DUALITY FOR THE LEVEL SUM OF QUASICONVEX
FUNCTIONS AND APPLICATIONS
M. VOLLE
Abstract. We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derive new duality results for the minimization of the max of two qua- siconvex functions. Following Barron and al., we show that the level sum provides quasiconvex viscosity solutions for Hamilton-Jacobi equa- tions in which the initial condition is a general continuous quasiconvex function which is not necessarily Lipschitz or bounded.
1. Introduction
The powerful properties of the Legendre-Fenchel conjugation for convex functions can be summarized by saying that, under some mild assumptions (also called qualication conditions), it exchanges the pointwise sum of func- tions with the inmal convolution of their conjugates and vice versa (15], 27]). The geometrical interpretation of the inmal convolution of two ex- tended real-valued functions is well known: it lies in the fact that the vecto- rial (or Minkowski) sum of the strict epigraphs of two extended real-valued functions is nothing but the strict epigraph of their inmal convolution. By the way such a property is at the origin of another terminology in which the inmal convolution is called epigraphical sum (2] 3]). When dealing with quasiconvex functions this operation suers from the fact that the epi- graphical sum of two quasiconvex functions is no longer quasiconvex (18]).
However, there exists an interesting substitute for the epigraphical sum in the eld of quasiconvexity, namely, the level sum of functions, that is the function whose strict lower level sets are the vectorial sum of the strict lower level sets of the initial functions (18] 28] 30] 33],...). Thus the level sum of two quasiconvex functions is still quasiconvex. Another way to introduce the level sum is obtained by perturbing the problem of minimizing the max of two extended real-valued quasiconvex functions. It appears that such a problem plays an analogous role to the one played by the minimization of the sum of two convex functions in convex optimization (35]). Moreover the level sum has also been recently used to obtain a viscosity solution of the Hamilton-Jacobi equation ut+H(D uu) = 0 arising in the minimization of the maximum cost in L1 spaces here the Hamiltonian H is sublinear in the rst variable and nondecreasing in the second variable (7], 8]). In this paper we introduce and study a conjugation for quasiconvex functions,
University of Avignon, Department of Mathematics, 33, rue Louis Pasteur, 84000 Avi- gnon, France. E-mail: [email protected].
Received by the journal September 2, 1997. Revised May 7, 1998. Accepted for publi- cation May 15, 1998.
c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
slightly dierent from those introduced by Crouzeix in 11] 12] and Barron and Liu in 8], that transforms the level sum of arbitrary extended real- valued functions into the pointwise sum of their conjugates. Such a property is not shared by the more or less classical conjugations already introduced in quasiconvex duality theory (19] 20] 25] 29] 31] 32] 34],...). Another advantage of the biconjugation we present is that it constitutes a Galois correspondence nevertheless it is not a generalized conjugation in the sense of Moreau (21]). In fact it enters in the general framework introduced by J.-P. Penot and the author (23]) but has never been studied for its own interest. The classes of regular functions related to this biconjugation are entirely characterized in theorems 3.4 and 3.5. The case of radial functions is studied in details in Propositions 4.10, 4.11 and Corollaries 4.12, 4.13.
Under a qualication condition involving the directions of majoration of the functions, we prove that the level sum of two quasiconvex lower semicon- tinuous (`.c.s.) extended real-valued functions coincides with the conjugate of the sum of their conjugates (Theorem 4.2). As a by-product we get dual problems with zero duality gap for the global minimization of the max of two quasiconvex `.s.c. extended real-valued functions (Theorem 4.5 and its corollaries). In this way we recapture a dual problem introduced in 35] with dierent conditions ensuring a zero duality gap. Finally, following Barron and al.(7]), we show that the level sum provides a quasiconvex viscosity so- lution of the Hamilton-Jacobi equation ut+H(D uu) = 0,u(0 ) =gwhen the initial condition g is a general continuous quasiconvex function which is not necessarily Lipschitz or bounded (Proposition 5.2, Theorem 5.5), a result announced in 36].
More generally, the present paper may be viewed as a contribution to minimax analysis 5], 16].
2. Definitions, notations
In the sequel X is a Hausdor locally convex topological vector space with topological dualY and we denote byhithe bilinear mapping hxyi:=
y(x)(x 2X y 2 Y). Given an extended real-valued function f :X ! R and t2R :=Rf;1gf+1g we set
ff <tg=fx2X:f(x)<tg
for the strict lower level set (at level t) of f. Observe that ff < tg is nonempty i t >inf
X
f. The function f is said to be quasiconvex if all its strict lower sets are convex. Of course this amounts to saying that all its lower sets ff tg(t2R) are convex, or that
f(tu+ (1;t)v)max(f(u)f(v)) :=f(u)_f(v) for all uv2X,t201] (4], 13], ...).
By adding strict lower sets of two extended real-valued functions fg :
X !R,
ff <tg+fg<tg:=fu+v:u2ff <tgv2fg<tgg one obtains the strict lower level set of the function
x2X 7;!(f4g)(x) := inf(f(u)_g(v) :u+v=x): (2.1)
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More precisely, one has
ff4g<tg=ff <tg+fg<tg (2.2) for all t2R. Here we adopt the convention
A+=+A= (2.3)
for any subset AX accordingly to the relation inf
X
(f4g) = inf
X
f_inf
X
g: (2.4)
The function f4g is called the level sum of f and g. It is quasiconvex whenever the functions f and g are so.
Support functions will play a central role in the paper. Recall that the support function of a subsetAX is dened by
y2Y 7;!i
A(y) := sup
x2A
hxyi
with the convention sup=;1. In other words the support function ofAis the Legendre-Fenchel conjugate of the indicator function iA of A(iA(x) = 0 if x 2 A, iA(x) = +1 if x 2 XnA)iA is positively homogeneous (p.h.) (iA(y) =iA(y) whenever >0 y2Y) and convex this means thatiA is sublinear (i.e. p.h. and subadditive). Of course iA is also`.s.c.. Notice that iA is either proper (i.e. iA(Y) Rf+1g and iA(Y) 6= f+1g ) or identically ;1. In fact, Hormander's theorem (17]) says that the class of support functions coincides with the class of `.s.c. sublinear extended real-valued functions which are either proper or identically ;1.
Taking 2.3 into account one has (for all y2Y),
i
A+B(y) =iA(y)+.iB(y) (2.5) where (+1)+.(;1) = (;1)+.(+1) =;1.
Of course we set (+1) _+(;1) = (;1) _+(+1) = +1. The operations +. and _+ enjoy calculus rules (21]): in particular we shall use the fact that, for all extended-real numbers rst2R, the following equivalence holds:
s_+t<r()9s1t1 2R :r=s1_+t1 s<s1 t<t1: (2.6)
3. A Galois correspondence for quasiconvex functions To each extended real-valued function f : X ! R let us associate the conjugate function fc dened by
f
c(yr) := sup
f(x)<r
hxyi= sup
ff<r g
hyi (3.1)
for all (yr) 2 Y R. Let us rst observe that this conjugation is closely related to the level sum of functions:
Theorem 3.1. For arbitrary extended real-valued functions f, g :X ! R one always has
(f4g)c=fc+.gc
Proof. An easy consequence of the denition (3.1) and of relations (2.2), (2.5).
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Among many other interesting properties of the correspondence f 7;! f we retain the following:
Proposition 3.2. Let f :X ! R be a general extended real-valued func- tion the conjugate function fc satises the following properties:
a) for any r2R,fc(r) is`.s.c., sublinear, proper or identically ;1 b) for any y2Y,fc(y ) is `.s.c., nondecreasing.
Moreover,
(inf
i2I f
i)c = sup
i2I f
c
i (3.2)
for any family (fi)i2I of extended real-valued functions dened on X. Proof. Let r 2 R if ff < r g = then f( r) is identically ;1 if ff <
r g 6= then f(r) is nothing but the support function of the nonvoid set
ff < r g which is sublinear `.s.c. and proper. It is evident that, for any
y 2Y, the function fc(y ) is nondecreasing, while the lower semicontinuity of this function follows from the easy but important property, valid for all
y 2Y,r,t2R:
f
c(yr)t() inf
hxy i>t
f(x)r: (3.3)
On the other hand, (3.2) is a trivial consequence of the relation finf
i2I f
i
<
r g=
i2I ff
i
<r g.
What formula (3.2) says is that the mappingf 7;!fcis a branch of a Galois correspondence (9]). Let us now describe the other branch.
Proposition 3.3. Let f : X ! R and ' : Y R! R be two extended real-valued functions. The following assertions are equivalent:
a) fc(yr)'(yr) for all (yr)2Y R b) f(x)sup
y 2Y
supfr2R:hxyi>'(yr)gfor allx2X.
Proof. By (3.3), a) amounts to saying that for anyx2X,y2Y,r2Rone has
hxyi>'(yr) =)f(x)r or, equivalently, for any x2X y2Y,
f(x)supfr2R:hxyi>'(yr)g that is exactly b).
Thus let us dene the conjugate of an arbitrary extended real-valued func- tion ':Y R!R by setting
'
(x) := sup
y 2Y
supfr2R:hxyi>'(yr)g (3.4) for allx2X. We then obtain the other branch of the Galois correspondence above mentioned and the related properties:
f c
'()f '
(inf
i2I '
i) = sup
i2I '
i
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for any extended real-valued functions f :X !R,':Y R!R,
'
i :Y R!R(i2I). Moreover the composite operators
f 2RX 7;!fc := (fc) 2RX
'2RYR7;!'c:= (')c2RYR are isotone:
f g=)fc gc ' =)'c c contracting:
f c
f ' c
'
and idempotent:
(fc)c =fc ('c)c ='c: More precisely one always has
f
cc =fc 'c =':
The class of functionsf 2RXthat coincide with their biconjugatefc (that we shall call regular in the sequel) is given byf' :'2RYRg=f' :'=
' c
g. Of course a similar property holds for the class of regular functions ' on Y R.
The biconjugate of f 2RX (resp. '2RYR) is nothing but the greatest regular minorant of f (resp. '):
f
c = supfh2RX :hc =h and hfg (resp:'c = supf2RYR:c = and 'g:
All the above considerations are mere consequences of the theory of Galois correspondences (5], 9]). The description of the classes of regular functions is crucial for our purpose. We begin with regular functions on X.
Theorem 3.4. Let f : X ! R be an extended real-valued function. Then
f =fc i f is`.s.c. quasiconvex.
Proof. It follows from (3.4) that for any '2RYRt2R,
f'
tg= \
y 2Y
\
r >t
fhyi'(yr)g
that is an intersection of closed convex sets hence the condition is necessary.
Let us prove that it is also sucient. Let x 2 X and t 2 Rbe such that
r <f(x) thus x does not belong to the closed convex set ff r g by the Hahn-Banach theorem there exists y 2 Y such that hxyi> sup
f(u)r
huyi
f
c(yr):By (3.4) we then havefc(x)rso that, by taking the supremum over r<f(x) fc(x) f(x). As the opposite inequality always holds the proof is complete.
Let us now describe the class of regular functions on Y R.
Theorem 3.5. Let ' : Y R ! R be an extended real-valued function.
Then, '='c i the following conditions are satised:
a) for any r2R,'(r) is `.s.c., sublinear, proper or identically;1 b) for any y2Y, '(y ) is `.s.c. nondecreasing.
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Proof. Necessity has been established in Proposition 3.2. Conversely, for all real number r there exists a closed convex set Ar such that '(r) =
sup
x2Ar
hx i Let us set f(x) := infft 2R: x 2 Atg and prove that ' = fc. First observe that ff <r g=
t<r A
t. On the other hand, as '(y ) is `.s.c.
nondecreasing,
sup
t<r
sup
x2At
hxyi= sup
x2Ar hxyi
so that
sup
f(x)<r
hxyi= sup(hxyi:x2
t<r A
t) = sup
x2A
r hxyi
that is to say fc ='.
4. Explicit dual formulas for the level sum of l.s.c.
quasiconvex functions.
Thank to Theorem 3.1 and Theorem 3.4, let us observe that for any extended real-valued functions f, g : X ! R, (fc+.gc) is just the `.s.c.
quasiconvex hull of the level sum f4g. In the case when f and g are quasiconvex, f4g is quasiconvex too so that (fc+.gc) is then the `.s.c.
hull of f4g. Now there exists a condition ensuring that the level sum of two quasiconvex `.s.c. extended real-valued functions is still `.s.c. and, moreover, exact (i.e. the inmum is attained in (2.1)). This condition involves the so-called directions of majoration introduced by Amara (1]).
Let us briey describe this notion. Given a quasiconvex functionf :X!R, a vector d 2 X is said to be a direction of majoration of f is there exists (x0r0)2XRsuch that
f(x0+td)r0 for all t0:
The set of directions of majoration of f is a convex cone we denote byCf. The function f is called inf-locally compact ifff r gis locally compact for anyr2R. We then have the following result:
Lemma 4.1. (1]) Let fg:X!R be two extended real-valued `.s.c. quasi- convex functions such that Cf \;Cg =f0g. Assume moreover that f or g is inf-locally compact. Then f4g is`.s.c., quasiconvex and exact.
It follows from the previous considerations that:
Theorem 4.2. Forf and g as in Lemma 4.1 one has
f4g= (fc+.gc):
Clearly the dual formulations of the level sum we look for will come from the computation of the conjugate of the sum of two functions onY R. For doing this, the use of quasi-inverses is essential. Let us recall (see for instance 24]) that two nondecreasing functions ab:R!R are quasi-inverses if for all real numbers r,sone has
a(r)<s=)rb(s)
a(r)>s=) rb(s):
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Among all the quasi-inverses of athere is a smallest one namely
a
e(s) = supfr2R:a(r)<sg= inffr2R:a(r)sg
which is also the only `.s.c. quasi-inverse of a. It is characterized by the relations
a(r)<s=)rae(s)r<ae(s) =)a(r)<s (4.1) for any real numbers rs.
Given a function f :X !R andy2Y, typical quasi-inverses are the fol- lowing (13], 24]): py andby,py andy,y andby,yandy wherepy(r) =
f
c(yr) y(r) = sup
f(x)r
hxyi
y(s) = inf
hxy i>s
f(x), by(s) = inf
hxy is f(x) (rs2R).
Now observe that, from Theorem 4.2 and (3.4), (4.1), one has (f4g)(x) = sup
y 2Y
(py+.qy)e(hxyi) (4.2) for all x2X, withpy(r) =fc(yr) andqy(r) =gc(yr). Hence, to apply in concrete terms formula (4.2), we need to estimate the smallest quasi-inverse of the sum of two nondecreasing extended real-valued functions. In the next two lemmasa1anda2 are such functions and we denote bybi a quasi-inverse of ai(i= 12). We also set
(b15b2)(s) = sup
s
1 +s
2
=s b
1(s1)^b2(s2) (s2R): Lemma 4.3. (a1+.a2)e (b15b2)_inf
R b
1 _inf
R b
2. Proof. Lets2R observe that
(a1+.a2)e(s) = supfr2R:a1(r) _+a2(r)<sg_r1_r2
where ri= supfr2R:ai(r) =;1g(i= 12). In fact, one easily getsri = inf
R b
i(i= 12). Now by (2.6): a1(r) _+a2(r)<s()9s1s22R:s1+s2 =s,
a
i(r) < si thus r <bi(si)(i = 12) therefore, supfr 2 R: a1(r) _+b1(r) <
sg(b15b2)(s) and Lemma 4.3 is proved.
The next relation is linked with a formula given in 22] (Lemma 2.4) for non-negative functions.
Lemma 4.4. (a1+.a2)e b14b2.
Proof. Let r>(b14b2)(s). Thus there exists1s2 2Rsuch thats1+s2=s and bi(si) <r(i= 12). We then have si ai(r) so that s a1(r)+.a2(r) and rinffr2R:a1(r)+.a2(r)sg= (a1+.a2)e(s).
We are now ready to state the main result of this section:
Theorem 4.5. Let f, g :X ! R be two extended real-valued `.s.c. quasi- convex functions such thatCf\;Cg =f0g. Assume thatf orgis inf-locally compact. Then for all x2X :
(f4g)(x) = sup
y 2Y
sup
s+t=hx y i
inf
huy is
f(u)^ inf
hv y it
g(v) (4.3)
= sup
y 2Y
inf
s+t=hxy i
inf
huy is
f(u)_ inf
hv y it
g(v): (4.4)
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Proof. For each y 2Y let us consider the quasi-inverse by(s) = inf
huy is f(u) (resp. cy(s) = inf
hv y is
g(v)) of py( resp. qy), and observe that inf
R b
y = inf
X finf
R c
y = inf
X
g. From Lemma 4.3, (4.2), and (2.4) we then have (f4g)(x) sup
y 2Y
(by5cy)(hx yi)_inf
X
f_inf
X g
sup
y 2Y
(by5cy)(hx yi)_inf
X
(f4g): (4.5) Now observe that for anyy 2Y
(by5cy)(hx yi)(f4g)(x): (4.6) In fact let st2R s+ t=hx yiuv2X u+ v= x. If hu yitthen
f(u)_g(v) inf
huy i
t
f(u) inf
huy i
t
f(u)^ inf
hv y i s g(v) If hu yi<tthenhv yisand
f(u)_g(v) inf
hv y i s
g(v) inf
huy i
t
f(u)^ inf
hv y i s g(v)
hence (4.6) holds. Now (2.3), (4.5) and (4.6) entail (4.3). Let us prove (4.4).
Lemma 4.4, (4.1) and (4.2) give the inequality in (4.4). Conversely, let
y 2Y, uv2X u+ v= x thus inf
s+t=hxy i b
y(s)_cy(s) by(hu yi)_cy(hv yi)
f(u)_g(v) so that the inequality in (4.4) is satised.
Remark 4.6. A careful reading of the proof of Theorem 4.5 shows that with the same hypothesis one has also
(f4g)(x) = sup
y 2Y
sup
s+t=hxy i
inf
huy i>s
f(u)^ inf
hv y it
g(v): (4.7) Formula (4.7) is particularly interesting whenf is the valley functionA of a closed convex setAX(A(x) =;1ifx2A A(x) = +1if x2XnA).
In such a case inf
huy i>s
A(u) = +1 if iA(y)s ;1 if iA(y)>s:
Denoting by 0+Athe recession cone ofAandb(A) :=fiA<+1gthe barrier cone of A we then obtain from (4.7) a duality formula for the minimization of a quasiconvex function over a convex set.
Corollary 4.7. LetAX be a closed locally compact convex set,g:X!
Ra `.s.c. quasiconvex function assume that 0+A\Cg =f0g then min
x2A
g(x) = sup
y 2b(A)
inf
hv y ii
A (y )
g(v):
More generally, Theorem 4.5 applies to the optimization problem minimizeg(x)_h(x) for x2X
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where gh :X ! R are two extended real-valued `.s.c. quasiconvex func- tions. Setting f(x) :=h(;x) and taking x= 0 in Theorem 4.5 and (4.7) we get:Corollary 4.8. Letg, h:X!R be two extended real-valued `.s.c. quasi- convex functions such that Cg\Ch=f0g. Assume that g or h is inf-locally compact then
min
x2X
g(x)_h(x) = sup
y 2Y
t2R
inf
huy it
h(u)^ inf
hv g it
g(v) (4.8)
= sup
y 2Y
t2R
inf
huy i<t
h(u)^ inf
hv g it
g(v) (4.9)
= sup
y 2Y
inf
t2R
inf
hv y it
h(u)_ inf
hv y it
g(v): (4.10) Remark 4.9. It should be emphasized that formulas quite similar to (4.8) and (4.9) has been established under totally dierent assumptions in 35]
where it is shown that they entail the usual convex duality formulas.
We conclude this section by describing the conjugates of an important class of quasiconvex functions, namely the radial quasiconvex functions. Let us consider a reexive Banach spaceXequipped with a normjjjjand denote by jjjj the dual norm: jjyjj= max
jjxjj=1
hxyi, for anyy in the topological dual
Y of X.
With any nondecreasing extended real-valued function a : 0+1! R is associated the quasiconvex function f = ajj jj. Setting ~a(r) = a(0) for
r2];10 and ~a(r) =a(r) forr20+1 we get a nondecreasing function
~
a : R ! R such that ~a(0) = inf
R
~
a. We then have ~ae(s) = ;1 if s 2 ];1a(0)]\R ~ae(s)0 ifs2]a(0)+1. Adopting the conventions
(;1)0 =;1 (+1)0 = 0 (4.11) the quasiconjugate off =ajjjjcan then be described as follows:
Proposition 4.10. Let aand ~abe as above then (ajjjj)c(ys) = ~ae(s)jjyjj for any y2Y s2R.
Proof. By (3.1) and (4.1) we get
f
c(ys) = sup
a(jjxjj)<s
hxyi sup
jjxjj ~a e
(s) hxyi
so that, taking (4.11) into account,
f
c(ys)~ae(s)jjyjj with equality for y= 0:
Conversely assume that y 6= 0 and let t 2 Rbe such thatt < a~e(s)jjyjj by (4.1): ~a(t=jjyjj) <s. If t 0 then s>a(0) and fc(ys) 0 if t>0
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let us choose x 2 X such that jjxjj = 1 and hxyi = jjyjj we then have
f
c(ys)thx=hxyiyi=t so that in any casefc(ys)~ae(s)jjyjj. What Proposition 4.10 says is that the conjugate of a radial quasicon- vex function can be written as the product of a nondecreasing function by the dual norm. Let us see how to compute the second conjugate of such functions.
Proposition 4.11. Let ':Y R!R be dened by
'(yr) =b(r)jjyjj with b:R!R nondecreasing then,
'
=bejjjj:
Proof. By (3.4) one has
'
(x) = sup
y 2Y
sup(r2R:b(r)jjyjj<hxyi): It follows from (4.11) that fory = 0 one has:
sup(r2R:b(r)jjyjj<hxyi) = sup(r2R:b(r) =;1) = inf
R b
e thus,
'
(x) = sup
y 6=0
sup(r2R:b(r)<hxyi=jjyjj)_inf
R b
e
= sup
y 6=0 b
e(hxyi=jjyjj):
As be is `.s.c. nondecreasing we then have '(x) = be(sup
y 6=0
hxyi=jjyjj
) =
b
e(jjxjj).
Corollary 4.12. Letf =ajjjjwitha: 0+1!R nondecreasing. Then
f
c = ajjjj where a is the `.s.c. hull of a.
Proof. By Proposition 4.10, fc(ys) = ~ae(s)jjyjj setting b = ~ae be is nothing but the `.s.c. hull ~a of ~a. By Proposition 4.11 we then have fc =
~
ajjjj= ajjjj.
Corollary 4.13. Let ' : Y R! R, '(ys) = b(s)jjyjj with b : R! 0+1] nondecreasing thus
'
c(ys) = b(s)jjyjj for any y2Rn, s2R.
Proof. By Proposition 4.11,' =bejjjj now let a(r) :=be(r) forr0 as b is nonnegative one has ~a=be and the result follows from Proposition 4.10 together with the relation bee= b.
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