J Optim Theory Appl (2015) 164:123–137
DOI 10.1007/s10957-014-0582-y
On Local Coincidence of a Convex Set and its Tangent
Cone
Kaiwen Meng · Vera Roshchina · Xiaoqi Yang
Received: 21 August 2013 / Accepted: 3 May 2014 / Published online: 23 May 2014
© Springer Science+Business Media New York 2014
Abstract In this paper, we introduce the exact tangent approximation property for a
convex set and provide its characterizations, including the nonzero extent of a convex
set. We obtain necessary and sufficient conditions for the closedness of the positive hull
of a convex set via a limit set defined by truncated upper level sets of the gauge function.
We also apply the exact tangent approximation property to study the existence of a
global error bound for a proper, lower semicontinuous and positively homogeneous
function.
Keywords Tangent approximation · Extent of a convex set · Positive hull · Error
bounds · Support functions · Gauge functions · Positively homogeneous functions
Mathematics Subject Classification (2000)
49J52 · 49J53
Communicated by Vaithilingam Jeyakumar.
K. Meng (B)
School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China
e-mail: [email protected]
V. Roshchina
Collaborative Research Network, Federation University Australia, Ballarat, VIC, Australia
e-mail: [email protected]
X. Yang
Department of Applied Mathematics, The Hong Kong Polytechnic University,
Kowloon, Hong Kong, China
e-mail: [email protected]
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1 Introduction
Polyhedral sets possess several nice and important properties, which are fundamental at
least in convex analysis and optimization. For instance, polyhedral sets admit exactness
of tangent approximations [1, Exercise 6.47] in the sense that they can be locally
identified with their tangent cones. In this paper, we will introduce and study such
a property for general convex sets, and we say that a convex set admits exactness of
tangent approximation (ETA, for short) at a given point iff it can be locally identified
with its tangent cone at this point; see Definition 2.1 below.
This paper aims at investigating the ETA property for a convex set and at building
connections between the ETA property and other known properties of a convex set. We
will show that the ETA property of a convex set implies the closedness of the positive
hull of a closed and convex set containing the origin but in general not vice versa. We
also apply the ETA property of a convex set to study the existence of a global error
bound for a proper, lower semicontinuous and positively homogeneous function. In
our study, the ETA property for a convex set is related to many existing properties of
a convex set such as the ones described by the nonzero extent of a convex set [2], the
global error bound property for the support function of a convex set and the relative
continuity of the gauge function of a convex set.
It is worth mentioning that our characterizations of the ETA property may find
potential applications in optimization. For instance, simplex-like methods have been
designed in [3] for solving linear semi-infinite programming problems with boundedly
polyhedral feasible sets, which are convex sets having the ETA property as shown in
Proposition 4.1. Another possible application of the ETA property (or equivalently the
continuity property of the gauge functions as shown by Theorem 4.1) can be found
in gauge optimization [4], which seeks the element of a convex set that is minimal
with respect to a gauge function. This conceptually simple problem can be used to
model many useful problems, including a special case of conic optimization, and
problems arising in machine learning and signal processing. The duality and variational
properties of gauge optimization problems have been recently studied in [5].
The outline of the paper is as follows. In Sect. 2, some notations and preliminary
results are recalled, and the ETA property for a convex set is formally introduced. For
a closed and convex set containing the origin, a characterization of the closedness of
its positive hull is given in Sect. 3 by virtue of a limit set defined via the truncated
upper level sets of its gauge function. In Sect. 4, various characterizations of the ETA
property of a convex set are given. In particular, a counterexample is constructed to
demonstrate that the ETA property is merely sufficient for a positive hull to be closed.
For an inequality system defined by a proper, lower semicontinuous and positively
homogeneous function to possess a global error bound, sharp necessary and sufficient
conditions are given in Sect. 5. Some conclusions are made in Sect. 6.
2 Notations and Preliminaries
We begin with recalling some notations and definitions, most of which are taken from
the book [1]. Throughout the paper, the discussion is in IRn and we use IR := IR∪{±∞}
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to denote the extended real line, and IR++ := {t ∈ IR | t > 0} to denote the set of
positive numbers. The Euclidean norm of a vector x is denoted by x, and the inner
product of vectors x and y is denoted by x, y. We let B be the closed unit ball and S
be the unit sphere. Let K ⊂ IRn . We denote by cl K , conv K , pos K := {0} ∪ {λx |
x ∈ K , λ > 0}, and K ∗ := {v | v, x ≤ 0} the closure, the convex hull, the positive
hull and the negative polar of K , respectively. The support function σ K : IRn → IR of
K is defined by
σ K (w) := sup v, w.
v∈K
The distance of x from K is defined by
d(x, K ) := inf x − y.
y∈K
For a convex set C, the extent of C is defined in [2] by
exte C := inf{x | x ∈ cl C, t x ∈ cl C ∀t > 1},
where the convention inf ∅ = +∞ is used. For a set A ⊂ IRn and one of its points
x, we use T A (x) to denote the tangent cone to A at x, i.e. w ∈ T A (x) iff there exist
some tk ↓ 0 and wk → w such that x + tk wk ∈ A for all k. Moreover, we denote by
A (x) := T A (x)∗ the regular normal cone to A at x, and by N A (x) the normal cone to
N
A at x , i.e. v ∈ N A (x) iff there are sequences xk → x with xk ∈ A, and vk → v with
A (xk ). We say that A is regular at x iff A is locally closed at x (i.e. A ∩ V is
vk ∈ N
A (x). Note that a convex
closed for some closed neighbourhood of x) and N A (x) = N
set C is regular at x ∈ C as long as C is locally closed at x̄.
A function h : IRn → IR is positively homogeneous iff h(0) < +∞ and h(λx) =
λh(x) for all x and all λ > 0. It is sublinear iff in addition
h(x + x ) ≤ h(x) + h(x ) ∀x, x ∈ IRn .
For a closed and convex set C with 0 ∈ C, the gauge of C is the function γC : IRn → IR
defined by
γC (x) := inf{λ ≥ 0 | x ∈ λC},
where the convention inf ∅ = +∞ is used. According to [1, Example 3.50], γC is a
non-negative, lower semicontinuous and sublinear function with level sets
τ C = {x | γC (x) ≤ τ } ∀τ > 0,
(1)
pos C = {x | γC (x) < +∞}.
(2)
and
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Let f : IRn → IR be a function with f (x̄) finite. We denote by dom f := {x |
f (x) < +∞} the domain of f , by ker f := {x | f (x) = 0} the kernel of f , by
epi f := {(x, α) | f (x) ≤ α} the epigraph of f , by
lev≥α f := {x ∈ IRn | f (x) ≥ α}
the upper level set of f at the level α ∈ IR, by lim inf f (x) the lower limit of f at x̄,
x→x̄
and by cl (conv f ) the closure of the convex hull of f , i.e. cl (conv f ) is a function,
whose epigraph is the closure of the convex hull of epi f . We say that f is proper iff
f (x) < +∞ for at least one x ∈ IRn , and f (x) > −∞ for all x ∈ IRn , and that f is
lower semicontinuous iff
lim inf f (x) = f (x̄) ∀x̄ ∈ IRn .
x→x̄
It is well known that f is lower semicontinuous if and only if epi f is closed. We
say that f is (subdifferentially) regular at x iff f (x) is finite, and epi f is regular at
(x, f (x) as a subset of IRn × IR. The vector v ∈ IRn is a regular subgradient of f at
x̄, written v ∈ ∂ f (x̄), iff,
f (x) ≥ f (x̄) + v, x − x̄ + o(x − x̄),
where o(x − x̄) is a term with the property that o(x − x̄)/x − x̄ → 0 when
x → x̄ with x = x̄. For any w ∈ IRn , the subderivative of f at x̄ for w is defined by
d f (x̄)(w) :=
lim inf
τ →0+, w →w
f (x̄ + τ w ) − f (x̄)
.
τ
Note that the subderivative d f (x̄) is a lower semicontinuous and positively homogeneous function and that the regular subdifferential set can be derived from the
subderivative as follows:
∂ f (x̄) = {v ∈ IRn | v, w ≤ d f (x̄)(w) ∀w ∈ IRn }.
For a set-valued mapping S : IRn ⇒ IRm and a point x̄ ∈ IRn , the outer limit of S at x̄
is defined by
lim sup S(x) := {u | ∃x ν → x̄, ∃u ν → u with u ν ∈ S(x ν )},
x→x̄
and the inner limit of S at x̄ is defined by
lim inf S(x) := {u | ∀x ν → x̄, ∃u ν → u with u ν ∈ S(x ν )}.
x→x̄
The limit of S at x̄ exists iff its outer and inner limits are equal:
lim S(x) = lim sup S(x) = lim inf S(x).
x→x̄
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It is well known [1] that, for a polyhedral set P at one of its points x̄, there exists a
neighbourhood V of x̄ such that
P ∩ V = [x̄ + TP (x̄)] ∩ V.
We now introduce such a property for a convex set in an analogous way.
Definition 2.1 We say that a convex set C ⊂ IRn admits exactness of tangent approximation (ETA, for short) at one of its points x̄, if and only if there exists a neighbourhood
V of x̄ such that
(cl C) ∩ V = [x̄ + TC (x̄)] ∩ V.
3 A Characterization of Closed Positive Hulls
In this section, let C ⊂ IRn be closed and convex with 0 ∈ C, and we shall give an
equivalent condition for the positive hull pos C to be closed. To show that pos C is
closed, it suffices to show that for any xk → x with xk ∈ pos C (that is, γC (xk ) < +∞)
for all k, it follows that x ∈ pos C (that is, γC (x) < +∞). Observing that γC is lower
semicontinuous, we get from [1, Lemma 1.7] that
γC (x) = min{α ∈ IR | ∃x ν → x with γC (x ν ) → α} ∀x ∈ IRn .
Therefore, the sequences xk → x with γC (xk ) < +∞ for all k such that γC (xk ) →
+∞ are important for characterizing the closedness of pos C. This motivates us to
introduce a limit set as follows:
pos∞ C := lim M(α),
α↓0
(3)
where M : IR++ ⇒ IRn is a set-valued mapping defined by
M(α) := {x |
1
≤ γC (x) < +∞} ∀α > 0.
α
Clearly, M(α) = pos C ∩ lev≥ 1 γC is the truncated upper level set of γC and it
α
decreases as α tends to 0 with α > 0 (making it possible for the limit set in (3) to
exist). From the positive homogeneity of γC , it follows that pos∞ C is a closed cone
(possibly nonconvex) contained in cl (pos C). From the facts that pos C = dom γC
[see (2)] and that x ∈ lev ≥ 1 γC if and only if x ∈ λC for all λ ∈]0, α1 [, it follows that
α
x ∈ pos∞ C if and only if
∃xk → x with γC (xk ) < +∞ such that γC (xk ) → +∞,
(4)
or equivalently
1
.
∀αk ↓ 0, ∃xk → x with xk ∈ pos C and xk ∈ tC ∀t ∈ 0,
αk
(5)
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We now illustrate the closed cone pos ∞ C by a simple example.
Example 3.1 Let C = {x ∈ IR2 | x2 ≥ x12 }. It is clear that
⎧ 2
x1
⎪
⎪
if x2 > 0,
⎨ x2 ,
γC (x) = 0,
if x1 = x2 = 0,
⎪
⎪
⎩
+∞, otherwise.
Clearly, we have pos C = dom γC = {x ∈ IR2 | x2 > 0} ∪ {(0, 0)}, and
pos∞ C ⊂ cl (dom γC ) = {x ∈ IR2 | x2 ≥ 0}.
Since the gauge function γC is sublinear (hence convex), it is continuous on the
interior of its domain, implying that {x ∈ IR2 | x2 > 0} ∩ pos ∞ C = ∅. Let xk =
(1, 1/k)T . It is clear to see that xk → (1, 0) with γC (xk ) < +∞ and γC (xk ) → +∞.
Thus, we have (1, 0) ∈ pos ∞ C. Similarly, we have (−1, 0) ∈ pos ∞ C. Since pos ∞ C
is a cone, we have pos ∞ C = {x ∈ IR2 | x2 = 0}. Note that in this example, pos C is
not closed and pos ∞ C is not contained in pos C.
It turns out that the closed cone pos∞ C encapsulates enough information for characterizing closedness of positive hulls. We capture this fact in the following theorem.
Theorem 3.1 For a closed and convex set C ⊂ IRn with 0 ∈ C, its positive hull pos C
is closed if and only if
pos∞ C ⊂ pos C.
(6)
Proof If pos C is closed and x ∗ ∈ pos∞ C, then it follows from (5) that x ∗ ∈ pos C
and hence (6) follows. Conversely, assume that (6) holds. Let xk → x be such that
xk ∈ pos C for all k (that is, γC (xk ) < +∞). If γC (xki ) ≤ τ for some τ > 0 and a
subsequence {xki } of {xk }, then xki ∈ τ C for all ki , implying that x ∈ τ C ⊂ pos C.
Alternatively, γC (xk ) → +∞, and from (4) and (6) it follows that x ∈ pos∞ C ⊂
pos C. Thus, in either case, we have x ∈ pos C, implying that pos C is closed.
It was shown by [6, Theorem 3.3] that pos C is closed if exte C > 0. In this case,
(6) holds trivially because it turns out in next section that exte C > 0 if and only if
pos∞ C = ∅. Moreover, our findings in next section suggest that for general convex
sets, the equality pos∞ C = ∅ can be used to define a nice property that polyhedral
sets have.
4 Characterizations of Exact Tangent Approximations
In this section, we first build connections among the ETA property of a convex set
and other properties of a convex set and then show by a counterexample that the ETA
property alone is merely sufficient for a positive hull to be closed in higher dimensional
spaces even though it is necessary in a two-dimensional space.
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Theorem 4.1 For a convex set C̃ ⊂ IRn and a point x̄ ∈ C̃, let C := cl C̃ − x̄. The
following properties are equivalent and entail pos C being closed:
(a) C̃ admits ETA at x̄.
(b) C admits ETA at 0.
(c) There is some τ > 0 such that the following equivalent conditions hold:
(c1) exte C ≥ τ1 .
(c2) pos C ∩ S ⊂ τ C.
(c3) d(x, ker σC ) ≤ τ σC (x) ∀x ∈ IRn .
(d) The gauge function γC of C is continuous at 0 relative to pos C.
(e) pos∞ C = ∅.
Proof To begin with, we point out the following facts: (I) C is a closed and convex
set containing the origin. (II) Under the assumption of (c1), the closedness of pos C
was shown in [6, Theorem 3.3]. (III) The equivalence of (c1) and (c3) was shown in
[6, Theorem 4.1].
[(a) ⇐⇒ (b)]: By definition, C̃ admits ETA at x̄ if and only if there exists some
neighbourhood V of x̄ such that (cl C̃) ∩ V = [x̄ + TC̃ (x̄)] ∩ V or equivalently
C ∩ (V − x̄) = TC̃ (x̄) ∩ (V − x̄). Observing that V − x̄ is a neighbourhood of 0, and
that
TC̃ (x̄) = Tcl C̃ (x̄) = T(cl C̃)−x̄ (0) = TC (0),
the equivalence of (a) and (b) follows readily.
[(c2) ⇒ (c1)]: Let x ∈ C be such that t x ∈ C for all t > 1. It is clear to see that
x
1
x = 0. By (c2), we have τ x
∈ C, which implies that τ x
≤ 1 (that is, x ≥ τ1 ).
Therefore, we have
exte C := inf{x | x ∈ C, t x ∈ C ∀t > 1} ≥
1
.
τ
[(c1) ⇒ (c2)]: Let x ∈ pos C ∩ S. Suppose by contradiction that x ∈ τ C (that
is, τx ∈ C). Since C is a closed and convex set containing 0, there must exist some
τ > τ such that τx ∈ C and t τx ∈ C for all t > 1. Thus, we have
exte C := inf{x | x ∈ C, t x ∈ C ∀t > 1} <
x
1
1
= < ,
τ
τ
τ
contradicting to (c1).
[(b) ⇒ (c2)]: It follows from (b) that there exists some neighbourhood V of 0
such that TC (0) ∩ V = C ∩ V . Since C is convex with 0 ∈ C, by [1, Theorem 6.9],
we have TC (0) = cl (pos C). Thus, we have pos C ∩ V ⊂ C. Let τ > 0 be such that
1
τ B ⊂ V . It then follows that pos C ∩ B ⊂ τ C. This implies that (c2) holds.
[(c2) ⇒ (b)]: Let V := τ1 B. Since C is a convex set containing 0, by [1, Theorem
6.9], we have TC (0) = cl (pos C) and can rewrite (c2) as pos C ∩ V ⊂ C. In view
of C ⊂ pos C, we have pos C ∩ V = C ∩ V . It follows from the fact (II) and the
equivalence of (c1) and (c2) that pos C is closed, implying that TC (0) ∩ V = C ∩ V .
That is, C admits ETA at 0.
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[(c2) ⇒ (d)]: Without loss of generality, we assume that there is a sequence
{xk } ⊂ pos C\{0} such that xk → 0 (otherwise, we have pos C = C = {0}, and
xk
∈ C for all k. In view of (1),
hence (d) holds trivially). By (c2), we have τ x
k
xk
we have γC τ xk ≤ 1 (that is, γC (xk ) ≤ τ xk ). Since τ xk → 0, we have
γC (xk ) → 0 = γC (0). Therefore, γC is continuous at 0 relative to pos C.
[(d) ⇒ (c2)]: Suppose by contradiction that (c2) does
not hold, i.e. there exists a
sequence {xk } ⊂ pos C ∩ S with xk ∈ kC (that is, γC xkk > 1) for all k. It is clear to
see that xkk → 0 with xkk ∈ pos C for all k. By (d), we have γC xkk → γC (0) = 0,
xk which is impossible because we have γC k > 1 for all k.
[(d) ⇒ (e)]: Suppose by contradiction that pos∞ C = ∅. Let x ∈ pos∞ C and
let αk ↓ 0. In view of (4), there exists a sequence {xk } converging to x such that
1
αk ≤ γC (x k ) < +∞ or 1 ≤ γC (αk x k ) < +∞ for all k. Since αk x k → 0 and
αk xk ∈ pos C for all k, it then follows from (d) that γC (αk xk ) → 0, contradicting to
the fact that 1 ≤ γC (αk xk ) for all k.
[(e) ⇒ (c2)]: Suppose by contradiction that (c2) does not hold, i.e. there exists a
sequence {xk } ⊂ pos C ∩ S with xk ∈ kC or γC (xk ) ≥ k for all k. Since xk ∈ pos C
for all k, we have γC (xk ) < +∞ for all k. Let αk ↓ 0. By taking a subsequence if
necessary, we may assume that xk → x ∗ ∈ S with α1k ≤ γC (xk ) < +∞ for all k. In
view of (4), we have x ∗ ∈ pos∞ C = ∅, contradicting to (e).
The proof is completed.
Corollary 4.1 Let C ⊂ IRn be a closed and convex set containing 0. We have
1
= inf{τ > 0 | d(x, ker σC ) ≤ τ σC ∀x ∈ IRn }
exte (C)
= inf{τ > 0 | pos C ∩ S ⊂ τ C},
where the conventions
1
0
= +∞,
1
+∞
= 0 and inf ∅ = +∞ are used.
Proof The first equality was proved in [6, Theorem 4.1]. The second equality follows
easily from the equivalence of (c2) and (c3) in Theorem 4.1.
Corollary 4.2 Let C ⊂ IRn be a closed and convex set containing 0. The following
properties are equivalent and entail that C admits ETA at 0:
(a) C is locally polyhedral at 0, i.e., C ∩ V is a polyhedron for some polyhedral
neighbourhood V of 0.
(b) pos C is a polyhedral cone.
Proof Since the positive hull of any polyhedron is also a polyhedron and pos C =
pos (C ∩ V ) for any neighbourhood V of 0, it is clear to see that (a) implies (b).
Assuming now that pos C is a polyhedral cone. It then follows from the continuity
properties of convex functions [1, Theorem 2.35] that γC is continuous relative to its
domain pos C. By Theorem 4.1, C admits ETA at 0. This implies that C is locally
polyhedral at 0.
Corollary 4.3 Let C ⊂ IR2 be a closed and convex set containing 0. Then, C admits
ETA at 0 if and only if pos C is closed.
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Fig. 1 The curve γ : [0, 2π ] → IR3 from Example 4.1 is shown in red solid line, which is the sharp
boundary of the solid grey set, the convex hull C. The semi-transparent cone corresponds to the set pos C.
Proof Observing that in IR2 , every closed cone is polyhedral, the result follows readily
from Theorem 4.1 and Corollary 4.2.
In Corollary 4.3, we show that a closed and convex set C containing the origin
in two-dimensional space admits ETA at 0 if and only if its positive hull pos C is
closed. However, this equivalence may not hold in higher dimensional spaces. This is
demonstrated by a three-dimensional example in which pos C is closed but C does
not admit ETA at 0.
Example 4.1 Let γ : [0, 2π ] → IR3 be a three-dimensional curve defined as follows:
γ (t) = (r (t) cos t, r (t) sin t, r (t))
with
t
r (t) = sin ,
4
and let C = conv {γ (t), t ∈ [0, 2π ]}. The set C and the curve γ are shown in Fig. 1.
In what follows, we shall show the following properties one by one:
(i) The set C is a compact and convex set containing 0.
(ii) The set pos C is a closed and convex cone; moreover,
2
2
pos C = (x, y, z) | z ≥ x + y .
(iii) The set of all extreme points of C is γ ([0, 2π ]), the extent of C is 0, and the
gauge function γC is not continuous at 0 relative to γ ([0, 2π ]).
(iv) C does not admit ETA at 0.
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Proof of (i) Observe that C is convex by definition; also, 0 = γ (0) ∈ C. Finally, C is
compact as a convex hull of the compact setγ ([0, 2π ]).
Proof of (ii) Let K := (x, y, z) | z ≥ x 2 + y 2 . First, we show that pos C ⊂ K .
Indeed, let p = (x, y, z) ∈ pos C. Then for some τ > 0, we have p ∈ τ C. Therefore,
p can be represented as
p=τ
l
αi γ (ti ),
ti ∈ [0, 2π ], αi > 0 ∀ i ∈ {1, . . . , l},
i=1
l
αi = 1.
i=1
We have
x 2 + y2 = τ
l
i=1
= τ2
ti
αi cos ti sin
4
l
i, j=1
= τ2
l
2
+ τ
i=1
l
αi α j cos(ti − t j ) sin
=τ
l
l
i=1 αi
sin
ti
4
tj
ti
sin
4
4
2
ti
αi sin
4
i=1
Since z = τ
2
tj
ti
sin
4
4
αi α j sin
i, j=1
2
ti
αi sin ti sin
4
tj
ti
αi α j cos ti cos t j + sin ti sin t j sin sin
4
4
i, j=1
≤ τ2
l
= z2.
> 0, this yields
z≥
x 2 + y2,
and hence p ∈ K .
It remains to show that K ⊂ pos C. Let p = (x, y, z) ∈ K . First, consider the case
when x 2 + y 2 = 0. Let t0 ∈]0, 2π ] be such that
cos t0 = x
x 2 + y2
,
sin t0 = y
x 2 + y2
.
If t0 ≤ π , let t1 = t0 + π , otherwise let t1 = t0 − π . Observe that
sin t0 = − sin t1 ,
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cos t0 = − cos t1 .
J Optim Theory Appl (2015) 164:123–137
133
Set
z + x 2 + y2
,
α :=
2 sin t40
z − x 2 + y2
β :=
.
2 sin t41
0; moreover, we
Observe that since p ∈ K , we have z ≥ x 2 + y 2 , and hence α, β ≥ have p = αγ (t0 )+βγ (t1 ), and α+β = 0 (otherwise p = 0 and hence x 2 + y 2 = 0).
Let τ := α + β, λ0 := α/τ , and λ1 := β/τ . We have
p = τ (λ0 γ (t0 ) + λ1 γ (t1 )), τ > 0, λ0 + λ1 = 1, λ0 , λ1 ≥ 0,
(7)
hence, p ∈ τ C. It remains to consider the case when x = y = 0, i.e. when p = (0, 0, z)
with z ≥ 0. Observe that if z = 0, we have√p = 0 = γ (0) ∈ C = 1 · C.√ Assume that
z > 0. Let t0 = π , t1 = 2π . Let λ0 := √ 2 , λ1 := √ 1 , τ := 1+2 2 z > 0. It is
2+1
2+1
straightforward to check that (7) holds, and hence p ∈ τ C. By the arbitrariness of p,
we have shown that K ⊂ pos C. Therefore, pos C = K , and since K is a closed and
convex cone, so is pos C.
Proof of (iii) Denote by extr (C) the set of all extreme points of C. First, we will show
that extr (C) ⊂ γ ([0, 2π ]), and that γ (0) is an extreme point of the closed cone pos C
and γ (t) with t ∈]0, 2π ] is an extreme direction of pos C. Thus, if there exist some
t0 , t1 , t2 ∈ [0, 2π ] and α ∈]0, 1[ such that
αγ (t1 ) + (1 − α)γ (t2 ) = γ (t0 ),
then t1 = t2 = t0 . This implies that γ ([0, 2π ]) ⊂ extr (C) and hence extr (C) =
γ ([0, 2π ]). Since γ (t) → 0 as t ↓ 0, the extent of C is by definition 0. By definition,
we have γC (γ (t)) = 1 for all t ∈]0, 2π ], implying that γC cannot be continuous at 0
relative to γ ([0, 2π ]) as we have γC (γ (0)) = γC (0) = 0.
Proof of (iv) By Theorem 4.1, the result follows directly from (iii).
Let C ⊂ IRn be closed and convex, and let x ∈ C. Consider the following conditions:
(a) C is locally polyhedral at x.
(b) C admits ETA at x.
(c) pos (C − x) is closed.
The results that we obtained so far show that (a) ⇒ (b) ⇒ (c). Example 4.1
demonstrates that (c) may not imply (b), and any intersection set of a non-polyhedral
closed and convex cone with a neighbourhood of the origin can be used to demonstrate
that (b) may not imply (a), because such a set admits ETA at the origin but it is
not locally polyhedral at the origin. However, the following proposition show that
conditions (a), (b) and (c) are the same whenever they holds on the whole set.
Proposition 4.1 Let C ⊂ IRn be closed and convex. The following properties are
equivalent:
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(a) C is locally polyhedral at every x ∈ C.
(b) C admits ETA at every x ∈ C.
(c) pos (C − x) is closed for all x ∈ C.
Proof From Corollary 4.2, we obtain that (a) implies (b). By Theorem 4.1, we know
that (b) implies (c). It was shown by [7, Lemma 3.2] that if (c) holds, then pos (C − x)
is a polyhedral cone for every x ∈ C. In view of Corollary 4.2, we know that (c)
implies (a). The proof is completed.
Remark 4.1 Let C ⊂ IRn be closed and convex. It was shown by [8, Proposition 2.17]
that C is locally polyhedral at every x ∈ C if and only if C is a boundedly polyhedral
set (i.e. any nonempty intersection of C with a bounded polyhedron is polyhedral).
Therefore, the ETA property on the whole set C actually gives an alternative description
of a boundedly polyhedral set. Note that a boundedly polyhedral set is not necessarily
polyhedral. For instance, the convex hull of the set {(x, x 2 ) | x = 0, ±1, ±2, ±3, . . .}
is not a polyhedral set but a boundedly polyhedral set.
5 Global Error Bounds for Positively Homogeneous Functions
In this section, we consider a proper, lower semicontinuous and positively homogeneous function h : IRn → IR, and give necessary and sufficient conditions for the
inequality system h(x) ≤ 0 to possess a global error bound, i.e. for the existence of
some τ > 0 such that
d(x, ker h + ) ≤ τ h + (x) ∀x ∈ IRn ,
(8)
where h + (x) := max{h(x), 0} for all x ∈ IRn . It is clear to see that h + is a nonnegative, lower semicontinuous and positively homogeneous function, and that
ker h + = {x ∈ IRn | h(x) ≤ 0}
is a nonempty and closed (possibly nonconvex) cone. Moreover, it is worth pointing
out that ker h + is closely related to a closed and convex set defined by
C := {v ∈ IRn | v, x ≤ h + (x) ∀x ∈ IRn },
(9)
of which the ETA property plays a key role in the sequel. Note that we have 0 ∈
C because h + is non-negative. From [1, Theorem 8.24], it follows that the support
function σC of C is the closure of the convex hull of h + , i.e.
σC = cl (conv h + ).
Recall from Theorem 4.1 that there exists some τ > 0 such that
d(x, ker σC ) ≤ τ σC (x) ∀x ∈ IRn ,
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if and only if pos C ∩ S ⊂ τ C (i.e. C admits ETA at 0). Surprisingly, it turns out that
(10) is a consequence of (8), as can be seen from the following theorem, in which we
present sharp necessary and sufficient conditions for the inequality system h(x) ≤ 0
to possess the global error bound (8).
Theorem 5.1 Let τ > 0 and let h : IRn → IR be a proper, lower semicontinuous and
positively homogeneous function. If (8) holds, then the following equivalent properties
hold:
ker h + (0) ∩ S ⊂ τ C.
(a) N
(b) pos C ∩ S ⊂ τ C and pos C = [ker h + ]∗ .
Conversely, if
(c) Nker h + (0) ∩ S ⊂ τ C,
then (8) holds. Moreover, if ker h + is regular at 0, then (a) ⇐⇒ (c).
ker h + (0) = Nker h + (0) and hence
Proof If ker h + is regular at 0, then we have N
(a) ⇐⇒ (c).
ker h + (0) =
[(a) ⇐⇒ (b)]: Since ker h + is a nonempty closed cone, we have N
∗
[ker h + ] . In view of (9), we have v, x ≤ 0 for all v ∈ C and x ∈ ker h + . Thus, we
have C ⊂ [ker h + ]∗ and hence
pos C ⊂ [ker h + ]∗ .
ker h + (0) ⊂ pos C and hence pos C = [ker h + ]∗ ,
If (a) holds, then [ker h + ]∗ = N
implying that (b) holds. Alternatively, if (b) holds, then (a) holds trivially. That is, (a)
and (b) are equivalent.
[(8) ⇒ (a)]: Let v ∈ [ker h + ]∗ ∩ S. By the definition of the polar cone, we have
−v, x ≥ 0 for all x ∈ ker h + . That is, the linear function −v, · admits a minimum
over ker h + at x̄ = 0. Note that the linear function −v, · is Lipschitz of rank 1 on
IRn , i.e.
|−v, x1 − −v, x2 | ≤ − vx1 − x2 = x1 − x2 ∀x1 , x2 ∈ IRn .
Then from (8) and [9, Proposition 2.4.3], it follows that there exists some τ > 1 such
that the function −v, · + τ h + (·) admits a minimum over IRn at x̄. That is,
h + (x) ≥
v
τ
,x
∀x ∈ IRn .
ker h + (0) ∩ S = [ker h + ]∗ ∩ S ⊂ τ C.
In view of (9), we have v ∈ τ C. Thus, N
[(c) ⇒ (8)]: Let x ∈ ker h + . From [1, Proposition 6.27], it follows that there exists
some v̄ ∈ Nker h + (0) with v̄ = 1 such that
d(x, ker h + ) = v̄, x.
(11)
By (c), we have v̄ ∈ τ C. In view of (9) and (11), we have
d(x, ker h + ) ≤ τ h + (x).
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This completes the proof.
Remark 5.1 Let C ⊂ IRn be a closed and convex set with 0 ∈ C and let be the set
of proper, lower semicontinuous and positively homogeneous functions h satisfying
cl (conv h + ) = σC . From [1, Theorem 8.24], it follows that σC ∈ = ∅, and that
any h ∈ can be used to define C via (9). In general, could be an infinite set. If C
admits no ETA at 0, then it follows from Theorem 5.1 that there is no h ∈ such that
the inequality system h(x) ≤ 0 possesses a global error bound as defined by (8). This
demonstrates from another angle that ETA is an important property of convex sets.
Moreover, by setting h to be σC , the equivalence of (c2) and (c3) in Theorem 4.1 can
be recovered from Theorem 5.1.
In the remainder of this section, we shall apply Theorem 5.1 to study variational
properties of local minima in terms of regular subgradients and subderivatives. Let
f : IRn → IR be a function with f (x̄) finite. If f has a local minimum at x̄, then the
generalized Fermat’s rule [1, Theorem 10.1] asserts that d f (x̄) ≥ 0 or equivalently
0 ∈
∂ f (x̄). Beyond the equivalence of these two conditions, we can exploit more
relations between the subdrivative d f (x̄) and the regular subdifferential set ∂ f (x̄) as
shown by the following proposition, which follows in a direct way from Theorem 5.1.
Proposition 5.1 Assume that the function f : IRn → IR has a local minimum at x̄
with f (x̄) finite, and that the closed cone ker d f (x̄) is convex. Then, the following
statements are equivalent:
(a) There is some τ > 0 such that
d(w, ker d f (x̄)) ≤ τ d f (x̄)(w) ∀w ∈ IRn .
(b) ∂ f (x̄) admits ETA at 0 and pos (
∂ f (x̄)) = [ker d f (x̄)]∗ .
6 Conclusions
In this paper, we introduced, for a convex set, the exact tangent approximation (ETA)
property that (boundedly) polyhedral sets possess, and provided its full characterizations by relating ETA with many existing properties of a convex set, such as the
relative continuity of the gauge function γC , and the nonzero extent of a convex set.
We also presented necessary and sufficient conditions for the closedness of the positive
hull of a convex set via a limit set defined by truncated upper level sets of the gauge
function. By virtue of the ETA property, we gave sufficient and necessary conditions
being of almost no gap for the existence of a global error bound for a proper, lower
semicontinuous and positively homogeneous function. To end this paper, we pose a
conjecture on the continuity property of the gauge function as follows.
Conjecture 6.1 For a closed and convex set C containing 0, its gauge γC is a nonnegative, lower semicontinuous and sublinear function. From the literature, it is well
known that γC is continuous at 0 if and only if γC is everywhere continuous. We
conjecture that γC is continuous at 0 relative to pos C (that is, C admits ETA at 0), if
and only if, γC is continuous at every x ∈ pos C relative to pos C.
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Acknowledgments This research was supported by the National Natural Science Foundation of China
(Grants: 71090402, 11201383) and the Research Grants Council of Hong Kong (PolyU 5295/12E). The
authors would like to thank the two anonymous reviewers for their valuable comments and suggestions,
which have helped to improve the paper.
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