Continuity Properties

3.1 Continuity on the Interior ofthe Domain

Convex functions turn out to enjoy sharp continuity properties. Simple pictures sug-gest that a convex function may have discontinuities at the endpoints of its interval of definition dom

f,

but has a continuous behaviour inside. This is made precise in the following result.

Theorem 3.1.1

If

f E Conv

R

then f is continuous on int dom f. Even more: for each compact interval [a, b] C int dom f, there is L ~ 0 such that

If(x) - f(x')1 ~ Llx - x'i for all x and x' in [a, b]. (3.1.1)

o

Property (3.1.1) is the Lipschitz continuity of f on [a, b]. What Theorem 3.1.1 says is that f is locally Lipschitzian on the interior of its domain. It follows that the difference quotients [f(x) - f(x')]/(x - x') are themselves locally bounded, i.e.

bounded on every bounded interval of int dom

f.

To prove Theorem 3 .1.1, the basic inequality (1.1.1) can be used on an enlargement of [a, b], thus exhibiting an appropriate Lipschitz constant L. We prefer to postpone the proof to Remark 4.1.2 below, where another argument, coming from the differential behaviour of

f,

yields the Lipschitz constant directly.

It remains to see how

f

can behave on the boundary of its domain (assumed to be

"at finite distance"). In the statement below, we recall that our notation x

i

a excludes the value x

=

a.

Proposition 3.1.2 Let the domain of f E Conv IR have a nonempty interior and call a E lR. its left endpoint. Then the right-limit f(a+) := limx.J..a f(x) exists in lR. U Hoo}, and f(a) ;;:: f(a+).

Similarly,

if

b E lR. is the right endpoint of dom f, the left-limit f (b _) exists in lR. U Hoo} and feb) ~ f(b_).

PROOF. Let Xo E int dom

f,

set d := -1, to := Xo - a > O. The increasing function f(xo

+

td) - f(xo)

o

< t 1-+ =: q(t)

t

has a limit

e

^E lR. U {+oo} for t

t

to, in which case Xo + td

i

a. It follows

3 Continuity Properties 17 f(xo + td) = f(xo) + tq(t) --+ f(xo) + (xo - a)l =: f(a+) E lR U {+oo} . Then let t

t

^to in the relation

f(a) - f(xo)

q(t) :,;; q(to) = for all t E ]0, toe Xo - a

to obtain

f(a+) - f(xo) f(a) - f(xo)

l= :,;; ,

Xo - a Xo - a

hence f(a+) :,;; f(a). The prooffor b uses the same arguments.

o

The function q of the above proof is the directional difference quotient, already encoun-tered in §2.3, and is nothing more than the slope-function [f(x) - f(xo)]/(x - xo). We took the trouble to use it as an illustration of Remark 1.3.3, to avoid the unpleasant division by x - Xo < O. Furthermore it will play an important role in several dimensions.

Among other things, Proposition 3.1.2 says that f is upper semi-continuous (relative to dom f) on the edge of its domain, hence on its whole domain. This property, however, is specific to the one-dimensional case, and is not true in several dimensions.

3.2 Lower Semi-Continuity: Closed Convex Functions

According to Proposition 3.1.2, the behaviour of a convex function at the endpoints of its domain has to resemble one of the cases illustrated on Fig. 3.2.1. We see that case (2) is somewhat "abnonnal"; it is ruled out by the following definition, which thus appears as "natural", and important for existence of solutions in minimization problems.

f(a) E [f(a+),+oo]

~

f(a+)

a a ^a

(1) Continuous (2) Upper semi-continuous (3) Unbounded Fig.3.2.1. Continuity properties of univariate convex functions

Definition 3.2.1 We say that

f

E Conv lR is closed, or lower semi-continuous, if liminf f(x) ~ f(xo) for all Xo E lR.

x ... xo (3.2.1)

The set of closed convex functions on lR is denoted by Conv R

o

Of course, (3.2.1) is an inequality in IR U {+oo}. A reader not totally alert may overlook a significant detail: both x and Xo are points in the whole of IR. A closed function is therefore lower semi-continuous on the whole line, and not only relative to its domain of definition, as is the usual practice in real analysis. This is why the terminology "closed" should be preferred to lower semi-continuous.

The requirement (3.2.1) demands nothing of

1

beyond the closure of its domain;

it is moreover true for Xo E int dom

1

(Theorem 3.1.1), the only possible problems are on the boundary of dom

I.

As far as the left endpoint is concerned, a closed convex function has to look like case (1) or (3) in Fig. 3.2.1 (and note: if a

=

-00, no trouble arises).

The closedness property can also be described geometrically, which by the same token justifies the terminology (note that convexity plays little role here).

Proposition 3.2.2 The function

1

is closed

if

and only

if

one of the following condi-tions holds:

(i) epi

1

is a closed set of 1R2;

(ii) the sublevel-sets

SrU) := {x E IR : I(x) ~ r}

are closed intervals of IR (possibly empty), for all r E R o The best way of proving this result is probably to look at Fig. 3.2.1. In practice, the closedness criterion (ii) is very useful. As an example, the function

I/x;

of §2.3 is always closed.

Example 3.2.3 Let

1

be a convex function whose domain is the whole oflR, and let C be a nonempty closed interval. Then the "convex restriction" of

1

to C:

fc(x) := I(x) if x E C. +00 otherwise

is closed and convex. Its epigraph is the intersection of epi

1

with the vertical stripe

generated by C. 0

Example 3.2.4 Let C be a nonempty interval of IR. The indicator function of C is

{ o

^ifxEC,

Ic(x):= +00 otherwise.

It is a closed convex function if and only if C is closed (its sublevel-sets are empty or C).

The above indicator function, of constant use in convex analysis, must not be confused with the characteristic function xc of measure theory. which is 1 on C and

o outside - in fact xc = exp( -Ic). 0

Let us return to Fig. 3.2.1. In case (2) - the only bad case - we see that it is not difficult to close I: it suffices to pull I(a) down to I(a+). The result is in ConvlR, and differs very little indeed from

I.

3 Continuity Properties 19 Definition 3.2.5 The closure of

I

E Conv lR is the function defined by:

{ liminf I(y) if x E cldom/, cl I(x) := y-+x

+00 if not. (3.2.2)

o

Remark 3.2.6 The construction (3.2.2) affects I only at the endpoints of its domain. Geo-metrically, cl I is the function whose epigraph is the closure (in the usual topological sense) of the set epi I C ~2. The closure of I is also the largest closed convex function minorizing

I:

cl I(x) = sup (g(x) : g E ConvlR and g ~ f}. (3.2.3) To close

I,

however, it is actually not necessary to scan the whole set of closed convex functions. It can be proved that, in (3.2.3), g can be restricted to being an affine function:

cl/(x) = sup{sx -r : sy - r ~ I(y) forally E JR}; (3.2.4) s,r

this will be established formally in Proposition Iv'I.2.8. Of course, it is the convexity of I

which allows this simplification.

The analytic operation (3.2.2), illustrated by Fig. 3.2.1, does not easily lend itself to generalizations in several dimensions (taking the lower semi-continuous hull of a function may be difficult). The operation (3.2.4) thus appears as a possible useful alternative. 0

3.3 Properties of Closed Convex Functions

Closed convex functions are offundarnental importance in convex analysis and opti-mization. For one thing, the existence of a solution for the problem

min {f(x) : x E C}

requires first

I

to be closed (i.e. lower semi-continuous), and also C to be closed (and of course C

n

dom

I #-

0). Just as §2.1 did with convexity, it is therefore useful to know which combinations of functions preserve closedness.

Proposition 3.3.1 Let II, ... , 1m be m closed convex functions and tl, ... , tm be positive numbers.

If

there exists Xo such that fj (xo) < +00 for j = 1, ... , m, then

the jUnction I :=

I:J=I

tj Ij is in Conv R 0

Here, convexity is taken care of by Proposition 2.1.1; as for closedness, recall that limes inferiores are stable under addition and positive multiplication, i.e.

liminft(Uk

+

^{Vk) ~ tliminfuk}

+

^tliminfvk

(an inequality in lR U (+oo}).

The next result also comes easily, since an intersection of closed sets is closed.

Proposition 3.3.2 Let (fj }jEJ be afamity of closed convexfunctions.

If

there exists Xo E lR such that SUPjEJ fj(xo) < +00, then the jUnction

I

:= SUPjEJ

Ij

is in

ConvR 0

Example 3.3.3 Let

I :

lR --+ lR U {+oo} be a function not identically +00 (but not necessarily convex) minorized by some affine function: there are So, ro such that

I (x) ~ Sox - ro for all x. Then the so-called conjugate function of I:

lR '3 S ~ sup (sx - I(x) : x E dom f}

is finite at least at So, and it is closed convex (as a supremum of affine functions!). 0 The case of the infimal convolution is much more delicate (it is an infimum, hence a limit, and the inequality signs go the wrong direction, in contrast with Proposition 3.3.2).

Remark 3.3.4 It can indeed be proved that the inf-convolution of two functions of Con v lR is still closed, but this is a specific result ofthe univariate case: its extension to several variables requires some additional assumption.

To accept the closedness of a one-dimensional inf-convolution, a first key is to realize that, if a is the left endpoint of the domain of I

=

^II t

12,

then a

=

^al

+

^a2 and I(a) = II (al)

+

^h(a2) where, for i = 1,2, ai is the left endpoint ofdomJi. Thus, for k = 1,2, ... , take x~ and x~ satisfying:

xf

^{E dom}

^Ii

^{and x~}

⁺

^{x~ = Xk ~ a;} a second key is then to see that

xf

^{,j. ai for i = 1,2. If,} in addition,

II

(x~)

+

h(x~) :::; (fl t h)(xk)

+ 1/

k,

it suffices to pass to the limit, using the properties I(xf) --+ I(ai) for i = 1,2. 0

Finally, the case of limit-functions of §2.2 is of course hopeless. The traditional example x ~

!k

(x) := Ix Ik converges pointwise when k --+

+00

to

x

~

^{I(x) =}

1 ~

+00

if x E]-I,+I[, if x E {-I, +l}, otherwise,

which is not closed. Some "uniformity" in the convergence is required, and this es-tablishes a link between Remark 2.2.2 and Theorem 3.1.1.

Dans le document Wissenschaften 305 A Series (Page 31-35)