Best Approximation and Perturbation Property in Hilbert Spaces

DOI: 10.1007/s10957-005-4714-2journal of optimization theory and applications: Vol. 126, No. 2, pp. 265–285, August 2005 (© 2005)

Best Approximation and Perturbation Propertyin Hilbert Spaces1

Y. R. He2 and K. F. Ng3

Communicated by P. Tseng

Abstract. We study the perturbation property of best approximationto a set defined by an abstract nonlinear constraint system. We showthat, at a normal point, the perturbation property of best approx-imation is equivalent to an equality expressed in terms of normalcones. This equality is related to the strong conical hull intersectionproperty. Our results generalize many known results in the literatureon perturbation property of best approximation established for a setdefined by a finite system of linear/nonlinear inequalities. The connec-tion to minimization problem is considered.

Key Words. Best approximation, abstract nonlinear systems, strongconical hull intersection property, Robinson constraint qualification.

1. Introduction

In recent years, there has been much interest centered on the follow-ing constraint approximation problem:

min ‖x −y‖, s.t. y ∈C and G(y)∈K, (1)

where C and K are nonempty closed convex subsets of a Hilbert spaceX and a Bananch space Y , respectively, x ∈ X, and G is a differentiable

1The authors thank the referees for valuable suggestions.2Associate Professor, Department of Mathematics, Sichuan Normal University, Chengdu,Sichuan, PRC. This author was partially supported by Grant A0324638 from the NationalNatural Science Foundation of China and Grants (2001) 01GY051-66 and SZD0406 fromSichuan Province.

3Professor, Department of Mathematics, Chinese University of Hong Kong, Shatin, NewTerritories, Hong Kong. This author was supported by a Direct Grant (CUHK) and anEarmarked Grant from the Research Grant Council of Hong Kong.

2650022-3239/05/0800-0265/0 © 2005 Springer Science+Business Media, Inc.

266 JOTA: VOL. 126, NO. 2, AUGUST 2005

mapping from X to Y . This problem has its origin relating more concreteproblems such as the shape-preserving interpolation problem that arises incurve and surface fitting (see Ref. 1). In view of the fact that, in manypractical situations of (1), C has a very simple structure, such as a boxor a nonnegative cone in a finite-dimensional space, many researchers (seeRefs. 2–8) were led to reformulate (1) (on its special cases) as a system ofnonlinear equations by using dual formulation of the difficult constraintG(x)∈K and thus to address the issue when (1) has the following pertur-bation property:

x0 ∈PS(x)⇐⇒x0 =PC(x −G′(x0)∗λ), for some λ∈N K(G(x0)), (2)

where PC denotes the projection onto C,NK(G(x0)) is the normal cone ofK at G(x0),G

′(x0) is the derivative of G at x0,G′(x0)

∗ is the adjoint oper-ator of G′(x0), and S denotes the feasible set of (1) defined by

S :=C ∩{x ∈X :G(x)∈K}. (3)

If Y = Rm and G(x) = Ax − d but K = Rm−, Deutsch, Li, and Ward(Ref. 8) showed that the perturbation property (2) is equivalent to thestrong CHIP (strong conical hull intersection property) of {C, {x :Ax ≤d}}.Moreover, it is shown in Ref. 9, Chapter 3 that, if the graph of the mul-tifunction G(x)−K is convex, then the Robinson constraint qualification(RCQ) of the system (3) at x0 implies that the perturbation property (2)holds.

In this paper, we make some further theoretical developments on theequivalent characterizations of the perturbation property for the generalnonlinear constraint system (3). For this purpose, we introduce the con-cept of normal point and prove that, if x0 ∈PS(x) is a normal point, thenthe perturbation property (2) holds if and only if a basic constraint qual-ification is satisfied at x0 (Theorem 3.2). It is also proved that, if S is theconstraint system studied in Refs. 2–8, then all the points in S are normalpoints and the basic constraint qualification is just the strong CHIP; seeTheorem 4.4. Moreover, we prove that, if the contingent cone of S at apoint x0 ∈ PS(x) coincides with the closure of the radial cone of S at x0[which holds particularly when the graph of G(x) − K is convex] and ifthe system (3) satisfies the Robinson constraint qualification (RCQ) at x0,then x0 is a normal point and the perturbation property (2) holds (Theo-rem 3.3). Thus, not only our results recapture many of the earlier results,but also demonstrate a natural nonconvex setting for the theory on per-turbation property.

This paper is organized as follows. Some notations and preliminaryresults are presented in Section 2. Several equivalent characteristics of

JOTA: VOL. 126, NO. 2, AUGUST 2005 267

the perturbation property for the general nonlinear constraint system (3)are established in Section 3. These are used to compare several earlierresults on perturbation property for finite system of inequalities/equalitiesin Section 4.

2. Notations, Preliminaries, and Robinson Constraint Qualification

Recall that S is defined by (3) with C,X,G as in the beginning of Sec-tion 1. For any x0 ∈S, we use T G(x0) to denote the closed convex cone inX defined by

T G(x0) :=G′(x0)−1TK(G(x0));

that is,

T G(x0)={h∈X :G′(x0)h∈TK(G(x0))}, (4)

where TK(G(x0)) denotes the tangent cone of K at G(x0). We show in Sec-tion 3 that T G(x0)∩TC(x0) coincides with TS(x0) if (RCQ) is satisfied. Inview of a standard result in duality (Ref. 10, Page 64.), the cone T G(x0)

defined in (4) equals the polar of G′(x0)∗NK(G(x0)); consequently, by the

bipolar theorem (Ref. 10, Theorem 2.4.3),

T G(x0)� =G′(x0)

∗NK(G(x0)). (5)

Note that the norm-closure in (5) is the same as the weak∗-closure by con-vexity and the fact that the sets are in the Hilbert space X.

Definition 2.1. x0 ∈ S is called a normal point of S [with respect tothe constraint system of (3)] if

S ⊆C ∩ (x0 +T G(x0))⊆C ∩ (x0 + coTS(x0)). (6)

Given a Banach space Z, let Z∗ denote its (topological) dual spaceand write 〈z∗, z〉 for z∗(z), where z∈Z and z∗ ∈Z∗. For a linear operatorA :X →Z,A∗ denotes the adjoint operator of A,

〈A∗z∗, x〉=〈z∗,Ax〉, for each x ∈X and z∗ ∈Z∗.

Since X is a Hilbert space, X and X∗ are identified and so 〈·, ·〉 is simplythe inner product. For a set W in Z, the interior (respectively, closure, con-vex hull, closed convex hull, conic hull, negative polar) of W is denoted by


int W (repectively W , coW , coW , coneW,W�). The normal cone of W atw0 ∈W is denoted by NW(w0) and is defined by

NW(w0)= (W −w0)�,

namely,

NW(w0) :={z∗ ∈Z∗ : 〈z∗,w −w0〉≤0, for each w ∈W }.

For a set K ⊆Z∗,

K� :={z∈Z : 〈z∗, z〉≤0, for all z∗ ∈K}

is the negative polar of K. The Bouligand contingent cone (cf. Refs. 9–11)of W at w0 is denoted by TW(w0); the radial cone of W at w0 is denotedby RW(x0) and is defined by

RW(w0) :=⋃

t>0

t−1(W −w0).

It is known and easy to prove that

TW(w0)⊆RW(w0). (7)

If W is convex, then

TW(w0)=RW(w0), (8)

and the cones TW(w0),NW(w0) are polars of each other. For z∈Z,PW(z)

denotes the projection of z to W ,

PW(z)={w ∈W :‖w − z‖=dist(z,W)},

where

dist(z,W) := infy∈W‖z−y‖.

In the case when Z is a Hilbert space, PW(z) contains at most one pointfor each z∈Z. The following result, valid in Hilbert spaces, is well-known(see Ref. 12, Page 23 and Proposition 1.10) and will be useful for ourdiscussions.

Proposition 2.1. Let W be a closed convex set in a Hilbert space H ;let w0 ∈W and d ∈H . Then, d ∈NW(w0) if and only if PW(w0 + td)=‖td‖for some t>0.


For the following proposition, we consider the case when W is notnecessarily convex; thus, TW(w0) is not necessarily convex.

Proposition 2.2. Let Z be a Banach space and let w0 ∈W ⊆Z. Then,

coTW(w0)= coRW(w0) (9)

if and only if

W ⊆w0 + coTW(w0). (10)

Proof. We need to prove only that (10) implies (9). In view of (7), toestablish (9), it suffices to show the inclusion

RW(w0)⊆ coTW(w0).

But this inclusion follows trivially from (10) as coTW(w0) is a cone.

For the study of nonconvex sets, the following definition will be use-ful for us.

Definition 2.2. Let Z be a Banach space and let C be a convex set inZ. Let w0 ∈W ⊆Z and let T be a convex set in Z containing the origin.W is said to be convexified at w0 by T with respect to C if it holds that

W ⊆C ∩ (w0 +T )⊆C ∩ (w0 + coTW(w0)). (11)

Thus, in the situation of Definition 2.1, x0 ∈S is a normal point of S

if and only if S is convexified at x0 by T G(x0) with respect to C. Needlessto say, if W is assumed to be convex, then W is convexified at any x0 ∈W

by W −x0.In the following, we return to the constraint system (3): the notations

are as at the beginning of Section 1. In particular, C and K are assumedto be closed and convex; S denotes the set defined in (3), that is,

S =C ∩G−1(K).

Definition 2.3. See Refs. 9, 13. Let x0 ∈S. The constraint system (3)is said to satisfy (RCQ) at x0 if

G′(x0)RC(x0)−RK(G(x0))=Y. (12)


Recalling the definition of T G(x0) given in (4), we note that

TS(x0)⊆T G(x0)∩TC(x0), for each x0 ∈S. (13)

Indeed, if d is a unit vector in TS(x0), then

limn

(sn −x0)/‖sn −x0‖=d,

for some sequence (sn) ⊆ S converging to x0. Consequently, d ∈ TC(x0);since

G(sn)=G(x0)+G′(x0)(sn −x0)+o(‖sn −x0‖),

we have

G′(x0)d = limn

([G(sn)−G(x0)]/‖sn −x0‖)∈TK(G(x0)),

because G(sn) ∈ K, for each n. Therefore, (13) is true. Part (a) of thefollowing proposition shows that (RCQ) implies that the equality in (13)holds. This result is proved in Ref. 9 (see its Corollary 2.91) for the casewhen C =X.

Proposition 2.3. Let x0 ∈S and suppose that the constraint system (3)satisfies (RCQ) at x0. Then, the following assertions hold:

(a) TS(x0) = T G(x0) ∩ TC(x0); in particular, TS(x0) is closed andconvex.

(b) S ⊆x0 +TS(x0)⇔TS(x0)=RS(x0).(c) C ∩ (x0 +T G(x0))=C ∩ (x0 +TS(x0)).(d) NC∩(x0+T G(x0))

(x0)=NC(x0)+G′(x0)∗NK(G(x0)). (14)

Proof. Let Z denote the product X×Y and let K denote C ×K. LetG denote the function x �→ (x,G(x)) from X into Z. Then,

G′(x0)h= (h,G′(x0)h), for each h,

and

TK(x0,G(x0))=TC(x0)×TK(G(x0));

consequently, it follows from (4) that

{h :G′(x0)h∈TK(x0,G(x0))}=T G(x0)∩TC(x0). (15)


On the other hand, by (12), it is known (cf. Ref. 9, Page 71) and easy toverify that

G′(x0)X −RK(G(x0))=Z.

Hence, by Corollary 2.91 in Ref. 9, the set on the left-hand side of (15)equals the tangent cone of G−1(K) at x0, which is exactly TS(x0) becauseclearly S =G−1(K). Thus, (a) is proved. (b) follows easily from (a) and (7).In view of (a), to prove (c), it suffices to show that the set on the left-handside of (c) is contained in the set on the right-hand side. Accordingly, let

ξ ∈C ∩ (x0 +T G(x0)).

Then,

ξ −x0 ∈T G(x0) and ξ ∈C.

Since C is convex, it follows that

ξ −x0 ∈TC(x0).

Hence, by (a),

ξ −x0 ∈TS(x0).

and so ξ belongs to the set on the right-hand side of (c).It remains to prove the assertion (d). Define

F(x) :=G′(x0)(x −x0), for each x ∈X,

D :=TK(G(x0)),

and let S′ denote the set of all x satisfying the new system

x ∈C and F(x)∈D. (16)

Note that

F ′(x0)=G′(x0), F (x0)=0, TD(F(x0))=TK(G(x0)).

Hence,

T F (x0)=T G(x0), S′ =C ∩ (x0 +T G(x0)).

Since the constraint system (3) satisfies (RCQ) at x0, so does the new sys-tem (16). It follows from part (a) that

TS′(x0)=TC(x0)∩T F (x0)=Tc(x0)∩T G(x0). (17)


Noting the convexity of S′, making use of (5), and taking polars on bothsides of (17), one has

NS′(x0)=NC(x0)+G′(x0)∗Nk(G(x0)).

Thus, to prove (14), it remains to show that

NS′(x0)⊆NC(x0)+G′(x0)∗NK(G(x0)). (18)

To do this, let ξ ∈NS′(x0). Consider the following minimization problem:

min 〈−ξ, h〉, (19a)

s.t. h∈TC(x0), (19b)

G′(x0)h∈TK(G(x0)), (19c)

and its dual problem (Ref. 9, Section 2.5.6)

max 〈λ,0〉, (20a)

s.t. λ∈NK(G(x0)), (20b)

ξ −G′(x0)∗λ∈NC(x0). (20c)

Since

NS′(x0)� =TS′(x0),

h0 :=0 solves the problem (19). By the (RCQ) assumption, Theorem 2.187in Ref. 9 implies that there is no duality gap between the primal problem(19) and the dual problem (20). This shows that the feasible set of the dualproblem (20) is nonempty. That is, there exits λ∈NK(G(x0)) such that ξ −G′(x0)

∗λ belongs to NC(x0). Thus,

ξ ∈NC(x0)+G′(x0)∗NK(G(x0)).

This proves (18).

Let {C1,C2, . . . ,Cm} be a collection of closed convex sets and let x0 ∈∩m

i=1Ci . Following Refs. 4, 8, the collection is said to have the strongCHIP (strong conical hull intersection property) at x0 if

N∩mi=1Ci

(x0)=m∑

i=1

NCi(x0).


3. Perturbation Property of Best Approximation

Consider the following minimization problem:

(Pf,G) min f (x), (21a)

s.t. x ∈C and G(x)∈K, (21b)

where f : X → R is differentiable and the other notations are as in (3).Recall that the set

S =C ∩G−1(K)

is not necessarily convex as G is not assumed to be convex with respect toK. Let x0 ∈S and let T0 be a convex set in X containing the origin. Recallfrom Definition 2.2 that S is convexified at x0 by T0 with respect to C if

S ⊆C ∩ (x0 +T0)⊆C ∩ (x0 + coTS(x0)). (22)

Proposition 3.1. Assume (22) and denote C ∩ (x0 +T0) by S0. Then,for any convex and differentiable f , the following statements are equivalent:

(i) x0 is an optimal solution to (Pf,G).

(ii) −f ′(x0)∈TS(x0)�.

(iii) x0 minimizes f over C ∩ (x0 + co(TS(x0))).

(iv) x0 minimizes f over S0.

(v) −f ′(x0)∈NS(x0).

Proof. (i) ⇒ (ii). See the proof of Lemma 3.7 (i) in Ref. 9 even with-out assuming the convexity of f .

(ii) ⇒ (iii). Since −f ′(x0)∈TS(x0)�, one has

〈f ′(x0), y −x0〉≥0, for each y ∈x0 + co(TS(x0)).

Since f is convex, the subgradient inequality gives

f (y)−f (x0)≥〈f ′(x0), y −x0〉≥0, for each y ∈x0 + co(TS(x0)).

Thus, (iii) holds.(iii) ⇒ (iv) ⇒ (i). This is evident as S ⊆S0 ⊆C ∩ (x0 +co(TS(x0))) and

x0 ∈S.(iv) ⇔ (v). This is obvious due to the convexity of f and S0.


Below, we specialize Proposition 3.1 by considering functions of theform fx defined by

fx(w)= (1/2)‖w −x‖2, for each w ∈X, (23)

where x ∈X.

Theorem 3.1. Assume (22) and denote C ∩ (x0 +T0) by S0. Let N ′ beany nonempty subset of T

�0 . Then, the following statements are equivalent:

(i) NS0(x0)=NC(x0)+N ′.(ii) For any x ∈X,

x0 ∈PS(x)⇐⇒x0 =PC(x −x∗), for some x∗ ∈N ′. (24)

(iii) For any differentiable convex function f : X → R, x0 minimizesf over S if and only if there exits x∗ ∈N ′ such that −f ′(x0)−x∗ ∈NC(x0).

Proof. (i) ⇒ (ii). Let x ∈X and consider the problem (Pfx,G). Then,in the following the first equivalence is clear by Proposition 3.1,

x0 ∈PS(x)⇐⇒x0 =PS0(x)⇐⇒x −x0 ∈NS0(x0)

⇐⇒x −x0 ∈NC(x0)+N ′

⇐⇒x −x∗ −x0 ∈NC(x0), for some x∗ ∈N ′

⇐⇒x0 =PC(x −x∗), for some x∗ ∈N ′,

while the second and the last equivalences are due to Proposition 2.1 andthe convexity of S0 and C.

(ii) ⇒ (i). Since S0 − x0 ⊆ T0, one has T�

0 ⊆ NS0(x0). Similarly,NC(X0)⊆NS0(x0). Thus, to establish (i), it suffices to show that

NS0(x0)⊆NC(x0)+N ′.

To do this, let ξ ∈NS0(x0). Then, as above, x0 ∈PS(ξ +x0). By (24), it fol-lows that

x0 =PC(ξ +x0 −x∗), for some x∗ ∈N ′.

Then, we have ξ − x∗ ∈NC(x0) and it follows that ξ ∈NC(x0)+N ′. (i) ⇒(iii). By (i), it follows from Proposition 3.1 that

−f ′(x0)−x∗ ∈NC(x0), for some x∗ ∈N ′

⇐⇒ −f ′(x0)∈NS0(x0)

⇐⇒ x0 minimizes f over S.


This shows that (iii) holds. Finally, applying (iii) to the function fx , it isclear that (iii) implies (ii).

Recall that T G(x0) is defined in (4). Recall also that x0 ∈S is a nor-mal point of S with respect to the constraint system (3) if and onlyif (22) holds with T0 replaced by T G(x0). By virtue of (5) and lettingN ′ = G′(x0)

∗NK(G(x0)), the following result follows immediately fromTheorem 3.1.

Theorem 3.2. Let x0 be a normal point of S with respect to the con-straint system (3). Then, the following statements are equivalent:

(i) NC∩(x0+T G(x0))(x0)=NC(x0)+G′(x0)

∗NK(G(x0)).(ii) For any x ∈ X,x0 ∈ PS(x) if and only if there exists λ ∈

NK(G(x0)) such that x0 =PC(x −G′(x0)∗λ).

(iii) For any differentiable convex function f : X →R, x0 minimizesf over S if and only if there exists λ∈NK(G(x0)) such that λ

is a Lagrangian multiplier of the problem (Pf,G).

Similarly, we have two more consequences of Theorem 3.1. For brev-ity sake, we omit the statement regarding the Lagrange rule.

Corollary 3.1. Let x0 ∈S and assume that S is convexified by G′(x0)−1

(K −G(x0)). Let N ′ be any subset of G′(x0)∗NK(G(x0)). Then, the follow-

ing statements are equivalent:

(i) NC∩(x0+G′(x0)−1(K−G(x0)))

(x0)=NC(x0)+N ′.(ii) For any x ∈X, x0 ∈PS(x) if and only if there exists x∗ ∈N ′ such

that x0 =PC(x −x∗).

Proof. Denote G′(x0)−1(K − G(x0)) by T0. Then, (22) is satisfied.

Since

K −G(x0)⊆TK(G(x0)),

by convexity of K, we have also

T0 ⊆T G(x0);it follows from (5) that

G′(x0)∗NK(G(x0))⊆T

�0 .

Hence, N ′ is also a subset of T�0 . Thus, one can apply Theorem 3.1 to

conclude the proof.


Corollary 3.2. Let x0 ∈S. Assume that S is convexified by

G′(x0)−1RK(G(x0)) and that

C ∩(x0 +G′(x0)

−1RK(G(x0)))

=C ∩ (x0 +G′(x0)

−1RK(G(x0))). (25)

Let N ′ be any subset of G′(x0)∗NK(G(x0)). Then, the following statements

are equivalent:

(i) NC∩(x0+G′(x0)−1(K−G(x0)))

(x0)=NC(x0)+N ′.(ii) For any x ∈X, x0 ∈PS(x) if and only if there exists x∗ ∈N ′ such

that x0 =PC(x −x∗).

Proof. Denote G′(x0)−1RK(G(x0)) by T0. Then, (22) is satisfied and

T0 ⊆T G(x0).

Hence, as before, (ii) is equivalent to

(i∗) NC∩(x0+T0)(x0)=NC(x0)+N ′,

or what is the same (thanks to (25)),

(i#) NC∩(x0+G′(x0)

−1RK(G(x0)))(x0)=NC(x0)+N ′.

Thus, to complete the proof, we need to prove only that the sets on theleft-hand side of (i) and of (i#) are the same. It suffices to prove that

NC∩(x0+T0)(x0)⊆NC∩(x0+G′(x0)−1(K−G(x0)))

(x0), (26)

since the other inclusion relations clearly holds. To do this, let

ξ ∈NC∩(x0+T0)(x0) and z∈C ∩ (x0 +G′(x0)−1RK(G(x0))).

Then, there exists t ∈ (0,1) such that

G′(x0)(z−x0)∈ t−1(K −G(x0)).

This implies that

t (z−x0)∈T0

and hence that

x0 + t (z−x0)∈C ∩ (x0 +T0).


Consequently,

〈ξ, t (z−x0)〉≤0,

and so (26) is proved. �For the following theorem, we do not assume (22) at the outset.

Theorem 3.3. Let x0 ∈S Suppose that the constraint system (3) satis-fies (RCQ) at x0. Then, each of the following statements implies the otherstatements and that x0 is a normal point of S.

(i) TS(x0)=RS(x0).(ii) For any x ∈ X,x0 ∈ PS(x) if and only if there exists λ ∈

NK(G(x0)) such that x0 =PC(x −G′(x0)∗λ).

(iii) For any differentiable convex function f : X → R, x0 minimizesf over S if and only if there exists λ∈NK(G(x0)) such that λ

is a Lagrange multiplier of the problem (Pf,G).

Consequently, if in addition S is assumed to be convex, then (i) holds andhence (ii), (iii) hold.

Proof. Assume (i). Together with (RCQ), it follows from Proposition2.3 that x0 must be a normal point of S. Furthermore, by virtue of theassumptions, one can apply Proposition 2.3(d) to conclude that (i) in The-orem 3.2 holds. Hence, by that theorem, we have (i) ⇒ (ii) and (iii).

(ii) ⇒ (i). In view of Proposition 2.3, to establish (i), it suffices toprove that

S ⊆x0 +TS(x0).

By Lemma 3.1 in Ref. 14, the inclusion will follow provided that the fol-lowing implication holds: for any x ∈X,

x0 =PC∩(x0+TS(x0))(x)�⇒x0 ∈PS(x). (27)

By Proposition 2.3, TS(x0) is convex and

NC∩(x0+TS(x0))(x0)=NC∩(x0+T G(x0))(x0)

=NC(x0)+G′(x0)∗NK(G(x0)). (28)

To prove (27), let

x0 =PC∩(x0+TS(x0))(x).


Then, by Proposition 2.1,

x −x0 ∈NC∩(x0+TS(x0))(x0)

and it follows from (28) that

x −x0 −G′(x0)∗λ∈NC(x0), for some λ∈NK(G(x0)).

By Proposition 2.1 and (ii), then we conclude that x0 ∈PS(x).The implication (iii) ⇒ (ii) is clear by considering the function fx

defined in (23).

4. Applications to a Finite System of Nonconvex Inequalities

In this section, we apply the results in Section 3 to study the follow-ing system:

x ∈C and (hi ◦Fi)(x)≤0, for each i =1,2, . . . ,m, (29)

where C is a closed convex set in a Hilbert space X, each hi :Zi →R is acontinuous convex function on a Banach space Zi , and each Fi : X →Zi

is a differentiable mapping.

Remark 4.1. The system (29) has been studied by many researchersfor some special cases:

(a) Reference 15 studied the case when X =Zi for each i and Fi isthe identity mapping, and also the case when each Zi =R and hi

is the identity mapping.(b) The special case when, for each i, X=Zi , Fi is the identity map-

ping and each hi is affine has been studied extensively; see Refs.1–7. In particular, for such a system, Deutsch, Li, and Wardestablished in Ref. 4 the equivalence of (BCQ) and the perturba-tion property.

(c) Reference 14 studied the case when the spaces Zi , i =1,2, . . . ,m,are the same. The perturbation property for the system (29),established in Ref. 14, generalizes those established in Refs. 8,15.

Note that the system (29) can be converted to a system of the form

x ∈C, G(x)∈K; (30)


see (3). Indeed, let Y be the product space∏m

i=1 Zi and define G : X →Y

and K ⊆Y by

G(x) := (F1(x), . . . , Fm(x)), for each x ∈X (31a)

K :=m∏

i=1

Ki, where Ki :={y ∈Zi :hi(y)≤0}. (31b)

Clearly, K is closed and the feasible set S of (29) is the same as the setC ∩G−1(K). For x0 ∈S, let

I (x0)={i : (hi ◦Fi)(x0)=0}.Recall that T G(x0) stands for the set G′(x0)

−1TK(G(x0)). Then, by (31) ,

T G(x0)=m⋂

i=1

F ′i (x0)

−1TKi(Fi(x0))=

⋂

i∈I (x0)

F ′i (x0)

−1TKi(Fi(x0)). (32)

Indeed, if i �∈ I (x0), then hi(Fi(x0))< 0 and so Fi(x0) is an interior pointof Ki . This implies that

TKi(Fi(x0))=Zi, for every i �∈ I (x0),

and (32) is clear. Consequently,

T G(x0)�=

∑

i∈I (x0)

{F ′i (x0)

−1TKi(Fi(x0))}�=

∑

i∈I (x0)

F ′i (x0)

∗NKi(Fi(x0)). (33)

We say that x0 ∈S is a normal point of S with respect to (29) if it is a nor-mal point with respect to (30). Thus, henceforth we may refer this case bysaying that x0 is a normal point of S.

By standard results in convex analysis (cf. Ref. 11, Corollary 1 ofTheorem 2.4.7 and Proposition 2.3.9), one has that for each i ∈ I (x0),

∂hi(Fi(x0))⊆NKi(Fi(x0)), (34)

∂(hi ◦Fi)(x0)=F ′i (x0)

∗∂hi(Fi(x0)), (35)

where ∂(hi ◦Fi) is the Clarke subdifferential. Consequently, it follows from(32) that

cone{∂(hi ◦Fi)(x0) : i ∈ I (x0)}⊆∑

i∈I (x0)

F ′i (x0)

∗NKi(Fi(x0))

⊆T G(x0)�. (36)

Thus, applying Theorem 3.1 to T G(x0) for T0 with the first two sets in(36), respectively for N ′, we have immediately the following two theorems.


Theorem 4.1. Let S be the feasible set of the system (29) and let x0 ∈S

be a normal point. Then, the following statements are equivalent:

(i) NC∩(x0+T G(x0))(x0)=NC(x0)+ cone{∂(hi ◦Fi)(x0) : i ∈ I (x0)}.

(ii) For any x ∈X, x0 ∈PS(x) if and only if there existξi ∈∂(hi ◦Fi)(x0) and λi ≥0, with λi =0 for all i �∈I (x0), such that

x0 =PC

x −∑

i∈I (x0)

λiξi

.

Theorem 4.2. Let S be the feasible set of the system (29) and let x0 ∈S

be a normal point. Then, the following statements are equivalent:

(i) NC∩(x0+T G(x0))(x0)=NC(x0)+∑

i∈I (x0)F ′

i (x0)∗NKi

(Fi(x0)).

(ii) For any x ∈X, x0 ∈PS(x) if and only if, for every i ∈ I (x0), thereexist ξi ∈NKi

(Fi(x0)) such that

x0 =PC

x −∑

i∈I (x0)

F ′i (x0)

∗ξi

.

Following the notations in Ref. 14, let

CFD(x0) :={d ∈X :Fi(x0)+F ′i (x0)d ∈Ki for each i ∈ I (x0)}

=⋂

i∈I (x0)

F ′i (x0)

−1(Ki −Fi(x0)).

Note that

CFD(x0)⊆⋂

i∈I (x0)

F ′i (x0)

−1TKi(Fi(x0)).

Taking polars of the sets on both sides and applying the bipolar theorem(Ref. 10, Theorem 2.4.3), we have

∑

i∈I (x0)

F ′i (x0)

∗NKi(Fi(x0))⊆CFD(x0)

�. (37)

The following result was proved in Ref. 14 for the special case whenZi , i =1,2, . . . ,m are the same.


Theorem 4.3. Let S be the feasible set of (29) and let x0 ∈S. Supposethat S is convexified at x0 by CFD(x0) with respect to C. Then, The fol-lowing statements are equivalent:

(i) NC∩(x0+CFD(x0))(x0)=NC(x0)+ cone{∂(hi ◦Fi)(x0) : i ∈ I (x0)}.(ii) For each x ∈X, x0 ∈PS(x) if and only if there exist

ξi ∈∂(hi ◦Fi)(x0) and λi ≥0, with λi =0 for all i �∈I (x0), such that

x0 =PC

x −∑

i∈I (x0)

λiξi

.

Proof. In view of (36) and (37), one can apply Theorem 3.1 toCFD(x0) and cone{∂(hi ◦Fi)(x0) : i ∈ I (x0)} for T0 and N ′.

Below we provide an example for which Theorems 4.1 and 4.2 areapplicable, but not Theorems 4.3.

Example 4.1. Let X = Y = R2; let C be the ball in R2 with center(1, 0) and radius 1. Let h,F be defined by

h(x1, x2)=x21 −x2 and F(x1, x2)= (x1, x

22 ).

Take

x0 = (0,0) and K :=h−1(−∞,0].

Then, TK(F (x0)) is the upper half-plane and so contains the range of themap

F ′(x0)=(

1 00 0

).

Hence,

T G(x0)≡F ′(x0)−1TK(F (x0))

is the whole space X. Note that

S :=C ∩F−1(K)={x ∈R2 : (x1 −1)2 +x2

2 ≤1, x21 ≤x2

2

};

hence, coTS(x0) is the right half-plane. Thus,

S ⊆C ∩ (x0 +T G(x0))=C ∩ (x0 + coTS(x0)).


On the other hand,

CFD(x0)=F ′(x0)−1(K −F(x0))

is clearly the singleton {(0,0)}. Hence,

S �C ∩ (x0 +CFD(x0)).

Thus, S is not convexified at x0 by CFD(x0) with respect to C.For the remainder of this paper, we further specialize (29) to study a

finite system of linear inequalities/equalities:

(LS) x ∈C,

〈ai, x〉≤bi, i =1, . . . , �,

〈ai, x〉≤bi, i =�+1, . . . ,m,

with obvious interpretation if �=0 or �=m. Here, x and each ai are vec-tors in a Hilbert space X, each bi is a real number and C ⊆X is a closedconvex set. Let S consist of all x satisfying (LS). We assume that S is non-empty; as before, for each x0 ∈S, let

I (x0)={i : 1≤ i ≤�, 〈ai, x〉−bi =0}∪ {�+1, . . . ,m};

let

I ′(x0) :={i: 1≤ i ≤�, i �∈ I (x0)}.

Denote

Y =Rm and G(x)=Ax −b,

where A is the linear operator from X into Y defined by

Ax = (〈ai, x〉)i=1,...,m.

Let K ⊆Rm be defined by

K :=R�− ×{0}m−�.

Then,

TK(G(x0))=RI (x0)− ×RI ′(x0);


more precisely, u= (u1, . . . , um) belongs to TK(G(x0)) if and only if

ui ≤0, for each i ∈ I (x0).

Therefore, λ= (λ1, . . . , λm)∈NK(G(x0)) if and only if

λi ≥0, for each i ∈ I (x0),

λi =0, for each i ∈ I ′(x0),

Consequently,

G′(x0)∗NK(G(x0))=

∑

i∈I (x0)

λiai :λi ≥0 for each i

; (38)

in particular, this set is closed. Recalling the notation

T G(x0)=G′(x0)−1TK(G(x0)),

one has

x0 +T G(x0)={x ∈X : 〈ai, x〉≤bi, for each i ∈ I (x0)}. (39)

Thus, the first inclusion in the following is clear,

S ⊆C ∩ (x0 +T G(x0))⊆C ∩ (x0 +TS(x0)). (40)

To see the second inclusion, let ξ ∈ T G(x0). Then, 〈ai, ξ〉≤ 0 for each i ∈I (x0), and the equality holds for each i = � + 1, . . . ,m. Consequently, forall ε >0 small enough, one has that

〈ai, x0 + εξ〉≤0, for all i =1, . . . , �,

and the equality holds for all i =�+1, . . . ,m. Thus,

x0 + εξ ∈S, for all such ε;hence, ξ ∈TS(x0). The equivalences of (i)–(iii) in the following theorem aredue to Deutsch, Li and Ward (Refs. 4, 8). We denote by Hi the set {x :〈ai, x〉≤bi} for 1≤ i ≤� and the set {x : 〈ai, x〉=bi} for �+1≤ i ≤m.

Theorem 4.4. Let S be the feasible set of (LS). Then, each point ofS is normal. Moreover, the following statements are equivalent:

(i) {C,Hi(i ∈ I (x0))} has the strong CHIP.


(ii) For any x ∈X, x0 ∈PS(x) if and only if there exist λi ≥0 for eachi ∈ I (x0) such that x0 =PC

(x −∑

i∈I (x0)λiai

).

(iii) For any differentiable convex function f : X → R, x0 minimizesf over S if and only if there exists λi ≥0 for each i ∈ I (x0) suchthat

−f ′(x0)−∑

i∈I (x0)

λiai ∈NC(x0).

Proof. The first assertion follows from (40). Note that the set x0 +T G(x0) is the same as the set ∩i∈I (x0)Hi . By (5), (38), and (39), the equi-valences of (i)–(iii) are clear from Theorem 3.2.

References

1. de Boor, C., On Best Interpolation, Journal of Approximation Theory, Vol.16, pp. 28–42, 1976.

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4. Deutsch, F., Li, W., and Ward, J. D., A Dual Approach to Constrained Inter-polation from a Convex Subset of Hilbert Space, Journal of ApproximationTheory, Vol. 90, pp. 385–414, 1997.

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8. Deutsch, F., Li, W., and Ward, J. D., Best Approximation from the Intersec-tion of a Closed Convex Set and a Polyhedron in Hilbert Space, Weak SlaterConditions, and the Strong Conical Hull Intersection Property, SIAM Journalon Optimization, Vol. 10, pp. 252–268, 1999.

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10. Aubin, J. P., and Frankowska, H., Set-Valued Analysis, Birkhauser, Boston,Massachusetts, 1990.


11. Clarke, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons,New York, NY, 1983.

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13. Robinson, S. M., Stability Theory for Systems of Inequalities, II: Differentia-ble Nonlinear Systems, SIAM Journal on Numerical Analysis, Vol. 13, pp.497–513, 1976.

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15. Li, C., and Jin, X, Q., Nonlinearly Constrained Best Approximation in HilbertSpaces: The Strong CHIP annd the Basic Constraint Qualification, SIAMJournal on Optimization, Vol. 13, pp. 228–239, 2002.

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Best Approximation and Perturbation Property in Hilbert Spaces