COPYRIGHT NOTICE
FedUni ResearchOnline
https://researchonline.federation.edu.au
This is the author’s accepted version of the following publication:
Kruger, A. (2006) About regularity of collections of sets. Set-Valued
Analysis, 14(2), 187-206.
The version displayed here may differ from the final published version.
The final publication is available at:
http://dx.doi.org/10.1007/s11228-006-0014-8
Copyright © 2006, Springer.
Set-Valued Analysis manuscript No.
(will be inserted by the editor)
Alexander Ya. Kruger
About Regularity of Collections of Sets
Received: date / Accepted: date
Abstract The paper continues investigations of stationarity and regularity prop-
erties of collections of sets in normed spaces. It contains a summary of different
characterizations (both primal and dual) of regularity and a list of sufficient con-
ditions for a collection of sets to be regular.
Keywords variational analysis, normal cone, optimality, extremality, stationarity,
regularity, set-valued mapping, Asplund space
Mathematics Subject Classification (2000) 90C46, 90C48, 49K27; Secondary:
58C20, 58E30
1 Introduction
This paper continues investigations of stationarity and regularity properties of col-
lections of sets in normed spaces started in [19]. See [19] for motivations as well
as comparisons of different approaches.
Starting with the pioneering work by Dubovitskii and Milyutin [10] it is quite
natural when dealing with optimality conditions to reformulate optimality in the
original optimization problem as a (some kind of) extremal behaviour of a certain
collection of sets. An easy example is a problem of unconditional minimization of
a real-valued function ϕ : X → R. If x
◦
∈ X one can consider the sets Ω
1
= epi ϕ =
{(x, µ) ∈ X × R : ϕ(x) ≤ µ} (the epigraph of ϕ) and Ω
2
= X × {µ : µ ≤ ϕ(x
◦
)}
(the lower halfspace). The local optimality of x
◦
is then equivalent to the condition
Ω
1
∩ int Ω
2
∩ (x
◦
+ ρB) = /0 for some ρ > 0.
The concept of extremality for a collection of sets was first defined in [21].
This definition was extended to stationarity in [14–16], where two definitions of
School of Information Technology and Mathematical Sciences, Centre of Information and Ap-
plied Optimization, University of Ballarat, POB 663, Ballarat, Vic 3350, Australia
E-mail: a.kruger@ballarat.edu.au
2 Alexander Ya. Kruger
stationarity (one of them was at first called “extended extremality”) were intro-
duced. The relation between extremality and stationarity for collections of sets is
similar to that between optimality and stationarity for optimization problems.
Regularity can be considered as the property opposite to stationarity (see [19]).
Regularity properties of collections of sets play an important role in different
fields of optimization and approximation: constraint qualifications, error bounds,
convergence analysis, etc. (see [3,7,26] for numerous examples). They are also
closely related to similar properties of multifunctions and can be used in analysis,
in particular, in nonsmooth calculus.
Following [16,18], extremality-stationarity-regularity properties are defined in
the paper with the help of some constants providing quantitative estimates of the
corresponding properties. A similar approach (for different properties) is under-
taken in the recent paper [2].
Examining and comparing different regularity concepts for collections of sets
has attracted recently considerable attention in the literature (see [2–5,7,19,26,
27]), although according to [2] some traces of such considerations can be found
in the 1940 paper by M. Krein.
The strong regularity concept investigated in the current paper is closely re-
lated to the metric regularity property (sometimes referred to as pseudo regularity)
of multifunctions. Different characterizations of strong regularity of collections of
sets are presented as well as some relations to other properties. The main emphasis
in the current paper is on primal space conditions.
The definitions of the constants, the relations between them and the corre-
sponding stationary and regularity concepts developed in the current paper are
very similar to those for nonsmooth functions and multifunctions (see [18–20]).
Actually these are different applications of the same variational approach.
In the convex case there exists another set of definitions of regularity properties
(based on linear regularity [3]) with numerous interesting equivalences and other
relations (see [3,7,26]). Linear regularity is defined as a global property. However,
local versions of this property can also be of interest. Note that (local) linear regu-
larity is in general weaker than the strong regularity property considered here (see
the example at the end of subsection 3.3). Thus, two sets of regularity conditions
exist in parallel with many similarities between them.
The paper is organized as follows. The definitions of extremality, stationarity
and regularity for the collection of sets are introduced in Section 2. Some pri-
mal space constants characterizing the mutual arrangement of sets in space are
used in the definitions. Section 3 contains a summary of different characterizations
(both primal and dual) of strong regularity with relations to other properties: met-
ric inequality, error bounds, weak sharp minima, Jameson’s property (G), strong
additive regularity. Some more constants for quantitative characterization of the
corresponding properties are introduced. The final Section 4 is devoted to consid-
ering examples of strongly regular collections of sets. It contains a list of sufficient
conditions for a collection of sets to be strongly regular.
Mainly standard notations are used throughout the paper. A closed unit ball
in a normed space is denoted by B. If Ω is a set then intΩ, bdΩ and clΩ are
respectively its interior, the boundary and the closure. When considering product
spaces we will always assume that they are equipped with the maximum-type
norm:
k
(x
1
,x
2
)
k
= max(
k
x
1
k
,
k
x
2
k
).
About Regularity of Collections of Sets 3
2 Extremality, stationarity and regularity
Let us consider a collection of sets Ω
1
, Ω
2
, .. ., Ω
n
(n > 1) in a normed space X
with x
◦
∈ ∩
n
i=1
Ω
i
.
The following constant can be used for characterizing the mutual arrangement
of the sets Ω
1
, Ω
2
, ..., Ω
n
near x
◦
([18,19]):
θ
ρ
[Ω
1
,...,Ω
n
](x
◦
) = sup{r ≥ 0 :
n
\
i=1
(Ω
i
− a
i
)
\
(x
◦
+ ρB) 6= /0, ∀a
i
∈ rB}. (1)
It shows how far the sets can be “pushed apart” while still intersecting in a neigh-
borhood of x
◦
. Evidently θ
ρ
[Ω
1
,...,Ω
n
](x
◦
) is nonnegative (and can be equal to
+∞) and nondecreasing as a function of ρ. Moreover,
lim
ρ→+0
θ
ρ
[Ω
1
,...,Ω
n
](x
◦
) = 0
unless x
◦
∈ int ∩
n
i=1
Ω
i
([19], Proposition 3).
A slightly more general form of (1) can be of interest:
θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
) = sup{r ≥ 0 :
n
\
i=1
(Ω
i
− ω
i
− a
i
)
\
(ρB) 6= /0, ∀a
i
∈ rB}. (2)
This constant corresponds to the case when instead of the common point
x
◦
∈ ∩
n
i=1
Ω
i
each of the sets Ω
i
is considered near its own point ω
i
∈ Ω
i
,
i = 1,2, . . . , n. The sets do not need to be intersecting. It is equivalent to consider-
ing the collection of translated sets Ω
1
− ω
1
, Ω
2
− ω
2
, ..., Ω
n
− ω
n
near 0:
θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
) = θ
ρ
[Ω
1
− ω
1
,...,Ω
n
− ω
n
](0).
If ω
1
= ω
2
= . . . = ω
n
= x
◦
then, of course,
θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
) = θ
ρ
[Ω
1
,...,Ω
n
](x
◦
).
When investigating stationarity-regularity properties it is important to know
how fast θ
ρ
[Ω
1
,...,Ω
n
](x
◦
) and θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
) approach 0 in com-
parison with ρ. This can be characterized by the following “linearized” constants:
θ[Ω
1
,...,Ω
n
](x
◦
) = liminf
ρ→+0
θ
ρ
[Ω
1
,...,Ω
n
](x
◦
)
ρ
. (3)
θ[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
) = liminf
ρ→+0
θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
)
ρ
. (4)
The last step in the process of defining the collection of constants characteriz-
ing local properties of the collection of sets Ω
1
, Ω
2
, ..., Ω
n
near x
◦
is to consider
4 Alexander Ya. Kruger
the limit in the right-hand side of (4) when the points ω
1
, ω
2
, . .., ω
n
are not fixed,
but approach x
◦
. We arrive at the next constant:
ˆ
θ[Ω
1
,...,Ω
n
](x
◦
) = liminf
ω
i
Ω
i
→x
◦
ρ→+0
θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
)
ρ
(5)
The notation ω
Ω
→ x in (5) means that ω → x with ω ∈ Ω.
Proposition 1 The following inequality holds true:
ˆ
θ[Ω
1
,...,Ω
n
](x
◦
) ≤ liminf
ω
i
Ω
i
→x
◦
θ[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
). (6)
Proof Denote for brevity
ψ
ρ
(ω
1
,...,ω
n
) =
θ
ρ
[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
)
ρ
.
By the definition of the lower limit one can write
ˆ
θ[Ω
1
,...,Ω
n
](x
◦
) = lim
δ →+0
inf
ω
i
∈Ω
i
∩(x
◦
+δ B)
0<ρ≤δ
ψ
ρ
(ω
1
,...,ω
n
) =
lim
δ →+0
inf
ω
i
∈Ω
i
∩(x
◦
+δ B)
inf
0<ρ≤δ
ψ
ρ
(ω
1
,...,ω
n
). (7)
The legality of the replacement of the “double” infimum in the above formula
by two separate ones is quite obvious. For any ω
i
∈ Ω
i
, i = 1,2,...,n, and any
0 < δ
0
≤ δ one has
inf
0<ρ≤δ
ψ
ρ
(ω
1
,...,ω
n
) ≤ inf
0<ρ≤δ
0
ψ
ρ
(ω
1
,...,ω
n
).
Consequently,
inf
0<ρ≤δ
ψ
ρ
(ω
1
,...,ω
n
) ≤ lim
δ
0
→0
inf
0<ρ≤δ
0
ψ
ρ
(ω
1
,...,ω
n
) =
liminf
ρ→+0
ψ
ρ
(ω
1
,...,ω
n
) = θ[Ω
1
,...,Ω
n
](ω
1
,...,ω
n
). (8)
(6) follows from (7) and (8). ut
Remark 1 The above proof is valid for the “combined” lower limit of any function
of several variables.
Inequality (6) can be strict.
Example 1 Consider two sets in R
2
: Ω
1
=
(x,y) ∈ R
2
: ϕ(x) ≤ y
and Ω
2
=
(x,y) ∈ R
2
: y ≤ 0
, where the function ϕ : R → R is defined in the following
way: ϕ(x) = x if x ≤ 0, ϕ(x) = x − 1/n if 1/n < x ≤ 1/(n − 1), n = 2,3,...,
ϕ(x) = x− 1/2 if x > 1/2.
Take ω
1
= (x
1
,y
1
) ∈ Ω
1
and ω
2
= (x
2
,y
2
) ∈ Ω
2
. If ϕ(x
1
) < y
1
or y
2
< 0
then θ
ρ
[Ω
1
,Ω
2
](ω
1
,ω
2
) ≥ y
1
− ϕ(x
1
) − y
2
> 0 for any ρ > 0 and consequently