Operations Preserving Closedness ( a) Positive Combinations of Functions

Proposition 2.1.1 Let II, ... , 1m be in Conv IRⁿ [resp. in Conv IRn], tl, ... , tm be positive numbers, and assume that there is a point where all the

/j s

are finite. Then

the function

m 1:= Ltj/j

j=1 is in Conv IRⁿ [resp. in Conv IRn ).

PROOF. The convexity of

I

is readily proved from the relation of definition (1.1.1).

As for its closedness, start from

liminftjlj(Y)

=

^tj liminf Ij(Y) ~ tj/j(x)

y-+x y-+x

(valid for tj > 0 and

/j

closed); then note that the lim inf of a sum is not smaller than

the sum of lim inf's. 0

As an example, let

f

E Conv Rⁿ and C C ]Rn be closed convex, with dom

f

n C

i=

^0.

Then the function f

+

Ie of Example 1. 3 (a) is in Conv IR.^{n •} This trick can be used to simplify the notation for constrained minimization problems:

inf {f(x) : x E C} and inf{(f

+

Ie)(x) : x E ]Rn}

are clearly equivalent in the sense that they have the same infimal value and the same solution-set.

(b) Supremum of Convex Functions

Proposition 2.1.2 Let {fj }jE} be an arbitrary family of convex [resp. closed con-vex] functions.

If

there exists Xo such that sup} Ij(xo) < +00, then their pointwise supremum

I

:= sup

{/j :

j E J}

is in Conv]Rn [resp. in Conv IRn).

PROOF. The key property is that a supremum of functions corresponds to an intersec-tion of epigraphs: epi I

=

^njE} epi Ij' which conserves convexity and closedness.

The only needed restriction is nonemptiness of this intersection. ⁰ In a way, this result was already announced by Proposition 1.2.8. It has also been used again and again in the examples of § 1.3.

Example :U.3 (Conjugate Function) Let f : JRn ~ iR U {+oo} be a function not identi-cally +~, minorized by an affine function (i.e., for some (so, b) E JRn x JR, f ~ (so, .) - b on ]Rn). Then,

f* :]Rn 3 s 1-+ sup {(s,x) - f(x) : x E dom!}

is called the conjugate function of f, to be studied thoroughly in Chap. X. Observe that I*(so):::; band I*(s) > -00 for all s because domf

!

121. Thus,

1*

E Conv]Rn; this is true without any further assumption on f, in particular its convexity or closedness are totally

irrelevant here. 0

Example 2.1.4 Let S be a nonempty set (not necessarily convex) and take ]Rn 3 x 1-+ IPS(x) :=

! [lIxlI2 -

d~(x)] ,

where dS is the distance function to S, associated with the Euclidean norm 11·11. A surprising fact is that IPs is always convex. To see it, develop

d~(x)

=

^inf

IIx - cll2 = IIxll2 -

sup [2(c, x)

-lIcU2]

cES ceS

to obtain

IPS(x) = sup {(c, x) -

!lIcll2 :

c E

S} ;

IPS thus appears as the pointwise supremum of the affine functions (c, .) - 1/211c1l2, and is closed and convex. In view of the previous example, the reader will realize that IPs is the

conjugate of the function 1/211 . 112

+

^{Is. 0}

(c) Pre-Composition with an Affine Mapping

Proposition 2.1.5 Let

f

^{E Conv]Rn} [resp. Conv]Rn

J

and let A be an affine mapping from ]Rm to ]Rn such that 1m A

n

dom

f

⁼¹⁼ 0. Then the function

f ⁰ A: ]Rm 3 X ~ (f ⁰ A)(x)

=

^f(A(x»

is in Conv]Rm [resp. Conv]Rm

J.

PROOF. Clearly (f 0 A)(x) > -00 for all x, and there exists by assumption y

=

A(x) E ]Rn such that f(y) < +00. To check convexity, it suffices to plug the relation A(ax

+

^{(1 - a)x') = aA(x)}

+

^{(1 - a)A(x')}

into the analytical definition (1.1.1) of convexity. As for closedness, it comes readily

from the continuity of A when

f

is itself closed. 0

Example 2.1.6 With f (closed) convex on lRn , take Xo E dom f, d E ]Rn and define A: lR 3 t 1-+ A(t) = Xo

+

td;

this A is affine, its linear part is t 1-+ Aot := td. The reSUlting f ⁰ A appears as (a parametriza-tion of) the restricparametriza-tion of f along the line Xo

+

Rd, which meets dom f (at xo).

This operation is often used in applications: think for example of the line-search problem, considered in §II.3. Even from a theoretical point of view, the one-dimensional traces of f

are important, in that f itself inherits many of their properties; Proposition 1.2.5 gives an

instance ofthis phenomenon. 0

160 IV. Convex Functions of Several Variables

Remark 2.1.7 With relation to this operation on f E ConvlRⁿ [resp. ConvlRn ], call V the subspace parallel to aff dom f. Then, fix ^{Xo E} dom f and define the convex function fo E Conv V [resp. Conv V] by

fo(Y) := f(xo + Y) for all y E V.

This new function is obtained from

f

by a simple translation, composed with a restriction (from ]Rn to V). As a result, dom fo is now full-dimensional (in V), the relative topology relevant for fo is the standard topology of V. This trick is often useful and explains why "flat"

domains, instead of full-dimensional, create little difficulties. 0

(d) Post-Composition with an Increasing Convex Function

Proposition 2.1.8 Let f E Conv IRn [resp. Conv IRn] and let g E Conv IR [resp.

Conv lR] be increasing. Assume that there is Xo E IRⁿ such that f (xo) E dom g, and set g(+oo) :=

+00.

Then the compositejunction g 0 f : x f-+ g(f(x)) is in ConvlRⁿ [resp. in Conv IRnj.

PROOF. It suffices to check the inequalities of definition: (1.1.1) for convexity, (1.2.3)

for closedness. 0

The exponential g(t) := exp t is convex increasing, its domain is the whole line, so exp f (x) is a [closed] convex function of x E IRⁿ whenever f is [closed1 convex. A function

f :

IRn ~ ]0, +00] is called logarithmically convex when log

f

^E Conv IRn (we set again log(+oo) = +00). Because f

=

^{explog f,} a logarithmically convex function is convex.

As another application, the square of an arbitrary nonnegative convex function (for ex-ample a norm) is convex: post-compose it by the function g(t) = (max{O, t})2.

2.2 Dilations and Perspectives of a Function For a convex function

t

and u > 0, the function

fu : IRn 3 x ^f-+ fu(x)

=

uf(x/u)

is again convex. This comes from Propositions 2.1.1 and 2.1.5 but can also be seen geometrically: since fu(x)/u = f(x/u), the epigraphs and sublevel-sets are related by

epi fu

=

u epi f, episfu

=

u epis

t,

^{Sr (fu)}

=

^U Sr /u (f), which express that fu is a "dilated version" of f.

More interesting, however, is to study fu as a function of both variables x and u, i.e. to consider the set of all dilations of f. We therefore define the perspective of f as the function from IR x IRn to IR U {+oo} given by

j(u, x) := {uf(x/U) if u > 0,

+00

ifnot.

Proposition 2.2.1

If t

^E Conv IRn, its perspective j is in Conv IRn

+

^I.

PROOF. Here also, it is better to look at

j

with "geometric glasses":

epij = {(u,x,r)ElRtxlRnxlR: f(x/u)~r/u}

= {u(l,x',r'): u > 0, (x',r') E epif}

= Uu>o{u({l} x epif)} = lRt({l} x epif) and epi

j

is therefore a convex cone.

~(I

Fig.2.2.1. The perspective of a convex function

o

Figure 2.2.1 illustrates the construction of epi

j,

as given in the above proof. Embed epi f into IR x JRn x JR, where the first JR represents the extra variable u; shift it horizontally by one unit; finally, take the positive multiples ofthe result. Observe that, following the same technique, we obtain

domj = 1R~({l} x domf) . (2.2.1)

Another observation is that, by construction, epi j [resp. dom j) does not contain the origin oflR x lRn x IR [resp.lR x lRn).

Convexity of a perspective-function is an important property, which we will use later in the following way. For fixed Xo E dom f, the function d ~ f (xo

+

^{d) -} f (xo) is obviously convex, so its perspective

r(u, d) := u(f(xo

+

d/u) - f(xo») (for u > 0) (2.2.2) is also convex with respect to the couple (u, d) E lRt x ]Rn. Up to the simple change of variable u ~ t = l/u, we recognize a difference quotient.

The next natural question is the closedness of a perspective-function: admitting that

f

itself is closed, troubles can still be expected at u = 0, where we have brutally set j(O, .)

=

⁺⁰⁰ (possibly not the best idea ... ) A relatively simple calculation of cl

j

is in fact given by Proposition 1.2.5:

162 IV. Convex Functions of Several Variables

Proposition 2.2.2 Let

I

E Conv]Rn and let x' E ri dom

I.

Then the closure cl

j

of its perspective is given as follows:

{ ul(x/u) if u > 0,

(cl j)(u, x) = lillla-l-oal(x' - x

+

x/a) if u = 0,

+00

if u < O.

PROOF. Suppose first u < O. For any x, it is clear that (u, x) is outside cl dom

j

and, in view of (1.2.8), cl j(u, x)

= ^+00.

Now let u ? O. Using (2.2.1), the assumption on x' and the results of §I11.2.1, we see that (1, x') E ri dom

j,

so Proposition 1.2.5 allows us to write

(cl j)(u. x) = lim j (u, x)

+

a [(1 , x') - (u, x)]) a-l-o

= lim [u

+

a(1 - u)] I(x+a(xl-x»)

a-l-o u+a(l-u) .

If u = 1, this reads cl j(l, x) = cl I(x) = I(x) (because I is closed); ifu = 0, we

just obtain our claimed relation. 0

Remark 2.2.3 Observe thatthe behaviour of j(u, -) for u

+

0 just depends on the behaviour of f at infinity. If x = 0, we have

cl ](0,0) = lirnaf(x') = 0 [f(x') < +oo!].

a./-O

For x =1= 0, suppose for example that domf is bounded; then f(x' - x

+

x/a)

= +00

for a small enough and cl ](0. x) =

+00.

On the other hand, when domf is unbounded, cl ](0, .) may assume finite values if, at infinity, f does not increase too fast.

For another illustration, we apply here Proposition 2.2.2 to the perspective-function r of (2.2.2). Assuming Xo E ri dom f, we can take d' = 0 - which is in the relative interior of the function d H- !(xo

+

d) - !(xo) - to obtain

(cl r)(O, d)

=

lim !(xo - d

+

.d) - !(xo)

r-++oo •

Because (. - 1)/. ~ 1 for. ~

+00,

the last limit can also be written (in lR U {+oo}) (clr)(O. d) = lim f(xo

+

td) - f(xo)

1-++00

We will return to all this in §3.2 below.

o

2.3 Infimal Convolution

Starting from two functions

II

and

/Z,

fonn the set epi

11 +

epi

h c IRn x IR:

C := {(XI

+

^{X2, rl}

+

^{'2) : rj ? !.i(Xj) for j = 1. 2}.}

Under a suitable minorization property, this C has a lower-bound function le as in (1.3.5):

lc(x)

=

inf {rl

+

r2 : rj ~ f.j(Xj) for j = 1,2, Xl

+

X2 = x}.

In the above minimization problem, the variables are rl, r2, Xl, X2, but the rj's can be eliminated; in fact, le can be defined as follows.

Definition 2.3.1 Let II and f2 be two functions from IRⁿ to IR U {+oo}. Their infimal convolution is the function from IRⁿ to IR U {±oo} defined by

(fl

t

h)(x) ;= inf {fl (Xl)

+

h(X2) ; Xl

+

X2 = X}

= infyERn[fI(Y)

+

hex - y)]. (2.3.1) We will also call "infimal convolution" the operation expressed by (2.3.1). It is called exact at X

=

Xl

+

X2 when the infimum is attained at (Xl, X2), not necessarily

~~. 0

We refer to Remark 1.2.1.4 for an explanation and some comments on the termi-nology "infimal convolution". To exclude the undesired value -00 from the range of an inf-convolution, an additional assumption is obviously needed: in one dimension, the infimal convolution of the functions X and -x is identically -00. Our next result proposes a convenient such assumption.

Proposition 2.3.2 Let the functions II and

12

be in Conv IRn. Suppose that they have a common affine minorant:forsome (s, b) ^E IRⁿ x R

Ij(x) ~ (s, x) - b lor j

=

1,2 and all X ^E IRn . Then their infimal convolution is also in Conv IRn.

PROOF. For arbitrary X ^E IRⁿ and x], X2 such that Xl +X2

=

x, we have by assumption II (Xl)

+

h(X2) ~ (s, X) - 2b > -00,

and this inequality extends to the infimal value (fl

t

^h)(x).

On the other hand, it suffices to choose particular values Xj ^E dom f.j, j

=

I, 2, to obtain the point XI

+

X2 ^E dom(fl

t h).

Finally, the convexity of II

t 12

results from the convexity of a lower-bound function, as seen in § 1.3(g). 0

Remark 2.3.3 To prove that an inf-convolution of convex functions is convex, one

Dans le document Wissenschaften 305 A Series (Page 172-177)