c 2005 Society for Industrial and Applied Mathematics
SIAM J. OPTIM.
Vol. 15, No. 2, pp. 488–512
ON CONSTRAINT QUALIFICATION FOR AN INFINITE SYSTEM
OF CONVEX INEQUALITIES IN A BANACH SPACE∗
CHONG LI† AND K. F. NG‡
Abstract. For a general infinite system of convex inequalities in a Banach space, we study the
basic constraint qualification and its relationship with other fundamental concepts, including various
versions of conditions of Slater type, the Mangasarian–Fromovitz constraint qualification, as well
as the Pshenichnyi–Levin–Valadier property introduced by Li, Nahak, and Singer. Applications are
given in the restricted range approximation problem, constrained optimization problems, as well as
in the approximation problem with constraints by conditionally positive semidefinite functions.
Key words. system of convex inequalities, basic constraint qualification, Slater condition,
MFCQ, PLV property, optimality condition, best approximation
AMS subject classifications. Primary, 90C34; Secondary, 90C46, 41A29
DOI. 10.1137/S1052623403434693
1. Introduction. Let X be a Banach space over the real field R, C be a closed
convex subset of X, and let {gi : i ∈ I} be a family of continuous convex functions
on X, where I is an arbitrary (but nonempty) index-set. We consider the following
system of convex inequalities:
gi (x) ≤ 0,
(1.1)
i ∈ I,
and the associated system for the sup-function G,
G(x) ≤ 0,
(1.2)
where G is defined by
(1.3)
G(x) := sup gi (x)
for all x ∈ X.
i∈I
In this paper, we always assume that the sup-function G(.) is continuous on X.
It is clear that if X is of finite dimension and G(x) is finite for each x ∈ X, or if
{gi : i ∈ I} is locally uniformly bounded, then the condition that G(.) is continuous
on X is automatically satisfied. Throughout we use S to denote the solution set of
(1.1) (equivalently, of (1.2)):
(1.4)
S := {x ∈ X : gi (x) ≤ 0
for all i ∈ I} = {x ∈ X : G(x) ≤ 0}.
Let
(1.5)
K := C ∩ S;
∗ Received by the editors September 17, 2003; accepted for publication (in revised form) June 28,
2004; published electronically February 19, 2005.
http://www.siam.org/journals/siopt/15-2/43469.html
† Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China (cli@zju.edu.
cn). This author was supported in part by National Natural Science Foundation of China grant
10271025.
‡ Department of Mathematics, Chinese University of Hong Kong, Hong Kong, P. R. China (kfng@
math.cuhk.edu.hk). This author was supported by a Direct Grant (CUHK) and an Earmarked Grant
from the Research Grant Council of Hong Kong.
488
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
489
that is, K is the solution set of (1.1) with the constraint
(1.6)
x ∈ C.
Many important problems in mathematical programming are based on relationships
between the normal cone NK (x) in relation to NC (x) and the subdifferentials of G, gi
with i ∈ I(x). Here x ∈ X, and I(x) is the set of “active indices” at x defined by
(1.7)
I(x) := {i ∈ I : gi (x) = G(x)}.
In the classical case (that is, when I is finite, X is a Euclidean space, and each gi is differentiable), many fundamental notions of so-called constraint qualifications for (1.1)
have been introduced, such as the basic constraint qualification (BCQ), the Slater
condition (SC), and the Mangasarian–Fromovitz constraint qualification (MFCQ);
their interrelationships are well known and play a very important part not only in
optimization problems but also in many other areas, such as best approximation theory. A big step forward from the classical theory was recently achieved by Li, Nahak,
and Singer in [16], where detailed studies on the BCQ and the SC have been made
for the case when the index-set I is not restricted to be finite. They introduced and
studied the Pshenichnyi–Levin–Valadier (PLV) property and the weak PLV property;
in particular, they made use of these properties in extending the classical theory to
the case when I is infinite. Another direction of extension was done in [12, 13], where
I was assumed to be finite but X was allowed to be an infinite dimensional Hilbert
space; moreover, the constraint (1.6) was considered.
In this paper, we propose to study the general case: X is a Banach space, C is
a closed convex subset, and I is an arbitrary set. In this general case we introduce
and study various notions of constraint qualifications. In particular, in the stated
general setting, the notions of the BCQ and the PLV are introduced in section 2. We
show in Theorem 2.4 that, in the presence of the PLV, (1.1) satisfies the BCQ if and
only if (1.2) satisfies the BCQ. Our main results are presented in section 3, where
we establish some sufficient conditions to ensure the BCQ. Along with the extended
notions of the SC and the MFCQ, the weak versions (weak SC and weak MFCQ) are
also introduced. Some of their relationships are established. We show in particular
that, under the weak SC, the PLV implies the BCQ for (1.1). We remark that many
of the results obtained in this paper, for example, Theorems 3.8 and 3.10, are new
even for the case when C = X is a finite dimensional space. In particular, Theorem
3.10 provides a set of conditions ensuring that the following implication holds:
Each finite subsystem of (1.1) satisfies the BCQ=⇒the system (1.1) does.
Meanwhile, Examples 3.2, 3.3, and 3.4 show that the above implication is no longer
true if any of the conditions in the theorem are dropped. The last three sections of
the paper are devoted to applications of the main results. In section 4, we study the
problem of minimizing a real-valued function f (x) subjected to x ∈ C ∩S and we show
that, for an optimal point x0 , the BCQ of (1.1) relative to C at x0 is closely related
to the KKT property. In section 6, we address an approximation problem studied by
H. Strauss. We give a counterexample to show that [22, Theorem 6.2] is not true,
and we provide a new version which is established under the added assumption of the
BCQ.
One of the reasons why we study the proposed general case instead of the more restrictive case is that there are many approximation and optimization problems which
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CHONG LI AND K. F. NG
are in the general Banach space setting with infinitely many convex constraints. For
example, it was shown in [14] that the restricted Chebyshev approximation problem
(see [11, 21] and references therein) in the Banach space of complex-valued continuous functions on a compact Hausdorff space can be reformulated as a constrained
approximation problem with a system of convex inequalities. Section 5 is devoted to
another application dealing with the best restricted range approximation problem in
Lp space studied by Levasseur and Levis: a main result from [10] is extended and
improved under much weaker conditions.
2. Preliminaries, the BCQ, and the PLV property. For a set Z in X
(or in Rn ), ext Z denotes the set of all extreme points of Z and the interior (resp.,
relative interior, closure, convex hull, convex cone hull, linear hull, negative polar,
boundary) of Z is defined by int Z (resp., ri Z, Z, conv Z, cone Z, span Z, Z , bd Z).
We use |Z|(≤ ∞) to denote the number of elements of Z and use dim Z to denote the
dimension of span (Z − z), where z is an element of Z. For x ∈ X, in the case when
Z is closed and convex, the normal cone of Z at x is denoted by NZ (x) and defined
by NZ (x) = (Z − x) . R− denotes the subset of R consisting of all nonpositive real
numbers.
For a proper convex function on X and x ∈ X, the subdifferential of f at x is
defined by
∂f (x) := {z ∗ ∈ X ∗ : f (x) + z ∗ , y − x ≤ f (y)
for all y ∈ X}.
In particular, NZ (z) = ∂IZ (z) for each z ∈ Z. Here and throughout, IZ denotes the
/ Z.
indicator function of Z: IZ (x) = 0 if x ∈ Z and IZ (x) = +∞ if x ∈
Recall that the directional derivative f (x, d) of the function f at x in the direction
d is defined by
(2.1)
f (x, d) = lim
t→0+
f (x + td) − f (x)
.
t
Then (cf. [6, Proposition 2.2.7]) if f is convex and if x is a continuity point of f ,
(2.2)
∂f (x) = {z ∗ ∈ X ∗ : z ∗ , d ≤ f (x, d)
for all d ∈ X}
and
(2.3)
f (x, d) = max{z ∗ , d : z ∗ ∈ ∂f (x)}.
Remark 2.1. For a continuous convex function f on X, it is easy to show that
cone ∂f (x) ⊆ Nf −1 (R− ) (x) if f (x) = 0 and that the equality holds in both of the following cases: (a) f is affine, (b) x is not a minimizer of f . See [6, Corollary 1, p. 56].
Definition 2.1. Let x ∈ X, and let I(x) be defined by (1.7). Assuming I(x) = ∅,
define D (x) and N (x) by
⎛
⎞
D (x) := conv ⎝
(2.4)
∂gi (x)⎠ ,
i∈I(x)
⎛
(2.5)
N (x) := cone ⎝
i∈I(x)
⎞
∂gi (x)⎠ ;
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
491
also we adopt the convention that
N (x) = {0} if I(x) = ∅.
(2.6)
Part (ii) of the following proposition is well known while the other parts can be
verified easily by definitions.
Proposition 2.2. The following assertions hold:
(i) If i ∈ I(x), then ∂gi (x) ⊆ ∂G(x). Hence
(2.7)
N (x) ⊆ cone ∂G(x)
for all x ∈ X
and
(2.8)
D (x) ⊆ ∂G(x)
for all x ∈ X.
D (x) = ∂G(x)
for all x ∈ X.
(ii) If I is finite, then
(2.9)
(iii) If G(x) = 0 (e.g., if x ∈ bd S), then
(2.10)
∂G(x) ⊆ NK (x)
and
(2.11)
NC (x) + N (x) ⊆ NC (x) + cone ∂G(x) ⊆ NK (x).
Below we define the conversed relations of the inclusions given in the preceding
proposition.
Definition 2.3. Let x ∈ X. The system (1.1) is said to satisfy
(a) the BCQ at x relative to C if
(2.12)
NC (x) + N (x) ⊇ NK (x);
(b) the PLV at x relative to C if
(2.13)
NC (x) + D (x) ⊇ NC (x) + ∂G(x).
The system (1.1) is said to satisfy the BCQ relative to C if it satisfies the BCQ at
each point of K. In addition, the wording “relative to C” need not be mentioned if
C = X.
The following theorem (the proof of which is routine and hence omitted) extends
a result of Li, Nahak, and Singer for the case when C = X is finite dimensional.
Theorem 2.4. Let x ∈ C ∩ bd S. Suppose that the system (1.1) satisfies the PLV
at x relative to C. Then the system (1.1) satisfies the BCQ at x relative to C if and
only if (1.2) satisfies the BCQ at x relative to C.
Part (ii) of the following theorem is well known; see, for example, [9, Theorem
4.4.2] and [16, Theorem 3.1]. The proof given in [16] is valid for any compact space
I satisfying the first axiom of countability (in the sense that there exists a countable
local base at each point of I). For wider scope of applications (such as that in section
5), we do not assume that I is metrizable.
Theorem 2.5. Suppose that I is a compact space satisfying the first axiom of
countability and that the function: i → gi (x) is upper semicontinuous for each x ∈ X.
Then I(x) = ∅ for each x ∈ X and the following assertions hold:
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CHONG LI AND K. F. NG
w∗
(i) ∂G(x) = D (x) .
(ii) ∂G(x) = D (x), provided that I(x) is finite or X is finite dimensional.
(iii) If x ∈ span C and if span C is finite dimensional, then the system (1.1)
satisfies the PLV relative to C at x.
Furthermore, if in addition {gi : i ∈ I} is equicontinuous on X, then the assumption on I about the first axiom of countability can be dropped.
Proof. Let x, d ∈ X. By assumption, it is easy to verify that I(x) = ∅. For each
t > 0, define
gi (x + td) − G(x)
(2.14)
≥ G (x, d) .
It := i ∈ I :
t
By convexity, I(x + td) ⊆ It and it follows that It is nonempty. Moreover, {It : t > 0}
has the “finite intersection property” (and hence ∩t>0 It = ∅) because the function
(2.15)
t→
gi (x + td) − gi (x) gi (x) − G(x)
gi (x + td) − G(x)
=
+
t
t
t
is nondecreasing on (0, +∞) (the functions represented in the two terms of the righthand side of (2.15) are nondecreasing by the convexity of gi and the fact that gi (x) −
G(x) ≤ 0). Pick i0 ∈ ∩t>0 It . Then passing to the limits as t → 0 in
gi0 (x + td) − G(x) ≥ tG (x, d),
(2.16)
we see that gi0 (x) = G(x) (that is, i0 ∈ I(x)) and so (2.16) entails gi0 (x, d) ≥ G (x, d).
Consequently, it follows from (2.3) that
∗
max
(2.17) G (x, d) = max {gi (x, d)} = max
x
,
d
= ∗max
x∗ , d.
∗
i∈I(x)
i∈I(x)
x ∈∂gi (x)
x ∈D (x)
Thus (i) must hold by [6, Proposition 2.1.4]. Note that we have not used the assumption on the first axiom of countability. For (iii), write X̃ for span C; let G̃ and g̃i ,
respectively, denote the restrictions of G and gi to X̃. Clearly, G̃ is the sup-function
of {g̃i i ∈ I} and it follows from (ii) that, for x ∈ X̃,
(2.18)
∂ G̃(x) = conv {∂g̃i (x) : i ∈ I(x)}.
To prove (iii), we need only show that
(2.19)
∂G(x) ⊆ NC (x) + D (x).
To do this, let y ∗ ∈ ∂G(x), and let ỹ ∗ denote the
of y ∗ to X̃. Then
restriction
m
∗
∗
∗
ỹ ∈ ∂ G̃(x), and so it follows from (2.18) that ỹ = j=1 λj ỹj for some {λj } ⊆ [0, 1]
m
and {ỹj∗ } ⊆ X̃ ∗ with j=1 λj = 1, ỹj∗ ∈ ∂g̃ij (x), and i1 , i2 , . . . , im ∈ I(x). Noting
(2.20)
ỹj∗ , z ≤ g̃ij (x, z) = gij (x, z)
for all z ∈ X̃,
one can apply the Hahn–Banach theorem to obtain a continuous linear extension yj∗
of ỹj∗ such that
yj∗ , · ≤ gij (x, ·)
on X.
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
493
Thus yj∗ ∈ ∂gij (x) and
m
y ∗ , · =
λj yj∗ , ·
on X̃.
j=1
m
Since C − x ⊆ X̃, this implies that y ∗ − j=1 λj yj∗ ∈ NC (x). Therefore y ∗ ∈ NC (x) +
N (x) and (2.19) is proved.
Now we drop the assumption that I satisfies the first axiom of countability but
assume that {gi : i ∈ I} is equicontinuous on X. By the above proof, (iii) will be
true if (ii) holds. Hence, to complete the proof it suffices to show that ∂G(x) = D (x)
when X is finite dimensional. For this purpose, we need to introduce a new topology
on the set I. For each i0 ∈ I, x ∈ X, and each r ∈ R, r > 0, set
Ox (i0 , r) = {i ∈ I : gi (x) − gi0 (x) < r}.
Since X is finite dimensional, there exists a countable subset A of X such that the
closure A is equal to X. Let τN denote the topology generated by {Ox (i0 , r) : x ∈
A, i0 ∈ I, r ∈ R, r > 0}. Then, under the topology τN , the following assertions hold:
(1) I is compact;
(2) I satisfies the first axiom of countability;
(3) for each x ∈ X, the function i → gi (x) is upper semicontinuous.
In fact, (2) is trivial, and (1) holds because I is compact under the original
topology, which is stronger than the new topology τN , as each function i → gi (x) is
upper semicontinuous. (3) follows from the density of A in X and the equicontinuity
of {gi : i ∈ I} on X. In fact, let x ∈ X and > 0. By the equicontinuity, there exists
δ > 0 such that
(2.21)
for each i ∈ I and each y with x − y < δ.
|gi (x) − gi (y)| <
3
Pick x ∈ A such that x − x < δ. Let i0 ∈ I. Noting that, by (2.21),
(2.22)
gi (x) − gi0 (x) <
2
+ gi (x ) − gi0 (x ),
3
we get
(2.23)
gi (x) − gi0 (x) < for each i ∈ Ox (i0 , /3).
This shows that the function i → gi (x) is upper semicontinuous at i0 , and hence (3)
holds. Thus, by the first part of the proof, it is easy to see that ∂G(x) = D (x) and
the proof is complete.
Remark 2.2. In general, D (x) need not be w∗ -closed, and hence the w∗ -closure
in (i) cannot be dropped. For example, take X = l2 , the Hilbert space consisting of
∞
all infinite real sequences (xi ) satisfying j=1 |xj |2 < ∞. Write
ej = (0, . . . , 0, 1, 0, . . .)
for all j = 1, 2, . . . ,
that is, the jth component of ej is 1 while all other components are 0. Let I =
{0, 1, 1/2, . . . , 1/j, . . .} and define {gi }i∈I as follows:
0,
i = 0,
gi (x) =
i ∈ I \ {0}.
ie1/i , x,
494
CHONG LI AND K. F. NG
Let x = 0. Then I(x) = I,
∂gi (x) =
and hence,
D (x) =
⎧
⎨
⎩
{0},
{ie1/i },
i = 0,
i = 0,
m
m
⎫
⎬
1
λj ej : m > 0, λj ≥ 0,
λj ≤ 1 .
⎭
j
j=1
j=1
Clearly, D (x) is not closed.
3. The SC and the MFCQ. We continue to consider the system (1.1) relative
to a closed convex set C in X. Notation is as in the introduction. A main aim of this
section is to provide sufficient conditions ensuring that (1.1) satisfies the BCQ relative
to C. These conditions are of two types (the SC and the MFCQ): one is expressed
in terms of the functions gi , while the other is expressed in terms of the directional
derivatives of gi . Let us first extend several known concepts to our present general
case from the semi-infinite case (cf. [2, 12, 15] for the former type and [12, 17, 23] for
the latter type). Recall that S and K are defined by (1.4) and (1.5).
Definition 3.1. Let x ∈ C ∩ bd S. An element d ∈ X is called
(a) a linearized feasible direction of (1.1) at x if
(3.1)
z ∗ , d ≤ 0
for all z ∗ ∈ ext ∂gi (x) and for all i ∈ I(x);
(b) a sequentially feasible direction of K at x if there exist a sequence (dk ) with
(dk ) → d and a sequence (δk ) of positive real numbers with (δk ) → 0 such that
{x + δk dk } ⊆ K.
Let LFD(x) (resp., SFD(x)) denote the set of all d satisfying (a) (resp., (b)). Note
that LFD(x) and SFD(x) are both closed convex cones.
Definition 3.2. Let KS (x), KL (x) be, respectively, defined by
KS (x) = (x + SFD(x)) C
(3.2)
and
(3.3)
KL (x) = (x + LFD(x))
C.
Note that these two sets are closed convex sets.
Proposition 3.3. Let x ∈ C ∩ bd S. Then SFD(x) ⊆ LFD(x), and hence
(3.4)
K ⊆ KS (x) ⊆ KL (x).
Proof. Since K ⊆ KS (x), (3.4) follows from the first assertion and the fact
that LFD(x) is closed convex. To prove the first assertion, let d ∈ SFD(x), and
let {dk }, {δk } be as in Definition 3.1(b). In particular, for each i ∈ I(x), one has
gi (x + δk dk ) ≤ 0 and hence, for each z ∗ ∈ ∂gi (x), we have
(3.5)
z ∗ , δk dk ≤ gi (x + δk dk ) − gi (x) ≤ 0
thanks to the fact that G(x) = 0 as x ∈ bd S. This implies that z ∗ , dk ≤ 0. Letting
k → +∞ we obtain z ∗ , d ≤ 0. Since i ∈ I(x) and z ∗ ∈ ∂gi (x) are arbitrary,
d ∈ LFD(x), and the proof is complete.
Definition 3.4. Let x ∈ C ∩ bd S. We say that the system (1.1) satisfies
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
495
(a) the M F CQ relative to C at x if the following conditions are satisfied:
(1) (x + LFD(x)) ∩ ri C = ∅;
(2) {d ∈ X : gi (x, d) < 0 for all i ∈ I(x)} ∩ span (C − x) = ∅;
(b) the weak MFCQ relative to C at x if there exists a finite subset I0 of I(x)
such that
(1) (x + LFD(x)) ∩ ri C = ∅;
(2) {d ∈ X : gi (x, d) < 0 for all i ∈ I(x) \ I0 } ∩ span (C − x) = ∅;
(3) gi is affine for each i ∈ I0 .
We shall also say that x satisfies the MFCQ (resp., the weak MFCQ) for (1.1) relative
to C when (a) (resp., (b)) occurs.
For the following definitions, we need additional notation. Let
(3.6)
˜
I(x)
= {i ∈ I(x) : gi (x) = 0}.
˜
Clearly, I(x)
= I(x) in the case when x ∈ bd S.
Definition 3.5. The system (1.1) is said to satisfy
(a) the SC relative to C if there exists x̄ ∈ ri C such that G(x̄) < 0;
(b) the weak SC relative to C if there exists x̄ ∈ ri C ∩ S such that
˜
˜
(1) I(x̄)
is finite and gi is affine for each i ∈ I(x̄);
(2) G0 is continuous and G0 (x̄) < 0, where G0 denotes the sup-function of
˜
{gi : i ∈ I \ I(x̄)}:
(3.7)
G0 (z) =
sup gi (z),
˜
i∈I\I(x̄)
z ∈ X.
A point x̄ with the property in (a) (resp., (b)) is called a Slater (resp., weak Slater)
point for the system (1.1) relative to C.
As the nomenclature suggests, it is clear that (a)=⇒(b) in Definitions 3.4 and
3.5.
Theorem 3.6. Suppose there exists a weak Slater point x̄ for the system (1.1)
relative to C, and let x ∈ C ∩ bd S. Then (1.1) satisfies the BCQ relative to C at x
if (1.1) satisfies the PLV relative to C at x.
Proof. Let G0 be defined as in Definition 3.5(b). Set
˜
S0 = {y ∈ X : gi (y) ≤ 0, i ∈ I \ I(x̄)}
and
˜
S1 = {y ∈ X : gi (y) ≤ 0, i ∈ I(x̄)}.
˜
˜
is finite, and gi is affine for each i ∈ I(x̄).
Hence
Then G0 (x̄) < 0, I(x̄)
˜
cone ∂gi (x)
if I(x) ∩ I(x̄)
= ∅,
˜
i∈I(x)∩I(x̄)
NS1 (x) =
˜
0
if I(x) ∩ I(x̄)
=∅
and
NS0 (x) =
cone ∂G0 (x)
0
if G0 (x) = 0,
if G0 (x) < 0.
Note in particular that NS1 (x) and NS0 (x) are contained in cone ∂G(x). Moreover,
by [8, Proposition 2.3 and Theorem 5.1] (the proofs given there are valid for Banach
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CHONG LI AND K. F. NG
spaces, though the results were stated in a Hilbert space setting), {C, S1 } and {C ∩
S1 , S0 } have the strong conical hull intersection property (the strong CHIP). Hence
NK (x) = NC∩S1 (x) + NS0 (x) = NC (x) + NS1 (x) + NS0 (x)
⊆ NC (x) + cone ∂G(x) ⊆ NC (x) + N (x),
where the last inclusion holds because of the PLV property.
For convenience of reference, we note the following immediate consequence of
Theorem 2.5(iii) and Theorem 3.6.
Corollary 3.7. Suppose that I is a compact space satisfying the first axiom
of countability, and that the function: i → gi (x) is upper semicontinuous for each
x ∈ X. Let x ∈ K. Suppose that dim C < ∞ or I(x) is finite and that there exists
a weak Slater point for the system (1.1) relative to C. Then (1.1) satisfies the BCQ
relative to C at x.
Furthermore, if in addition {gi : i ∈ I} is equicontinuous on X, then the assumption on I about the first axiom of countability can be dropped.
Theorem 3.8. For the system (1.1) and relative to C, the following assertions
hold:
(i) If the weak SC is satisfied, then at each point in C ∩ bd S the weak MFCQ is
satisfied. If I is finite and if x ∈ C ∩ bd S satisfies the weak MFCQ, then there exist
weak Slater points arbitrarily near x.
(ii) If the SC is satisfied then, at each point in C ∩ bd S, the MFCQ is satisfied.
Conversely, if x ∈ C ∩ bd S satisfies the MFCQ and if in addition x satisfies the PLV,
then there exist Slater points arbitrarily near x (in particular, (1.1) satisfies the BCQ
relative to C).
Proof. (i) Let x̄ ∈ ri C ∩ S be a weak Slater point for the system (1.1) relative to
˜
C, and let x ∈ C ∩ bd S (hence G(x) = 0). Set I0 = I(x) ∩ I(x̄).
Then I0 is finite (as
˜
I(x̄)), and x with this I0 satisfies (1)–(3) of Definition 3.4(b). Indeed, (3) is evident.
˜
Suppose I(x) ⊆ I(x̄).
Then (2) of Definition 3.4(b) holds trivially. (1) also holds as
˜
x̄ ∈ (x + LFD(x)) ∩ ri C because, for each i ∈ I0 = I(x) ⊆ I(x̄),
one has
gi (x) = 0 and gi (x̄) = G(x̄).
It follows from the convexity that gi (x, x̄ − x) ≤ 0; that is, x̄ − x ∈ LFD(x). We
˜
can therefore assume henceforth that I(x) ⊆ I(x̄).
Consequently G0 (x) = G(x) = 0,
where G0 is the function on X defined by (3.7). Noting G0 (x̄) < 0, we have for each
i ∈ I(x) \ I0 that
gi (x, x̄ − x) ≤ gi (x̄) − gi (x) ≤ G0 (x̄) − G0 (x) < 0.
This implies that the intersection in (2) of Definition 3.4(b) is a nonempty set (containing x̄ − x) and that x̄ − x ∈ LFD(x) as gi is affine for each i ∈ I0 (because
˜
I0 ⊆ I(x̄)
and x̄ is a weak Slater point). Therefore x is a weak MFCQ point relative
to C. Conversely, we suppose that x satisfies the weak MFCQ relative to C and, in
addition, that I is finite. Let I0 ⊆ I(x) satisfy (2) and (3) of Definition 3.4(b). Then
by definition, x satisfies the MFCQ relative to C for the subsystem defined by
(3.8)
gi (·) ≤ 0
for all i ∈ I \ I0 .
Note that x satisfies the PLV for the subsystem (3.8) as I is finite. Hence, assuming
the results in (ii), we can apply (ii) to conclude that there exists x̄ ∈ ri C arbitrarily
near x such that sup{gi (x̄) : i ∈ I \ I0 } < 0. Clearly then x̄ is a weak Slater point for
(1.1) relative to C.
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
497
(ii) Let x̄, x be as in the first part of (i) but assume the stronger assumption that
˜
x̄ is a Slater point relative to C. Then I(x̄)
must be empty and so must I0 . In this
case, (2) of Definition 3.4(b) coincides with (2) of Definition 3.4(a). Therefore x is an
MFCQ point relative to C by (i). Conversely, suppose that x satisfies the MFCQ for
the system (1.1) relative to C and, in addition, that x also satisfies the PLV. Then,
by the former, there exist d1 , d2 such that
x + d1 ∈ (x + LFD(x)) ∩ ri C, d2 ∈ span(C − x),
and
gi (x, d2 ) < 0
(3.9)
for all i ∈ I(x).
Also, by the PLV property,
∂G(x) ⊆ NC (x) + conv {∂gi (x) : i ∈ I(x)} .
(3.10)
Take t0 > 0 small enough such that x+(1−t0 )d1 +t0 d2 ∈ ri C. Set d := (1−t0 )d1 +t0 d2 .
Then, for any i ∈ I(x),
(3.11)
gi (x, d) ≤ (1 − t0 )gi (x, d1 ) + t0 gi (x, d2 ) ≤ t0 gi (x, d2 ) < 0.
It follows that
(3.12)
G (x, d) < 0.
G (x, d) = z ∗ , x. By (3.10),
Indeed, by (2.3) there exists z ∗ ∈ ∂G(x) such that m
∗
∗
∗
there exist t1 , . . . , tm > 0 and z1 , . . . , zm ∈ X with j=1 tj = 1, and zj∗ ∈ ∂gij (x)
m
where i1 , . . . , im ∈ I(x) such that z ∗ − j=1 tj zj∗ ∈ NC (x); in particular, z ∗ −
m
∗
j=1 tj zj , d ≤ 0. It follows from (3.11) and (2.2) that
m
z ∗ , d ≤
tj gij (x, d) < 0;
j=1
this proves (3.12). Noting G(x) = 0 (as x ∈ bd S), (3.12) implies that G(x + t̄d) < 0
for all sufficiently small t̄ ∈ (0, 1). Clearly x + t̄d ∈ ri C (provided t̄ < t0 ) and so x + t̄d
is a Slater point for the system (1.1) relative to C.
Remark 3.1. Example 3.1(a) below will show that the finiteness assumption
cannot be dropped in the second assertion of Theorem 3.8(i), and the condition of the
PLV in the second assertion of Theorem 3.8(ii) cannot be dropped. Example 3.1(b)
provides an example of (1.1) which satisfies the PLV and the weak MFCQ relative to
C at some point x0 ∈ C ∩ bd S but fails for the weak SC and the BCQ relative to C
at x0 . (In particular, the finiteness assumption in the second assertion of Theorem
3.8(i) cannot be replaced by the PLV.)
Example 3.1. (a) Let X = C = R, I = [−1, 0) ∪ (0, +1], and let {gi : i ∈ I} be
defined by
ix,
i ∈ [−1, 0)
gi (x) =
for all x ∈ R.
2ix − i2 , i ∈ (0, +1]
Then the corresponding sup-function is given by
⎧
x≥1
⎨ 2x − 1,
0≤x≤1
G(x) = x2 ,
(3.13)
⎩
−x,
x≤0
for all x ∈ R.
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CHONG LI AND K. F. NG
Take x0 = 0. Then, S = {0}, I(x0 ) = [−1, 0), and
∂gi (x0 ) = {i} for all i ∈ [−1, 0).
(3.14)
Moreover,
(3.15)
gi (x0 , 1) = i < 0
for all i ∈ [−1, 0)
and
(3.16)
LFD(x0 ) = [0, +∞).
For the system (1.1) relative to C at the point x0 , the following assertions hold:
(1) The MFCQ is satisfied.
(2) The PLV is not satisfied.
(3) The weak SC is not satisfied.
(4) The BCQ at x0 is not satisfied.
Indeed, (1) is evident by (3.15) and (3.16). Next, by (3.14),
(3.17)
D (x0 ) = [−1, 0),
N (x0 ) = (−∞, 0],
so that
∂G(x0 ) = [−1, 0] = D (x0 ) ⊆ (−∞, 0] = N (x0 ),
∂G(x0 ) = D (x0 ).
Hence, (2) holds. (3) is also evident because G0 (x̄) = 0 whenever x̄ ∈ S and G0 is the
sup-function of {gi : i ∈ I \ I0 } for some finite subset I0 of I.
Finally, (4) holds because
NK (x0 ) = NS (x0 ) = R,
N (x0 ) = (−∞, 0].
(b) Let g0 be defined by g0 (x) = 0 for all x ∈ X. We add this function g0 to
the system in (a) above. The new system satisfies the PLV relative to C at x0 and
the weak MFCQ relative to C at x0 . However, the weak SC on C does not hold;
hence the BCQ relative to C at x0 is not satisfied.
It is well known (see, e.g., [16]) that the following implication is not true in
general:
(∗)
Each finite subsystem of (1.1) satisfies the BCQ=⇒the system (1.1) does.
Therefore the remainder of this section is devoted to studying the following natural
question: When does this implication hold? The following theorem (which will be
useful for our discussion in section 5) provides a set of conditions ensuring the above
implication (∗). To facilitate the proof of this theorem, we recall the following proposition which was established independently by Deutsch [7] and Rubenstein [20] (see
also [4]).
For a closed convex subset Z of X, let PZ denote the projection operator defined
by
PZ (x) = {y ∈ Z : x − y = dZ (x)},
where dZ (x) denotes the distance from x to Z. Recall that the duality map J from
∗
X to 2X is defined by
J(x) := {x∗ ∈ X ∗ : x∗ , x = x2 , x∗ = x}.
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
499
In fact, J(x) = ∂φ(x), where φ(x) := 12 x2 . Thus a Banach space X is smooth if
and only if for each x ∈ X the duality map is single-valued.
Proposition 3.9. Let Z be a closed convex set in X. Then for any x ∈ X, z0 ∈
PZ (x) if and only if z0 ∈ Z and there exists x∗ ∈ J(x − z0 ) such that x∗ , z − z0 ≤ 0
for any z ∈ Z, that is, J(x − z0 ) ∩ NZ (z0 ) = ∅. In particular, when X is smooth,
z0 ∈ PZ (x) if and only if z0 ∈ Z and J(x − z0 ) ∈ NZ (z0 ).
Theorem 3.10. Let I be a compact metric space and let C be a closed convex
subset of X such that dim C := l < +∞. Suppose that
(i) for each x ∈ X, the function i → gi (x) is continuous;
(ii) for any finite subset J of I, the subsystem
gj (x) ≤ 0,
(3.18)
j ∈ J,
satisfies the BCQ relative to C;
(iii) for any finite subset J of I with |J| ≤ l, the subsystem (3.18) satisfies the SC
relative to C.
Then (1.1) satisfies the BCQ relative to C.
Proof. Since I is compact and metrizable, there exists a sequence {Ik } of subsets
of I such that
(a) each Ik is finite;
(b) Ik ⊆ Ik+1 for k = 1, 2, . . .;
(c) ∪∞
k=1 Ik = I.
Set Sk = ∩i∈Ik {x ∈ X : gi (x) ≤ 0} for each k. Then
K = C ∩ S ⊆ C ∩ Sk .
(3.19)
Let x0 ∈ K and x∗ ∈ NK (x0 ). We have to show that x∗ ∈ NC (x0 ) + N (x0 ). We
will first show that there exist{xk } ⊆ C with xk → x0 and x∗k ∈ NC∩Sk (xk ) such that
{x∗k } is bounded and
(3.20)
lim x∗k , y = x∗ , y for each y ∈ span (C − x0 ).
k→∞
In fact, since Z := span (C −x0 ) is of finite dimension, we may assume, without loss of
generality, that the norm restricted on Z is both strictly convex and smooth. Clearly,
we may assume that x∗ |Z = 0. Take z0 ∈ Z such that x∗ , z0 = x∗ |Z ·z0 = z0 2 .
Write x = x0 + z0 . Then, x∗ |Z = J(x − x0 )|Z and, by Proposition 3.9, x0 = PK (x).
Let xk = PC∩Sk (x). Then
xk ≤ xk − x + x ≤ x − x0 + x;
hence {xk } ⊂ C is bounded. Without loss of generality, assume that xk → x̄. Let
i ∈ I and > 0. By (b) and (c), there exists {ik } ⊆ I with ik ∈ Ik for each k such
that ik → i. By assumption (i), there exists an integer k0 such that
(3.21)
gi (x̄) − gik0 (x̄) < .
By (b), for each k > k0 , ik0 ∈ Ik0 ⊆ Ik ; hence, by the fact that xk ∈ Sk ⊆ Sk0 ,
(3.22)
gik0 (x̄) = gik0 (x̄) − gik0 (xk ) + gik0 (xk ) ≤ gik0 (x̄) − gik0 (xk ).
Consequently, by (3.21) and (3.22), we have that, for each k > k0 ,
(3.23)
gi (x̄) < + gik0 (x̄) − gik0 (xk ).
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CHONG LI AND K. F. NG
Taking the limit as k → ∞ we get gi (x̄) ≤ ; hence x̄ ∈ K as > 0 and i ∈ I are
arbitrary. Because
x − x̄ = lim x − xk ≤ x − y
for each y ∈ K,
k→∞
x̄ = PK (x) = x0 . Let x∗k ∈ X ∗ with x∗k |Z = J(x − xk )|Z and x∗k = x − xk . Then,
by Proposition 3.9, x∗k ∈ NC∩Sk (xk ). By the smoothness, the mapping x → J(x)|Z is
continuous and so (3.20) holds.
Now let Gk denote the sup-function corresponding to the subsystem (3.18) with
J = Ik ; that is,
Gk (x) := sup gi (x)
for each x ∈ X.
i∈Ik
If there exists a subsequence {kj } of {k} such that Gkj (x0 ) < 0 for each j, then
NC∩Skj (xkj ) = NC (xkj ), and hence x∗kj ∈ NC (xkj ). This implies that x∗ ∈ NC (xkj )
by (3.20) (and so x∗ ∈ NC (x0 ) + N (x0 )). Therefore we may assume that, for each k,
x∗k ∈
/ NC (xk ), and that Gk (xk ) = 0. Then Ik (xk ) := {i ∈ Ik : gi (xk ) = Gk (xk )} = ∅.
By assumption (ii), x∗k ∈ NC∩Sk (xk ) = NC (xk ) + cone {∂gi (xk ) : i ∈ Ik (xk )}, and
it follows from [19, Corollary 17.2.2] that there exist zk∗ ∈ NC (xk ), ikj ∈ Ik (xk ),
yi∗k ∈ ∂gikj (xk ), and λkj ≥ 0 (j = 1, 2, . . . , l) such that
j
l
x∗k = zk∗ +
(3.24)
λkj yi∗k
on Z,
k = 1, 2, . . . .
j
j=1
Let λk =
l
j=1
λkj . Then {λk } is bounded. Indeed, if not, by considering a subx∗
sequence if necessary, we have that limk→∞ λk = +∞. Thus λkk → 0 as k → ∞.
Furthermore, without loss of generality, we may assume that, as k → ∞,
ikj → ij
(3.25)
Note in particular that
and
l
j=1
λkj
→ λj ,
λk
j = 1, 2, . . . , l.
λj = 1. Since yi∗k ∈ ∂gikj (xk ) ⊆ ∂G(xk ) and since
j
z∗
xk → x0 , {yi∗k } is bounded. Consequently, by (3.24), { λkk |Z } is bounded too. Thus,
j
we may also assume that there exist z̃0∗ , ỹi∗j ∈ Z ∗ such that
zk∗
→ z̃0∗ ;
λk
(3.26)
yi∗k → ỹi∗j
on Z
j
as k → ∞. Then
(3.27)
z̃0∗ , z − x0 ≤ 0
for each z ∈ C
and, by assumption (i), (3.25), and (3.26), for each j,
(3.28)
ỹi∗j , z − x0 ≤ gij (z) ≤ gij (z) − gij (x0 )
for each z ∈ Z + x0 .
Hence by the Hahn–Banach theorem, z̃0∗ and ỹi∗j can be extended to z0∗ ∈ NC (x0 ) and
yi∗j ∈ ∂gij (x0 ). By (3.24), (3.25), and (3.26),
l
(3.29)
0 = z0∗ +
λj yi∗j
j=1
on Z.
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
501
By assumption (iii), take ȳ ∈ C such that gij (ȳ) < 0 for each j = 1, 2, . . . , l. Consequently, yi∗j , ȳ − x0 < 0 for each j = 1, 2, . . . , l, and hence,
l
z0∗ +
λj yi∗j , ȳ − x0
< 0,
j=1
which contradicts (3.29). Hence {λk } is bounded. Thus, taking the limits on the
two sides of (3.24) and using the similar arguments as above (if necessary, use subsequences), we get that
l
x∗ = z0∗ +
(3.30)
λj yi∗j
on Z
j=1
for some λj ≥ 0, z0∗ ∈ NC (x0 ), and yi∗j ∈ ∂gij (x0 ) (j = 1, 2, . . . , l). Note that, by
(3.28), yi∗j ∈ NC∩g−1 (R− ) (x0 ) for each j = 1, 2, . . . , l. Since yi∗j , ȳ − x0 < 0, it follows
ij
that yi∗j |Z = 0, and hence ij ∈ I(x0 ) for each j = 1, 2, . . . , l. Let y ∗ = x∗ − z0∗ −
l
∗
∗
∗
j=1 λj yij . Then y ∈ NC (x0 ) by (3.30), and so x ∈ NC (x0 ) + N (x0 ). The proof is
complete.
Remark 3.2. Example 3.2 below will show that the condition dim C < ∞ cannot
be dropped, while Examples 3.3 and 3.4 will show that conditions (i) and (iii) also
cannot be dropped. In each of these examples, I is a compact subset of R: I =
{0, 1, 12 , . . . , 1i , . . .}.
Example 3.2. Let C = X = {x = (x1 , x2 , . . . , xk , . . .) : limk xk exists} with the
norm defined by
x = sup |xk |,
x = (xk ) ∈ X.
k
Define
gi (x) =
limk xk ,
x 1i ,
i = 0,
i ∈ I \ {0},
x = (xk ) ∈ X.
Since |gi (x) − gi (y)| ≤ x − y for each i ∈ I, the corresponding sup-function G is
continuous on X. Note that G(x̄) = −1 < 0 for x̄ = (−1) ∈ X. Hence conditions (ii)
and (iii) in Theorem 3.10 hold. clearly, condition (i) in Theorem 3.10 is satisfied. Let
x0 = 0. Then x0 ∈ C ∩ S = S and I(x0 ) = I. It is easy to see that N (x0 ) is not
closed; hence this system does not satisfy the BCQ at x0 .
Example 3.3. Let C = X = R2 . Define
i = 0,
x1 + x2 ,
gi (x) =
x = (x1 , x2 ) ∈ X.
x1 + ix2 ,
i ∈ I \ {0},
Note that |gi (x) − gi (y)| ≤ 2x − y for each i ∈ I; thus the corresponding supfunction G is continuous on X. Let x0 = 0. Then K := C ∩ S = S = {(x1 , x2 ) : x1 ≤
0, x1 + x2 ≤ 0}, x0 ∈ bdS, and I(x0 ) = I. Since G((−1, 12 )) = − 12 < 0, conditions (ii)
and (iii) in Theorem 3.10 hold. However, NK (x0 ) = NS (x0 ) = {(t1 , t2 ) : 0 ≤ t2 ≤ t1 }
and N (x0 ) = {(t1 , t2 ) : 0 < t2 ≤ t1 } ∪ {(0, 0)}. Therefore this system does not satisfy
the BCQ at x0 . Note that condition (i) is not satisfied.
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CHONG LI AND K. F. NG
Example 3.4. Let C = X = R2 and define
i = 1,
x2 ,
gi (x) =
ix1 − x2 − i2 , i ∈ I \ {1},
x = (x1 , x2 ) ∈ X.
It is easy to verify that |gi (x)−gi (y)| ≤ 2x−y for each i ∈ I; thus the corresponding
sup-function G is continuous on X. Let x0 = 0. Then I(x0 ) = {0, 1} and K = S =
{(x1 , 0) ∈ R2 : x1 ≤ 0}. Hence
NK (x0 ) = {(t1 , t2 ) ∈ R2 : t1 ≥ 0}
and
N (x0 ) = cone {(0, −1), (0, 1)} = {(t1 , t2 ) ∈ R2 : t1 = 0}}.
Consequently, this system does not satisfy the BCQ at x0 . Note that conditions (i)
and (ii) in Theorem 3.10 are satisfied but condition (iii) is not.
4. Optimality conditions for minimization problems. The main purpose
of this section is to apply our results of the earlier sections to provide optimality
conditions for optimization problems of the following type. For a real-valued function
f on X, a typical problem is to
(4.1)
minimize
subject to
f (x)
x ∈ C ∩ S,
where C and S are as at the beginning of section 1. We will consider several kinds of
functions on X. Let A(X) (resp., C(X)) denote the class of all continuous real-valued
affine (resp., convex) functions on X. Moreover, for x0 ∈ X, let Gx0 (X) denote the
class of real-valued functions f on X such that f is Gateaux differentiable (with the
Gateaux derivative Df (x0 )) at x0 and the restriction of f to C is convex. Recall that
K := C ∩ S.
Theorem 4.1. Let x0 ∈ K. Then the following statements are equivalent:
(i) The system (1.1) satisfies the BCQ relative to C at x0 .
(ii) For any f ∈ Gx0 (X), x0 is an optimal solution of (4.1) if and only if
(4.2)
Df (x0 ) + x∗ +
λi x∗i = 0
on
X
i∈I0
for some finite subset I0 of I(x0 ), x∗ ∈ NC (x0 ), λi ≥ 0, and x∗i ∈ ∂gi (x0 ) (f or all i ∈
I0 ).
(ii∗ ) Same as (ii) but with (4.2) replaced by
(4.3)
Df (x0 ) + x∗ +
λi x∗i = 0
on
span C.
i∈I0
(iii) For any f ∈ C(X), x0 is an optimal solution of (4.1) if and only if
(4.4)
y ∗ + x∗ +
λi x∗i = 0
on
X
i∈I0
for some finite subset I0 of I(x0 ), y ∗ ∈ ∂f (x0 ), x∗ ∈ NC (x0 ), λi ≥ 0, and x∗i ∈ ∂gi (x0 )
(f or all i ∈ I0 ).
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
503
(iii∗ ) Same as (ii∗ ) but with (4.4) is replaced by
y ∗ + x∗ +
(4.5)
λi x∗i = 0
on
span C.
i∈I0
(iv) Same as (iii) but with f ∈ A(X).
(iv∗ ) Same as (iii∗ ) but with f ∈ A(X).
Proof. Since each of the statements in the above list is true (with I0 = ∅) when
x0 ∈ C ∩ int S, we assume henceforth that x0 ∈ C ∩ bd S.
(i)=⇒ (ii) and (ii∗ ). Let f ∈ Gx0 (X) define
F (x) = Df (x0 ), x − x0 + IK (x),
x ∈ X.
Then, by [6, Corollary 1, p. 105]
∂F (x0 ) = Df (x0 ) + NK (x0 ).
(4.6)
It follows from (i) that
(4.7)
∂F (x0 ) = Df (x0 ) + NC (x0 ) + N (x0 ).
If x0 is an optimal solution of (4.1), x0 is a minimal point of F on X. In fact, it is
clear that F (x) ≥ F (x0 ) for any x ∈ X \ K. Moreover, if x ∈ K, then
(4.8) F (x) = Df (x0 ), x − x0 = lim
t→0+
f (x0 + t(x − x0 )) − f (x0 )
≥ 0 = F (x0 )
t
because x0 + t(x − x0 ) ∈ K for each 0 < t < 1 (and x0 is an optimal solution of (4.1)).
Hence, by [6, Proposition 2.4.11], 0 ∈ ∂F (x0 ). Consequently, (4.7) entails (4.2) (and
(4.3)) for appropriate x∗ , λi , x∗i as stated in (ii) and (ii∗ ). Conversely, suppose
that
(4.3) holds for some I0 , x∗ , λi , x∗i as stated in (ii∗ ). Writing z ∗ for x∗ + i∈I0 λi x∗i ,
one has from (2.11) that z ∗ ∈ NK (x0 ) and from (4.3) that
Df (x0 ), · = −z ∗ , · on span C.
Therefore, by convexity, we have for each x ∈ K that
f (x) − f (x0 ) ≥ Df (x0 ), x − x0 = −z ∗ , x − x0 ≥ 0.
Thus the implications (i)=⇒ (ii) and (ii∗ ) are proved.
The implication (i)=⇒ (iii) is a direct consequences of [6, Corollary, p. 52]. The
necessity part of (iii∗ ) follows from (iii) while the sufficiency part of (iii∗ ) is proved
similarly to the proof of the corresponding part of (ii∗ ).
Further, since the implications (ii)=⇒ (iv), (ii∗ )=⇒ (iv∗ ), (iii)=⇒ (iv), and (iii∗ )
=⇒ (iv∗ ) are trivial, it remains to show that (iv)=⇒ (i) and (iv∗ )=⇒ (i). To show
these, let y ∗ ∈ NK (x0 ). Then −y ∗ ∈ A(X) and x0 is an optimal solution of (4.1) with
f := −y ∗ . Thus ∂f (x0 ) = −y ∗ and, assuming (iv) or (iv∗ ), there exist a finite subset
I0 of I(x0 ), x∗ ∈ NC (x0 ), λi > 0, and x∗i ∈ ∂gi (x0 ) (i ∈ I0 ) such that
−y ∗ + x∗ +
λi x∗i = 0
on span C,
i∈I0
that is,
y ∗ , · = z ∗ , · on span C,
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CHONG LI AND K. F. NG
where z ∗ := x∗ +
Consequently,
i∈I0
λi x∗i . Since x0 ∈ C, it follows that y ∗ − z ∗ ∈ NC (x0 ).
y ∗ ∈ NC (x0 ) + x∗ +
λi x∗i ⊆ NC (x0 ) + N (x0 ),
i∈I0
and (i) holds. This completes the proof.
5. Best restricted range approximation problem in Lp spaces. Let Ω :=
T ∪A, where A is a compact Hausdorff space and T is a measure space with a measure
µ. As usual, let CR (A) denote the Banach space of all real-valued continuous functions
on A equipped with the uniform norm denoted by · A .
Let X be the space consisting of all real functions x on Ω such that the restrictions
x|T ∈ Lp (T, µ) and x|A ∈ CR (A), where p ∈ [1, +∞). Then X is a Banach space under
the norm · defined by
x = x|T p + x|A A
for each x ∈ X.
For the so-called restricted range approximation problem in Lp spaces (studied by
Levasseur and Levis in [10]), one is given the following data: Let l and u be a pair
of real functions which are, respectively, upper and lower semicontinuous on A such
that l(t) ≤ u(t) for each t ∈ A. Let L be a finite dimensional vector subspace of X
generated by h1 , h2 , . . . , hn . Let z ∈ X be given. Then the problem to be considered is
(5.1)
|z(t) − x(t)|p dµ
minimize
T
subject to
(5.2)
x ∈ L,
l(t) ≤ x(t) ≤ u(t)
for all t ∈ A
(one can of course replace µ by µ of the form dµ = wdµ for some positive w ∈
L∞ (T, µ)).
The aim of this section is to apply results of the preceding section to establish
the following theorem, which was proved for the case when p = 2 (or p is an even
integer) in [10] using a very different method and under assumptions H1 –H6 given in
the following:
H1 : A is a compact subset of R and T is a finite union of compact intervals of R.
H2 : z is a real continuous function on Ω.
H3 : l and u are continuous on A with l(t) < u(t) for all t ∈ A.
H4 : {h1 , h2 , . . . , hn } is linearly independent on T .
H5 : There exists h ∈ X such that (5.2) is satisfied with x = h.
H6 : For each k = 1, 2, . . . , n, the set {h1 , h2 , . . . , hk } is a Chebyshev system of
order k on A; that is, if t1 , t2 , . . . , tk are distinct points in A, then the determinant
det(hj (ti )) = 0.
Instead of assumptions H1 –H6 , we consider the following two conditions:
D1 : There exists an element h0 ∈ L satisfying l(t) < h0 (t) < u(t) for all t ∈ A.
D2 : A is metrizable (in addition to being compact Hausdorff), l and u are continuous on A, and, for any k ≤ n and any distinct points t1 , t2 , . . . , tk in A, there exists
an element h̄ ∈ L satisfying l(ti ) < h̄(ti ) < u(ti ) for all i = 1, 2, . . . , k.
We remark that H1 + H6 =⇒ D2 .
505
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
We use Z(x) to denote the set of all points in T such that x(t) = 0. Note that if
x = x̃ a.e. on T , then the “symmetric difference” (Z(x) \ Z(x̃)) ∪ (Z(x̃) \ Z(x)) is of
µ-measure zero; hence Z(x) h(t)dµ = Z(x̃) h(t)dµ for each integrable function h on T .
We shall need the following well-known results; see, for example, [3, Example 2.19] for
a proof of Lemma 5.1. The other two lemmas follow immediately from Lemma 5.1
and the Riesz representation theorem.
Lemma 5.1. Let W be a normed vector space. Then, for any w ∈ W \ {0}, the
subdifferential ∂ · (w) of the norm function at w equals the set
{w∗ ∈ W ∗ : w∗ , w = w}.
Lemma 5.2. Let W = L1 (T, µ), where here and throughout, T denotes a measure
space with measure µ. Let w ∈ W \ {0}. Then, as a subset of L∞ (T ), ∂ · 1 (w)
consists of all β ∈ L∞ (T ) satisfying the properties that
|β(t)| ≤ 1
a.e. on T,
β(t) = sgn[w(t)] a.e. on T \ Z(w).
Lemma 5.3. Let W = Lp (T, µ) with p ∈ (1, +∞). Let w ∈ W \ {0}. Then, the
norm function is Fréchet differentiable at w with the derivative · p (w), which is
represented by ξ ∈ Lq (T, µ) ( p1 + 1q = 1) with
· |w(t)|p−1 · sgn[w(t)]
ξ(t) = w1−p
p
a.e. on T.
In particular, ∂ · p (w) = {ξ}.
Our main result in this section is now stated as follows.
Theorem 5.4. Suppose that D1 or D2 holds. Let x0 ∈ X satisfy (5.2) and let
z ∈ X. Then x0 is an optimal solution of the minimization problem (5.1) subject to
(5.2) if and only if there exist a finite subset {t1 , t2 , . . . , tk } of A and a finite subset
{λt1 , λt2 , . . . , λtk } of R (0 ≤ k ≤ n) with the following properties:
(a) x0 (ti ) = u(ti ) or x0 (ti ) = l(ti ) for each i = 1, 2, . . . , k;
(b) for each i = 1, 2, . . . , k,
1
if x0 (ti ) = u(ti ),
sgnλti =
−1
if x0 (ti ) = l(ti );
(c) for each j = 1, 2, . . . , n,
k
|z(t) − x0 (t)|p−1 sgn[z(t) − x0 (t)] hj (t)dµ =
(5.3)
T
sgn[z(t) − x0 (t)] hj (t)dµ +
(5.4)
T
λti hj (ti ),
p > 1;
i=1
k
β(t) hj (t)dµ =
Z(z−x0 )
λti hj (ti ),
p = 1,
i=1
for some β ∈ L∞ with |β(t)| ≤ 1 a.e. on Z(z − x0 ). (Note: we adopt the convention
that, if k = 0, the sums of the right-hand sides of (5.3) and (5.4) are read as zero.)
In order to prove this theorem, we first do some preparation. We make two
homeomorphic copies of A: one is still denoted by A while the other is denoted by
Ǎ = {ǎ : a ∈ A}, where a → ǎ is a bijection. Let I = A ∪ Ǎ. Thus I can be
506
CHONG LI AND K. F. NG
topologized such that A, Ǎ are two disjoint compact subsets of I. For each t ∈ A,
define gt : X → R by
gt (x) = et (x) − u(t),
(5.5)
x ∈ X,
where et : X → R is the “unit point functional” at t defined by
x ∈ X.
et (x) = x(t),
Similarly, for each ť ∈ Ǎ, define gť : X → R by
gť (x) = l(t) − et (x),
(5.6)
x ∈ X.
Thus {gi : i ∈ I} is a family of affine functions on X such that, for each i ∈ I,
(5.7)
|gi (x) − gi (y)| ≤ x − y,
x, y ∈ X.
In particular, {gi : i ∈ I} is equicontinuous on X, and the corresponding sup-function
x → G(x) := sup{gi (x) : i ∈ I} is continuous. Note furthermore that the function
i → gi (x) is upper semicontinuous for each x ∈ X. Clearly, x ∈ X satisfies (5.2) if
and only if x ∈ L and gi (x) ≤ 0 for each i ∈ I. Assuming D1 , it is easy to verify
that h0 ∈ L given in D1 satisfies the strict inequality G(h0 ) < 0 (thanks to the
compactness of A); thus h0 is a Slater point for the system gi (·) ≤ 0, i ∈ I. Thus by
Corollary 3.7, this system satisfies the BCQ relative to L at each point x in L ∩ S,
where S := {x ∈ X : gi (x) ≤ 0, i ∈ I} (the assertion being trivial if x ∈ L ∩ int S).
Similarly, if D2 holds, then it follows from Theorem 3.10 that this system satisfies the
BCQ relative to L at each point x in L ∩ S.
Note that if x ∈ bd S, then G(x) = 0, and I(x) defined in (1.7) is equal to the set
˜
I(x)
= {i ∈ I : gi (x) = G(x) = 0} defined in (3.6); hence
(5.8)
I(x) = {t ∈ A : x(t) = u(t)} ∪ {ť ∈ Ǎ : x(t) = l(t)}
whenever x ∈ bd S. Note that, since l(·) < u(·) on A, t ∈ I(x) implies ť ∈
/ I(x);
similarly, ť ∈ I(x) implies t ∈
/ I(x). For easy reference, we record the following trivial
fact:
i = t ∈ A,
et ,
gi (x) =
(5.9)
−et , i = ť ∈ Ǎ.
Now define f : X → R by
(5.10)
p1
|z(t) − x(t)| dµ
p
f (x) =
for each x ∈ X.
T
Then f is a continuous convex function on X, and x0 is a solution of the minimization
problem (5.1) subject to (5.2) if and only if it is a solution of the problem of minimizing
f subject to
(5.11)
x ∈ L,
gi (x) ≤ 0
for each i ∈ I,
where {gi : i ∈ I} is defined by (5.5) and (5.6).
Since f (x) defined in (5.10) depends only on the restriction of x to T , the following
result follows from Lemmas 5.2 and 5.3 together with the chain rule.
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
507
Proposition 5.5. Suppose that (z − x0 )|T is not the zero-element of Lp (T, µ).
Then the following assertions hold:
(a) The case when p = 1. A functional y ∗ ∈ ∂f (x0 ) if and only if there exists
β ∈ L∞ (T, µ) such that
|β(t)| ≤ 1
(5.12)
(5.13)
β(t) = sgn[z(t) − x0 (t)]
and
(5.14)
y ∗ , x = −
a.e. on T,
a.e. on T \ Z(z − x),
x(t)β(t)dµ
for each x ∈ X.
T
(b) The case when p ∈ (1, +∞). A functional y ∗ ∈ ∂f (x0 ) if and only if
(5.15) y ∗ , x = −c
x(t)|z(t) − x0 (t)|p−1 sgn[z(t) − x0 (t)]dµ for each x ∈ X,
T
where c := (z − x0 )|T 1−p
.
p
Proof of Theorem 5.4. Let f, gi , I, and S be the same as those introduced after
the statement of Theorem 5.4. Since the proof for the case when x0 ∈ L ∩ int S is
easier (in this case one can take k = 0, and the right-hand sides of (5.3) and (5.4)
are replaced by zero), it suffices to consider the case when x0 ∈ L ∩ bd S. As already
noted, by assumption D1 or D2 , the system of inequalities gi (·) ≤ 0, i ∈ I, satisfies
the BCQ at x0 relative to L. Noting that x∗ = 0 on L whenever x∗ ∈ NL (x0 ), it
follows from Theorem 4.1 that the following statements are equivalent:
(i) x0 is a minimizer of f subject to (5.11).
(ii) There exist a finite subset I0 of I(x0 ), y ∗ ∈ ∂f (x0 ), µi > 0, and x∗i ∈ ∂gi (x0 )
(f or all i ∈ I0 ) such that
(5.16)
y∗ +
µi x∗i = 0
on L.
i∈I0
(ii∗ ) Same as (ii) but with (5.16) replaced by
(5.17)
cy ∗ +
µi x∗i = 0
on L,
i∈I0
where c is a positive constant.
In view of (5.8) and (5.9), we note that, in (ii) and (ii∗ ), each x∗i can be expressed
as either et or −et :
if i = t ∈ I(x0 ) ∩ A (i.e., if x0 (t) = u(t)),
et
x∗i =
−et if i = ť ∈ I(x0 ) ∩ Ǎ (i.e., if x0 (t) = l(t)).
Thus µi x∗i can be expressed in the form λti eti for some λti ∈ R and some ti ∈ A
satisfying x0 (ti ) = u(ti ) or x0 (ti ) = l(ti ). Explicitly, λti is defined by
if x0 (ti ) = u(ti ) and ti ∈ I0 ,
µti
λti =
−µti if x0 (ti ) = l(ti ) and ťi ∈ I0 .
Moreover, when (ii) (or (ii∗ )) holds, we can suppose I0 satisfies an additional property
that |I0 |, the number of elements of I0 , is not greater than dim L, the dimension of
508
CHONG LI AND K. F. NG
L. To see this, we suppose, without loss of generality, that h1 , h2 , . . . , hn are linearly
independent, so dim L = n. Let V denote the subset of Rn defined by
{(x∗i , h1 , x∗i , h2 , . . . , x∗i , hn ) : i ∈ I0 }
and U := V ∪ {(y ∗ , h1 , y ∗ , h2 , . . . , y ∗ , hn )}. Since L is spanned by h1 , h2 , . . . , hn ,
(ii) implies that 0 ∈ co U. Note that
(5.18)
x∗i , h0 − x0 < 0
for each i ∈ I0 .
Indeed, if i = ť ∈ I0 with some t ∈ A, then x0 (t) = l(t) and
x∗i , h0 = −et , h0 = −h0 (t) < −l(t) = −et , x0 = x∗i , x0 ;
thus (5.18) is true, as the case when i = t ∈ I0 with some t ∈ A can be considered
similarly. Since h0 − x0 ∈ L and L is spanned by h1 , h2 , . . . , hn , it follows from (5.18)
that 0 ∈
/ co V; thus 0 ∈ co U \ co V.
By the
Carathéodory theorem (cf. [5]), there exist µ̄0 > 0, µ̄ij ≥ 0 (j = 1, 2, . . . , n)
n
with µ̄0 + j=1 µ̄ij = 1 and i1 , i2 , . . . , in ∈ I0 such that
n
µ̄0 y ∗ +
µij x∗ij = 0
at h1 , h2 , . . . , hn .
j=1
Therefore, I0 in (ii) can be replaced by its subset with at most n elements. Making
use of Proposition 5.5 and noting that L is spanned by h1 , h2 , . . . , hn , we now see that
(ii) (or (ii∗ )) holds if and only if (a), (b), and (c) of Theorem 5.4 hold for some finite
subsets {t1 , t2 , . . . , tk } ⊆ A and {λt1 , λt2 , . . . , λtk } ⊆ R, where k ≤ n. Finally, as we
already noted, (i) holds if and only if x0 is an optimal solution of the minimization
problem (5.1) subject to (5.2). This completes the proof of Theorem 5.4.
6. Approximation problem with constraints by conditionally positive
semidefinite functions. This problem is studied in a recent paper [22] by Strauss.
Following his approach, let Q be a nonempty subset of a Euclidean space Rd and let
F (Q) denote the class of all real-valued functions on Q. Let H be a vector subspace
of F (Q) and suppose that H is equipped with a seminorm induced by a semi-inner
product ·, · on H ×H. Let U = span {u1 , u2 , . . . , um } be an m-dimensional subspace
of H, and let K be a reproducing kernel with respect to U (for undefined terms, see
[1, 18]). Let A = {z1 , z2 , . . . , zn } ⊆ Q and let CA denote the (m×n)-matrix defined by
CA = (uj (zi ))m,n
j=1,i=1 .
Let Wn , Wm , respectively, denote an (n×n)-matrix and an (m×m)-matrix; we assume
that Wn and Wm are positive semidefinite. Define a seminorm · on Rn × Rm by
(a, b)2W = aT Wn a + bT Wm b for all a ∈ Rn , b ∈ Rm .
Suppose that we are given the following data: h ∈ H, {ξi : i ∈ I} ⊆ R, {pi : i ∈ I} ⊆
Rn , and {qi : i ∈ I} ⊆ Rm , where I is an index set. Let
ZI = (a, b) : a ∈ Rn , b ∈ Rm , CA a = 0, pTi a + qTi b ≤ ξi , i ∈ I
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
509
and let dˆ : Rn × Rm → R be defined by
⎛
⎞
2
n
m
2 ⎟
ˆ b) = 1 ⎜
h−
ai K(·, zi ) −
bj u j d(a,
⎝
+ (a, b)W ⎠ ,
2 i=1
j=1
where a = (a1 , a2 , . . . , an ) ∈ Rn and b = (b1 , b2 , . . . , bm ) ∈ Rm . Under the obvious
modifications, ZI and dˆ can be defined when m = 0, and our discussions below are
valid for this case as well as for the case when m = 0.
The problem to be considered here is
(6.1)
ˆ b)
minimize d(a,
subject to
(a, b) ∈ ZI .
Define
(6.2)
UA⊥ = {(a, b) ∈ Rn × Rm : CA a = 0},
(6.3)
KA = (K(zi , zj ))ni,j=1 ,
(6.4)
hA = (h(z1 ), . . . , h(zn ))T ,
and
(6.5)
ê(a, b) =
1 T
1
a (KA + Wn )a + bT Wm b − hTA a + h, h.
2
2
By Proposition 2.3 and the proof of Proposition 4.1 in [22], we have
(6.6)
ˆ b) = ê(a, b)
d(a,
for all (a, b) ∈ UA⊥
and
(6.7)
KA is positive semidefinite with respect to UAT
in the sense that aT KA a ≥ 0 whenever (a, b) ∈ UAT . This implies of course that
KA + Wn is also positive semidefinite, and so the quadratic function ê is convex on
UAT .
We always assume that (6.1) has a solution: let (a0 , b0 ) denote a solution of
(6.1). Let r(CA ) denote the rank of the matrix CA . Finally, |J| denotes the number
of elements of a finite set J. The following result was asserted as a theorem ([22,
Theorem 6.2]):
Suppose that (a0 , b0 ) is a solution of (6.1), r(CA ) = m, and that there exists
(ā, b̄) ∈ Rn × Rm such that
(6.8)
CA ā = 0,
pTi ā + qTi b̄ < ξi
for all i ∈ I.
Then there exists a finite subset I0 of I with |I0 | ≤ n such that (a0 , b0 ) also minimizes
ˆ b) subject to (a, b) ∈ ZI .
d(a,
0
While we will give a counterexample below to show this result is false, we put
forward the following corrected version.
510
CHONG LI AND K. F. NG
Theorem 6.1. Let (a0 , b0 ) be a solution of (6.1). Assume that the system of
linear inequalities
pTi a + qTi b ≤ ξi ,
(6.9)
i ∈ I,
satisfies the BCQ relative to UA⊥ at (a0 , b0 ). Then there exists a finite subset I0 of
ˆ b) subject to
I with |I0 | ≤ m + n − r(CA ) such that (a0 , b0 ) also minimizes d(a,
(a, b) ∈ ZI0 .
Proof. By assumption and (6.6), (a0 , b0 ) is a solution of the following minimization problem:
subject to (a, b) ∈ UA⊥ and (6.9).
Minimize ê(a, b)
We shall use Dê(a0 , b0 )|U ⊥ to denote the restriction to UA⊥ of the Gateaux (equivA
alently, Fréchet) derivative of the function ê at (a0 , b0 ). Since, by assumption, the
system (6.9) in X := Rn × Rm satisfies the BCQ relative to UA⊥ at (a0 , b0 ), one can
apply the implication (i)=⇒ (ii∗ ) of Theorem 4.1 to conclude that there exist a finite
subset I0 of I(a0 , b0 ) and a collection {λi : i ∈ I0 } of nonnegative real numbers such
that
λi (pTi , qTi )|U ⊥ = 0,
Dê(a0 , b0 )|U ⊥ +
(6.10)
A
A
i∈I0
because x∗ |U ⊥ = 0 whenever x∗ ∈ NU ⊥ (a0 , b0 ). We write µ for the sum
A
A
i∈I0
λi and
ˆ on U ⊥ by
assume that µ = 0 (otherwise, (a0 , b0 ) must be a minimal point of ê(= d)
A
⊥
convexity of ê on UA ). Thus
−Dê(a0 , b0 )|U ⊥ ∈ cone{(pTi , qTi )|U ⊥ : i ∈ I0 }.
(6.11)
A
A
Since dim UA⊥ = n + m − r(CA ), it follows from [19, Corollary 17.1.2] that I0 in (6.11)
can be replaced by a subset I0 with |I0 | ≤ n + m − r(CA ). Note that (6.10) continues
to hold when I0 is replaced by I0 (replace the family {λi } by a suitable new one if
necessary). Consequently, by (ii∗ ) of Theorem 4.1, (a0 , b0 ) minimizes ê(a, b) on UA⊥
subject to the system of linear inequalities
pTi a + qTi b ≤ ξi , i ∈ I0 .
This completes the proof as before.
Example 6.1. Let A = Q consist of two distinct elements z1 , z2 . We identify
F (Q) with R2 in the obvious manner. Let H = F (Q) with the usual inner product.
Take h = (1, 0)T ∈ H, m = 0, Wm = 0, U = {0}, CA = 0, and UA⊥ = R2 . Define Wn
1
by Wn = ( 04 01 ) and fix the reproducing kernel K : Q → R defined by
4
K(x, y) =
1
0
if x = y,
if x = y.
Take I = {1, 2, . . .}, ξi = 0, and pi = (1, 1i ) for i ∈ I. Then, (6.8) is satisfied for
appropriate ā. Note
!
1
1
2
2
ˆ
ˆ
h − a + a
d(a) := d(a, b) =
2
4
CONSTRAINT QUALIFICATION FOR INFINITE SYSTEMS
and
511
ZI =
1
a ∈ R2 : a1 + a2 ≤ 0 for all i ∈ I .
i
Clearly, ZI is the convex set of R2 containing the 4th quadrant and bounded by the
two half-lines, respectively, contained in a1 + a2 = 0 and a1 = 0. It follows easily
that the origin 0 ∈ R2 is the nearest point to h from ZI . On the other hand, if I0
is a finite subset of I, then ZI0 is also bounded by the two half-lines, respectively,
contained in a1 + i1− a2 = 0 and a1 + i1+ a2 = 0, where i− := min I0 , i+ := max I0 .
1
T
Let aI0 denote the nearest point to h from ZI0 . Then aI0 = 1+i
and
2 (1, −i+ )
+
h2 = aI0 2 + h − aI0 2 . This implies that
!
1 1
1
2
2
2
ˆ I ),
ˆ
aI0 + h − aI0 = d(a
d(0) = h >
0
2
2 4
while
ˆ = 1 h − 02 ≤ 1 h − a2 ≤ d(a)
ˆ
d(0)
2
2
for all a ∈ ZI .
Therefore [22, Theorem 6.2] mentioned earlier is not true.
Remark 6.1. Let li be the function on Rn × Rm defined by
li (a, b) = pTi ai + qTi b − ξi ,
(a, b) ∈ Rn × Rm .
Suppose that I can be topologized in such a way that it is a compact space satisfying
the first axiom of countability and that i → li (a, b) is upper semicontinuous on I for
each (a, b) ∈ UA⊥ . Then, by Corollary 3.7, the linear inequality system (6.9) satisfies
the BCQ relative to UA⊥ at (a0 , b0 ) if (6.8) is satisfied. In this particular case, the
conclusion of [22, Theorem 6.2] mentioned earlier does hold. Similarly, if I can be
topologized in such a way that it is a compact metric space and that i → li (a, b) is
continuous on i ∈ I for each (a, b) ∈ UA⊥ , then, by Theorem 3.10, the aforesaid result
is also valid even if (6.8) is replaced by a weaker condition given in the following: for
each subset J of I with |J| ≤ m + n − r(CA ), there exists (ā, b̄) ∈ Rn × Rm such that
(6.12)
CA ā = 0,
pTi ā + qTi b̄ < ξi
for all i ∈ J.
Acknowledgments. We wish to thank two referees for their valuable comments
and suggestions.
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