn =dist(xn, A), and bn−v−xn =dist(xn, B−v). Putλn= _en−^ε_xn ∈(0,1) and definezn=λnxn+(1−λn)en,a_n =λnan+(1−λn)en∈A, andb_n=λnbn+(1− λn)(en+v)∈B. By our construction, it holds that

dist(zn, A)≤ zn−a_n ≤ xn−an

and

dist(z_n, B−v)≤ b_n −v−z_n ≤ b_n−v−x_n .

Hence, dist(zn, E) =εand max{dist(zn, A),dist(zn, B−v)} →0. Moreover, sinceEis bounded{zn}is a bounded sequence. This contradicts (ii) and the proof is concluded.

The following theorem follows by [1, Theorem 3.7].

Theorem 3.3 Let X be a Hilbert space and A, B closed convex nonempty subsets of X. Suppose that the couple (A, B) is regular. Then the alternating projections method converges.

We present here below a proof of this theorem, slightly different from that contained in [1]. This proof is based on a simplified version of the argument that we will use in our main result Theorem 4.9. Let us point out that this proof is not essential for sequel of the paper, but it can be useful to visualize the geometrical idea behind the proof of Theorem 4.9. We consider only the case whereA∩Bis nonempty since the general case is similar but some unavoidable details would have made it more difficult to follow the outline of the proof.

Proof Let us consider the sequences {cn = PA(dn)} and {d_n+1 = PB(cn)}. By the nonexpansivity of the projections onto convex sets, for everyh∈A∩B, we have:

c_n−h = P_A(d_n)−P_A(h) ≤ d_n−h d_n+1−h = P_B(c_n)−P_B(h) ≤ c_n−h It follows immediately that:

(α) dist(c_n, A∩B)≤dist(d_n, A∩B)and dist(d_n+1, A∩B)≤dist(c_n, A∩B), whenevern ∈ N. This condition implies that [1, Theorem 3.3, (iv)] holds, therefore the following fact holds: “The sequence{c_n}converges to a point inA∩Biffdist(c_n, A∩B)→ 0.” Hence, it is sufficient to prove that dist(c_n, A∩B)→0.

Forε >0, by the equivalence(i)⇔(iii)in Proposition 3.2, there existsK >0 such that

dist(x, A∩B)≤Kmax{dist(x, A),dist(x, B)}, (3)

whenever dist(x, A∩B)≥ε. Observe thatK≥1 and defineη=

1−_K¹2. We claim that, for eachn∈N, the following condition holds:

(β) if dist(cn, A∩B)≥εthen dist(cn, A∩B)≤ηdist(dn, A∩B).

To prove this, lethn:=P_A∩B(dn)and observe that

[dist(c_n, A∩B)]²+[dist(d_n, A)]²≤ c_n−h_n ²+[dist(d_n, A)]². (4) By the variational characterization of the projection and sincecn=PA(dn), we obtain that the angleθn:=ang(dn−cn, hn−cn)is such that

π

2 ≤θn≤π.

Hence, (4) implies

[dist(cn, A∩B)]²+[dist(dn, A)]² ≤ cn−hn 2+[dist(dn, A)]²

−2 cn−hn dist(dn, A)cosθn

= [dist(d_n, A∩B)]²

where the last equality is obtained by applying the law of cosines to the triangle with vertices dn,cnand andhn. This gives

[dist(c_n, A∩B)]²≤[dist(d_n, A∩B)]²−[dist(d_n, A)]²

Finally, since, by(α), dist(dn, A∩B)≥dist(cn, A)≥ε, (3) and the last inequality gives [dist(c_n, A∩B)]²≤[dist(d_n, A∩B)]²− 1

K²[dist(d_n, A∩B)]², and the claim is proved.

Now, if there existsn₀∈Nsuch that dist(c_n0, A∩B)≤ε, then, (α) implies dist(c_n, A∩ B)≤εfor everyn≥n₀. On the other hand, if dist(c_n, A∩B)≥εfor alln∈N, then, by combining subsequently (β) and (α) we obtain

dist(cn, A∩B)≤ηdist(dn, A∩B)≤ηdist(cn−1, A∩B) (n∈N), a contradiction sinceη <1. Therefore we conclude that eventually dist(c_n, A∩B)≤ε. By the arbitrariness ofε >0 the proof is concluded.

4 Regularity and Perturbed Alternating Projections

This section is devoted to prove our main result. Indeed, here we show that if a couple (A, B) of convex closed sets is regular then not only the alternating projections method converges but also the couple(A, B)satisfies certain “stability” properties with respect to perturbed projections sequences.

Let us start by making precise the word “stability” by introducing the following two notions of stability for a couple(A, B)of convex closed subsets ofX.

Definition 4.1 LetAandBbe closed convex subsets ofXsuch thatE, Fare nonempty. We say that the couple(A, B)isstable[d-stable, respectively] if for each choice of sequences {An},{Bn} ⊂ c(X)converging with respect to the Attouch-Wets convergence to A and B, respectively, and for each choice of the starting pointa0, the corresponding perturbed alternating projections sequences{a_n}and{b_n}converge in norm [satisfy dist(a_n, E)→0 and dist(b_n, F )→0, respectively].

Remark 4.2 We remark that the couple(A, B)is stableif and only if for each choice of sequences{An},{Bn} ⊂ c(X)converging with respect to the Attouch-Wets convergence toAandB, respectively, and for each choice of the starting pointa0, there existse ∈ E such that the perturbed alternating projections sequences{a_n}and{b_n}satisfya_n→eand b_n→e+vin norm.

Proof Without any loss of generality, we can suppose that 0∈B. Let us start by proving that ifan→ethene∈E.

We claim that the sequence

P_AnP_Bn

uniformly converges on the bounded sets toP_AP_B. To see this observe that:

• since 0∈B, we have P_Bx ≤ x , wheneverx∈X;

• since projections are nonexpansive, we have

PA_nPB_nx−PA_nPBx ≤ PB_nx−PBx , wheneverx∈Xandn∈N;

• for eachx∈Xandn∈N, we have

PAnPBnx−PAPBx ≤ PAnPBnx−PAnPBx + PAnPBx−PAPBx . The previous observation implies that, forN >0, , we have

sup

x ≤N

P_AnP_Bnx−P_AP_Bx ≤ sup

x ≤N

P_Bnx−P_Bx + sup

x ≤N P_AnP_Bx−P_AP_Bx

≤ sup

x ≤N P_Bnx−P_Bx + sup

y ≤N P_Any−P_Ay . SinceA_n → A, B_n → Bfor the Attouch-Wets convergence, by Lemma 2.6,{P_An}uni- formly converges on bounded set toPAand{PBn}uniformly converges on bounded set to PB. The claim follows by the previous inequality.

Now, since{an}is bounded and

a_n+1=P_AnP_Bna_n=P_AP_Ba_n+(P_AnP_Bn−P_AP_B)a_n,

passing to the limit asn→ ∞, we obtaine=PAPBe. By Fact 2.4, (ii), we have thate∈E.

Similarly, it is easy to see that

b_n+1 =P_Bna_n=P_Ba_n+(P_Bn−P_B)a_n→P_Be=e+v, and the proof is concluded.

It is clear that if the couple(A, B)is stable, then it isd-stable. Moreover, ifE, F are singletons then also the converse implication holds true. The following basic assumptions will be considered in the sequel of the paper.

Basic assumptions 4.3 LetA, Bbe closed convex nonempty subsets ofX. Suppose that:

(i) E, Fare nonempty and bounded;

(ii) {A_n}and{B_n}are sequences of closed convex sets such thatA_n →AandB_n→B for the Attouch-Wets convergence.

Now, let us prove a chain of lemmas and propositions that we shall use in the proof of our main result, Theorem 4.9 below.

Lemma 4.4 LetGbe a closed convex subset ofX. Suppose that there existε, K >0such thatεBX⊂G⊂KBX. Then, ifu, w∈∂Gandθ:=cos(u, w) >0, we have

u−w ²≤K²(^K_ε2² +1)^1−θ2

θ² .

Proof The proof involves only the plane containing the origin and the vectorsuandv.

Henceforth, without any loss of generality we can suppose thatX=R²andu=( u ,0).

Let us denotew =(x, y), withx, y ∈R, and suppose thatu, w∈∂G. Observe that, since θ >0,xis positive.

We claim that it holds

|y| ≥ ε

u (x− u )

. (5) To prove our claim, suppose on the contrary that

|y|<

ε

u (x− u )

. (6) and let us consider two cases. First, letxbe such that 0 < x ≤ u . SinceεB_X ⊂Gand u=( u ,0)∈G, the set

L:= {(z, v)∈R²;0< z < u , |v|<| _u ^ε (z− u )|}

is contained in the interior ofG andw ∈ L, a contradiction. We now turn to the case x > u . Leth:=(0,−_{x− u} ^u y)∈X, then it holds

u= u x w+

1− u

x

h.

By (6), we have h < ε. Thenubelongs to the interior of the set conv({w} ∪εBX), and hence to the interior ofG. This contradiction proves the claim.

Sinceθ=cos(u, w)= _w ^x , we havey²= ^1−θ_θ2²x². Hence, by (5), we have (x− u )²≤^1−θ_θ2² u ²

ε² x². Finally,

u−w ²=(x− u )²+y²≤x² _{u 2}

ε² +1

1−θ²

θ² ≤K²

K² ε² +1

1−θ² θ² .

Proposition 4.5 Let Basic assumptions 4.3 be satisfied and, for eachn∈N, leta_n ∈A_n andb_n∈B_n. Suppose that the couple(A, B)is regular. Letε >0, then there existη∈(0,1) andn1∈Nsuch that for eachn≥n1we have:

(i) ifdist(a_n, E)≥2εanddist(b_n, F )≥2εthencos

a_n−e, b_n−(e+v)

≤η,whenever e∈E+εB_X.

(ii) ifdist(an, E) ≥2εanddist(b_n+1, F ) ≥ 2εthencos

b_n+1−f, an+v−f

≤ η, wheneverf ∈F+εBX.

Proof Let us prove that there existη ∈(0,1)such that eventually (i) holds, the proof that there existη ∈(0,1)such that eventually (ii) holds is similar. Suppose that this is not the

case, then there exist sequences{e_k} ⊂E+εB_X,{θ_k} ⊂(0,1)and an increasing sequence of the integers{nk}such that dist(ank, E)≥2ε, dist(bnk, F )≥2ε, and

cos

a_nk−e_k, b_nk−(e_k+v)

=θ_k→1.

LetG = E+2εB_X and observe thatG is a bounded body inX. Sincee_k ∈ intGand ank ∈intG, there exists a unique pointa_k ∈ [ek, ank] ∩∂G. Similarly, there exists a unique pointb_k∈ [ek, bn_k−v] ∩∂G. Moreover, by construction, we have that

cos

a_k −e_k, b_k−e_k

=θ_k.

Lemma 4.4 implies that (a_k−v)−(b_k−v) = a_k−b_k →0. SinceGis bounded and An_k →A, Bn_k →Bfor the Attouch-Wets convergence, there exist sequences{a_k} ⊂A and{b_k} ⊂B−vsuch that a_k −a_k →0 and b_k −b_k →0. Hence, by the triangle inequality, a_k−b_k →0 and eventually dist(a_k, E)≥ε, a contradiction since the couple (A, B)is regular.

Proposition 4.6 Let Basic assumptions 4.3 be satisfied, suppose that the couple(A, B)is regular, and letδ, ε > 0. For eachn∈ N, leta_n, x_n ∈A_nandb_n, y_n ∈B_nbe such that dist(x_n, E)→0anddist(y_n, F )→0. Then there existsn₂ ∈Nsuch that for eachn≥n₂ we have:

(i) ifdist(a_n, E)≥2ε,dist(b_n, F )≥2ε, anda_n=P_Anb_nthen cos

xn−an, bn−(an+v)

≤δ; (ii) ifdist(an, E)≥2ε,dist(bn+1, F )≥2ε, andbn+1=PB_n+1anthen

cos

y_n+1−b_n+1, a_n+v−b_n+1

≤δ.

Proof Let us prove that eventually (i) holds, the proof that eventually (ii) holds is similar.

Since dist(a_n, E) ≥ 2ε and dist(x_n, E) → 0 we have that eventuallyx_n−a_n = 0. By Proposition 4.5, there existsη∈(0,1)andn₁∈Nsuch that

cos

an−e, bn−(e+v)

≤η, (7)

whenevern≥n1ande∈E+εB_X. Observe that, if dist(a_n, E)≥2εand dist(b_n, F )≥2ε, we have that a_n−e ≥εand b_n−(e+v) ≥ε(n∈N). By the law of cosines we have

u−w ²= u ²+ w ²−2 cos(u, w) u w ≥2 u w

1−cos(u, w)

, u, w∈X.

By (7) and the previous inequality, applied tow = bn−(e+v)andu = an−e, there exists a constantη>0 such that b_n−(a_n+v) ≥η, whenevern≥n₁. By the above, x_n−a_n=0 andb_n−(a_n+v)=0 for alln≥n₁, then eventually cos

x_n−a_n, b_n−(a_n+v) is well-defined. Ifv = 0, the thesis is trivial since, by the variational characterization of an=PAnbn, it holds

x_n−a_n, b_n−a_n ≤0, whenevern∈N.

Suppose thatv=0. We claim that, ifvdenotes the displacement vector for the couple (A, B), eventually we have

v, an−xn ≤δη an−xn . (8) To prove our claim observe that, since dist(x_n, E)→0, we can suppose without any loss of generality that dist(x_n, E) ≤ε(n∈N). Moreover, we can consider a sequence{x_n} ⊂ E such that x_n −x_n →0. LetG=E+2εB_Xand observe thatGis a bounded body inX.

Sincex_n∈intGanda_n∈intG, there exists a unique pointa_n ∈ [x_n, a_n] ∩∂G. SinceGis bounded andAn→Afor the Attouch-Wets convergence, there exists a sequence{a_n} ⊂A such that a_n−a_n →0. Since{x_n} ⊂E, it follows that a_n−x_n ≥2εand, by taking into account the variational characterization of the projection, thatv, a_n−x_n ≤0. Hence, eventually we have

v, a_n−x_n −δη a_n−x_n ≤ −δηε.

Since x_n −xn →0 and a_n−a_n →0, eventually we have v, a_n−xn −δη a_n−xn ≤0.

By homogeneity ofv,·and of the norm, and by our construction, the claim is proved.

Now, by our claim, sincean=PA_nbnandxn∈An(n∈N), we have

xn−an, bn−(an+v) = xn−an, bn−an + an−xn, v ≤δη an−xn . Eventually, since bn−(an+v) ≥η, we have

cos

x_n−a_n, b_n−(a_n+v)

≤ δη

b_n−(a_n+v) ≤δ.

Now, we need a simple geometrical result whose proof is a simple application of the definition of cosine combined with the triangle inequality.

Fact 4.7 Letη, η ∈ (0,1)be such thatη < η. Ifδ ∈ (0,1)satisfies ^δ_1−δ^+η ≤ η and if x, y∈Xare linearly independent vectors such thatcos(x, y)≤ηandcos(y−x,−x)≤δ then x ≤η y .

Proof By our hypotheses and the definition of cosine, we have

• −y, x ≥ −η x y ;

• −y, x + x ²≤δ y−x x . Combining the two inequalities, we obtain

x ( x −η y )≤δ y−x x , and hence, by the triangular inequality,

x −η y ≤δ y−x ≤δ y +δ x . Finally,

x ≤δ+η

1−δ y ≤η y .

Proposition 4.8 Let Basic assumptions 4.3 be satisfied. For eachM > 0there existθ ∈ (0, M)andn₀∈Nsuch that ifn≥n₀we have:

(i) ifbn∈Bn,an=PA_nbn, anddist(bn, F )≤θthen dist(an, E)≤2M; (ii) ifan∈An,bn+1=PB_n+1an, anddist(an, E)≤θthen

dist(bn+1, F )≤2M.

Proof LetM >0 andρ= v , wherevis the displacement vector. By Lemma 2.5 (where we takeH = 3M), there exists ε ∈ (0,3M) such that, if C is a convex set such that ρ−ε≤ c ≤ρ+ε, wheneverc ∈C, then diam(C) ≤M. Putθ =ε/3, since Basic assumption 4.3 are satisfied, there existsn0∈Nsuch that ifn≥n0we have:

(a) ifw∈Anthen dist(w, F )≥ρ−3θ;

(b) ife∈E, there existsx∈A_nsuch that e−x ≤θ.

Now, letn ≥ n0,bn ∈ Bn, an = PAnbn, and dist(bn, F ) ≤ θ. Letfn ∈ F be such that fn−bn ≤ θ and puten = fn−v ∈ E. By (b), there existsxn ∈ Ansuch that

x_n−e_n ≤θ. Hence, sincea_n=P_Anb_nand e_n−f_n =ρ, we have an−fn ≤ an−bn + fn−bn

≤ xn−bn + fn−bn

= x_n−e_n+e_n−f_n+f_n−b_n + f_n−b_n

≤ xn−en +ρ+2 fn−bn ≤ρ+3θ. Let us consider the convex setC= [xn−fn, an−fn]. Observe that, since

xn−fn ≤ en−xn + en−fn ≤ρ+θ,

we have that c ≤ρ+3θ, wheneverc∈C. Moreover, since[x_n, a_n] ⊂A_nandf_n ∈F, by (a) we have c ≥ρ−3θ, wheneverc∈C. Hence, we can apply Lemma 2.5 to the set Cand we have an−xn =diam(C)≤M. Then

dist(an, E)≤ an−en ≤ an−xn + en−xn ≤M+θ≤2M.

The proof that eventually (ii) holds is similar.

We are now ready to state and prove the main result of this paper.

Theorem 4.9 LetA, B be closed convex nonempty subsets ofXsuch thatE andF are bounded. Suppose that the couple(A, B)is regular, then the couple(A, B)isd-stable.

Proof Let a₀ ∈ X and let {a_n} and {b_n} be the corresponding perturbed alternating projections sequences, i.e,

a_n=P_An(b_n) and b_n=P_Bn(a_n−1).

First of all, we remark that it is enough to prove that dist(a_n, E)→0 since the proof that dist(b_n, F ) → 0 follows by the symmetry of the problem. Therefore our aim is to prove that for eachM >0, eventually we have

dist(a_n, E)≤M.

Let us considerM > 0, then, by (ii) in Proposition 4.8, there exist 0 < θ < ^M₂ and n ∈ Nsuch that, for eachn > n, if dist(a_n, E) ≤θ then dist(b_n+1, F ) ≤ M. Now, by (i) in Proposition 4.8, there exist 0 < ε < ^θ₄ andn ∈ Nsuch that, for eachn > n, if dist(b_n, F )≤2εthen dist(a_n, E)≤θ. Therefore, we conclude that there exist 0< ε < ^M₈ andn₀=max{n, n} ∈Nsuch that, for eachn > n₀we have:

(α₁) if dist(b_n, F )≤2εthen dist(a_n, E)≤Mand dist(b_n+1, F )≤M.

Again, by applying Proposition 4.8 twice and a similar reasoning as above, we can suppose that, for eachn≥n₀: