ESAIM: Control, Optimisation and Calculus of Variations
URL: http://www.emath.fr/cocv/
September 1998, Vol. 3, 329{343
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 quasiconvex functions. Following Barron and al., we show that the level
sum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuous quasiconvex
function which is not necessarily Lipschitz or bounded.
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 functions with the inmal convolution of their conjugates and vice versa (15],
27]). The geometrical interpretation of the inmal convolution of two extended real-valued functions is well known: it lies in the fact that the vectorial (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 epigraphical 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 (Du u) = 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,
1.
University of Avignon, Department of Mathematics, 33, rue Louis Pasteur, 84000 Avignon, France. E-mail: [email protected].
Received by the journal September 2, 1997. Revised May 7, 1998. Accepted for publication May 15, 1998.
c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
330
M. VOLLE
slightly dierent from those introduced by Crouzeix in 11] 12] and Barron
and Liu in 8], that transforms the level sum of arbitrary extended realvalued 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 semicontinuous (`.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 solution of the Hamilton-Jacobi equation ut + H (Du u) = 0, u(0 ) = g when
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].
Definitions, notations
In the sequel X is a Hausdor locally convex topological vector space
with topological dual Y and we denote by h i the bilinear mapping hx y i :=
y(x)(x 2 X y 2 Y ). Given an extended real-valued function f : X ! R
:= R f;1g f+1g we set
and t 2 R
ff < tg = fx 2 X : f (x) < tg
for the strict lower level set (at level t) of f . Observe that ff < tg is
nonempty i t > inf
f . The function f is said to be quasiconvex if all its
X
strict lower sets are convex. Of course this amounts to saying that all its
lower sets ff tg(t 2 R) are convex, or that
f (tu + (1 ; t)v) max(f (u) f (v)) := f (u) _ f (v)
for all u v 2 X , t 2 0 1] (4], 13], ...).
By adding strict lower sets of two extended real-valued functions f g :
X ! R ,
ff < tg + fg < tg := fu + v : u 2 ff < tg v 2 fg < tgg
one obtains the strict lower level set of the function
x 2 X 7;! (f 4g)(x) := inf(f (u) _ g(v) : u + v = x):
(2.1)
2.
ESAIM: Cocv, September 1998, Vol. 3, 329{343
LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS
More precisely, one has
f
f g<t = f <t + g<t
4
g
f
g
f
g
331
(2.2)
. Here we adopt the convention
for all t 2 R
A+ = +A = (2.3)
for any subset A X accordingly to the relation
inf
(f 4g ) = inf
f _ inf
g:
(2.4)
X
X
X
The function f 4g 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 subset A X is dened by
y 2 Y 7;! iA (y) := sup hx yi x2A
with the convention sup = ;1. In other words the support function of A is
the Legendre-Fenchel conjugate of the indicator function iA of A(iA (x) = 0
if x 2 A, iA (x) = +1 if x 2 X nA) iA is positively homogeneous (p.h.)
(iA (y ) = iA(y ) whenever > 0 y 2 Y ) and convex this means that iA
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 ) R f+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 y 2 Y ),
iA+B (y) = iA(y)+.iB (y)
(2.5)
where (+1)+.(;1) = (;1)+.(+1) = ;1.
_ ;1) = (;1)+(+
_ 1) = +1. The operations
Of course we set (+1)+(
_
+. and + enjoy calculus rules (21]): in particular we shall use the fact that,
, the following equivalence holds:
for all extended-real numbers r s t 2 R
s+_ t < r () 9s1 t1 2 R : r = s1+_ t1 s < s1 t < t1:
(2.6)
A Galois correspondence for quasiconvex functions
let us associate the
To each extended real-valued function f : X ! R
c
conjugate function f dened by
f c(y r) := sup hx yi = sup h yi
(3.1)
3.
f(x)<r
ff<rg
for all (y r) 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
(f 4g )c = f c +.g c
Proof. An easy consequence of the denition (3.1) and of relations (2.2),
(2.5).
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332
M. VOLLE
Among many other interesting properties of the correspondence f 7;! f c
we retain the following:
be a general extended real-valued funcProposition 3.2. Let f : X ! R
tion the conjugate function f c satises the following properties:
a) for any r 2 R, f c ( r) is `.s.c., sublinear, proper or identically ;1
b) for any y 2 Y , f c (y ) is `.s.c., nondecreasing.
Moreover,
(inf
f )c = sup fic
(3.2)
i2I i
i2I
for any family (fi )i2I of extended real-valued functions dened on X .
Proof. Let r 2 R if ff < rg = then f ( r) is identically ;1 if ff <
rg 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 2 Y , the function f c(y ) is nondecreasing, while the lower semicontinuity
of this function follows from the easy but important property, valid for all
y 2 Y , r, t 2 R:
f c (y r) t () inf f (x) r:
(3.3)
hxyi>t
On the other hand, (3.2) is a trivial consequence of the relation finf
f <
i2I i
rg = ffi < rg.
i2I
What formula (3.2) says is that the mapping f 7;! f c is a branch of a Galois
correspondence (9]). Let us now describe the other branch.
and ' : Y R ! R
be two extended
Proposition 3.3. Let f : X ! R
real-valued functions. The following assertions are equivalent:
a) f c (y r) '(y r) for all (y r) 2 Y R
b) f (x) sup supfr 2 R : hx y i > '(y r)g for all x 2 X .
y 2Y
Proof. By (3.3), a) amounts to saying that for any x 2 X , y 2 Y , r 2 R one
has
hx y i > '(y r ) =) f (x) r or, equivalently, for any x 2 X y 2 Y ,
f (x) supfr 2 R : hx yi > '(y r)g
that is exactly b).
Thus let us dene the conjugate of an arbitrary extended real-valued func by setting
tion ' : Y R ! R
' (x) := sup supfr 2 R : hx yi > '(y r)g
(3.4)
y 2Y
for all x 2 X . We then obtain the other branch of the Galois correspondence
above mentioned and the related properties:
fc '
f ' (inf
' ) = sup 'i i2I i
i2I
()
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LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS
333
, ' : Y R ! R
,
for any extended real-valued functions f : X ! R
'i : Y R ! R (i 2 I ). Moreover the composite operators
f 2 R X 7;! f c := (f c) 2 R X
' 2 R Y R 7;! 'c := (' )c 2 R Y R
are isotone:
f g =) f c gc ' =) 'c c
contracting:
and idempotent:
f c f 'c '
(f c )c = f c ('c )c = 'c :
More precisely one always has
f cc = f c 'c = ' :
X that coincide with their biconjugate f c (that
The class of functions f 2 R
Y Rg = f' : ' =
we shall call regular in the sequel) is given by f' : ' 2 R
c
' g. Of course a similar property holds for the class of regular functions '
on Y R.
X (resp. ' 2 R
Y R) is nothing but the greatest
The biconjugate of f 2 R
regular minorant of f (resp. '):
f c = supfh 2 R X : hc = h and h f g
Y R : c = and 'g:
(resp: 'c = supf
2 R
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 .
be an extended real-valued function. Then
Theorem 3.4. Let f : X ! R
c
f = f i f is `.s.c. quasiconvex.
Y R t 2 R,
Proof. It follows from (3.4) that for any ' 2 R
f' tg =
fh y i '(y r )g
\\
y2Y r>t
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 R be such that
r < f (x) thus x does not belong to the closed convex set ff rg by the
Hahn-Banach theorem there exists y 2 Y such that hx y i > sup hu y i f(u)r
f c(y r) : By (3.4) we then have f c (x) r so that, by taking the supremum
over r < f (x) f c (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.
be an extended real-valued function.
Theorem 3.5. Let ' : Y R ! R
c
Then, ' = ' i the following conditions are satised:
a) for any r 2 R, '( r) is `.s.c., sublinear, proper or identically ;1
b) for any y 2 Y , '(y ) is `.s.c. nondecreasing.
ESAIM: Cocv, September 1998, Vol. 3, 329{343
334
M. VOLLE
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 hx i Let us set f (x) := inf ft 2 R : x 2 At g and prove that ' = f c .
x2Ar
First observe that ff < rg = At . On the other hand, as '(y ) is `.s.c.
t<r
nondecreasing,
sup sup hx y i = sup hx y i
so that
that is
t<r x2At
x2Ar
sup hx y i = sup(hx y i : x 2
f(x)<r
to say f c = '.
t<r
At) = sup x y x2Ar
h
i
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
, (f c +.g c ) is just the `.s.c.
extended real-valued functions f , g : X ! R
quasiconvex hull of the level sum f 4g . In the case when f and g are
quasiconvex, f 4g is quasiconvex too so that (f c +.g c) is then the `.s.c.
hull of f 4g . 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 function f : X ! R
a vector d 2 X is said to be a direction of majoration of f is there exists
(x0 r0) 2 X R such that
f (x0 + td) r0 for all t 0:
The set of directions of majoration of f is a convex cone we denote by Cf .
The function f is called inf-locally compact if ff rg is locally compact for
any r 2 R. We then have the following result:
be two extended real-valued `.s.c. quasiLemma 4.1. (1]) Let f g : X ! R
convex functions such that Cf \ ;Cg = f0g. Assume moreover that f or g
is inf-locally compact. Then f 4g is `.s.c., quasiconvex and exact.
It follows from the previous considerations that:
Theorem 4.2. For f and g as in Lemma 4.1 one has
f 4g = (f c+.gc) :
4.
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 on Y R. For
doing this, the use of quasi-inverses is essential. Let us recall (see for instance
are quasi-inverses if for
24]) that two nondecreasing functions a b : R ! R
all real numbers r, s one has
a(r) < s =) r b(s)
a(r) > s =) r b(s):
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LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS
335
Among all the quasi-inverses of a there is a smallest one namely
ae(s) = supfr 2 R : a(r) < sg = inf fr 2 R : a(r) sg
which is also the only `.s.c. quasi-inverse of a. It is characterized by the
relations
a(r) < s =) r ae(s) r < ae(s) =) a(r) < s
(4.1)
for any real numbers r s.
and y 2 Y , typical quasi-inverses are the folGiven a function f : X ! R
lowing (13], 24]): py and by , py and y , y and by , y and y where py (r) =
f c(y r) y (r) = sup hx yi y(s) = hxyinfi>s f (x), by (s) = hxyinfis f (x)
f(x)r
(r s 2 R).
Now observe that, from Theorem 4.2 and (3.4), (4.1), one has
(f 4g )(x) = sup(py +.qy )e (hx y i)
(4.2)
y2Y
c
py (r) = f (y r) and qy (r) = gc(y r).
for all x 2 X , with
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 lemmas a1 and a2 are such functions and we denote by bi a quasi-inverse
of ai (i = 1 2). We also set
(b1 5 b2 )(s) = sup b1(s1 ) ^ b2(s2 ) (s 2 R):
s1 +s2 =s
e
Lemma 4.3. (a1 +. a2 ) (b1 5 b2) _ inf b1 _ inf b2.
R
R
Proof. Let s 2 R observe that
(a1+.a2 )e (s) = supfr 2 R : a1 (r)+_ a2 (r) < sg _ r1 _ r2
where ri = supfr 2 R : ai (r) = ;1g(i = 1 2). In fact, one easily gets ri =
inf
b (i = 1 2). Now by (2.6): a1(r)+_ a2(r) < s () 9s1 s2 2 R : s1 + s2 = s,
R i
ai(r) < si thus r < bi(si)(i = 1 2) therefore, supfr 2 R : a1(r)+_ b1(r) <
sg (b1 5 b2)(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 exist s1 s2 2 R such that s1 + s2 = s
and bi (si ) < r(i = 1 2). We then have si ai (r) so that s a1 (r)+.a2 (r)
and r inf fr 2 R : a1 (r)+.a2 (r) sg = (a1 +.a2 )e (s).
We are now ready to state the main result of this section:
be two extended real-valued `.s.c. quasiTheorem 4.5. Let f , g : X ! R
convex functions such that Cf \ ;Cg = f0g. Assume that f or g is inf-locally
compact. Then for all x 2 X :
(f 4g )(x) = sup sup inf f (u) ^ inf g (v )
(4.3)
y2Y s+t=hxyi huyis
hvyit
= sup inf
inf f (u) inf g (v ):
hvyit
y2Y s+t=hxyi huyis
_
(4.4)
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336
M. VOLLE
Proof. For each y 2 Y let us consider the quasi-inverse by (s) = inf f (u)
huyis
(resp. cy (s) = inf g (v )) of py ( resp. qy ), and observe that inf
b =
R y
hvyis
inf
f inf cy = inf
g. From Lemma 4.3, (4.2), and (2.4) we then have
X
X
R
(f 4g )(x)
g
sup(by 5 cy )(hx y i) _ inf
f _ inf
X
X
sup (by 5 cy )(hx
y i) _ inf
(f 4g ):
X
y 2Y
y 2Y
(4.5)
Now observe that for any y 2 Y
(by 5 cy )(hx y i) (f 4g )(x):
(4.6)
In fact let s t 2 R s + t = hx y i u v 2 X u + v = x. If hu y i t then
f (u) _ g(v) inf f (u) inf f (u) ^ inf g(v)
huyit
s and
huyit
hvyis
If hu y i < t then hv y i f (u) _ g(v) hvyinfis g(v) huyinfit f (u) ^ hvyinfis 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 2 Y , u v 2 X u + v = x thus
inf by (s) _ cy (s) by (hu y i) _ cy (hv y i)
s+t=hxyi
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
(f 4g )(x) = sup sup inf f (u) ^ inf g (v ):
(4.7)
y2Y s+t=hxyi huyi>s
hvyit
Formula (4.7) is particularly interesting when f is the valley function A of a
closed convex set A X (A(x) = ;1 if x 2 A A (x) = +1 if x 2 X nA).
In such a case
inf A (u) = +1 if iA (y ) s ;1 if iA (y ) > s:
huyi>s
Denoting by 0+ A the recession cone of A and b(A) := fiA < +1g the 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. Let A X be a closed locally compact convex set, g : X !
R a `.s.c. quasiconvex function assume that 0+A \ Cg = f0g then
min
g(x) = sup inf g(v):
x2A
y2b(A) hvyiiA (y)
More generally, Theorem 4.5 applies to the optimization problem
minimize g (x) _ h(x) for x 2 X
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337
are two extended real-valued `.s.c. quasiconvex funcwhere g h : X ! R
tions. Setting f (x) := h(;x) and taking x = 0 in Theorem 4.5 and (4.7) we
get:
be two extended real-valued `.s.c. quasiCorollary 4.8. Let g , h : X ! R
convex functions such that Cg \ Ch = f0g. Assume that g or h is inf-locally
compact then
min
g(x) _ h(x) = sup inf h(u) ^ inf g(v)
x2X
(4.8)
= sup inf h(u) ^ inf g (v )
(4.9)
hvgit
y2Y huyit
t2R
y2Y huyi<t
hvgit
t2R
= sup tinf
2R inf h(u) _ inf g(v):
y2Y
hvyit
hvyit
(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 space X equipped with a norm jj jj and denote
by jj jj the dual norm: jjy jj = max hx y i, for any y in the topological dual
jjxjj=1
Y of X .
is
With any nondecreasing extended real-valued function a : 0 +1! R
associated the quasiconvex function f = a jj jj. Setting ~a(r) = a(0) for
r 2] ; 1 0 and a~(r) = a(r) for r 2 0 +1 we get a nondecreasing function
a~ : R ! R such that a~(0) = inf
a~. We then have a~e (s) = ;1 if s 2
R
] ; 1 a(0)] \ R a~e (s) 0 if s 2]a(0) +1. Adopting the conventions
(;1) 0 = ;1 (+1) 0 = 0
(4.11)
the quasiconjugate of f = a jj jj can then be described as follows:
Proposition 4.10. Let a and a~ be as above then
(a jj jj)c(y s) = a~e (s)jjy jj
for any y 2 Y s 2 R.
Proof. By (3.1) and (4.1) we get
f c(y s) = sup hx yi sup hx yi
a(jjxjj)<s
jjxjja~e(s)
so that, taking (4.11) into account,
f c(y s) a~e (s)jjyjj
with equality for y = 0:
Conversely assume that y 6= 0 and let t 2 R be such that t < a~e (s)jjy jj
by (4.1): a~(t=jjy jj) < s. If t 0 then s > a(0) and f c (y s) 0 if t > 0
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338
M. VOLLE
let us choose x 2 X such that jjxjj = 1 and hx y i = jjy jj we then have
f c(y s) thx=hx yi yi = t so that in any case f c (y s) a~e (s)jjyjj.
What Proposition 4.10 says is that the conjugate of a radial quasiconvex 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.
be dened by
Proposition 4.11. Let ' : Y R ! R
'(y r) = b(r) y jj jj
nondecreasing then,
with b : R ! R
' = be
:
jj jj
Proof. By (3.4) one has
' (x) = sup sup(r R : b(r) y < x y ):
y2Y
It follows from (4.11) that for y = 0 one has:
be sup(r R : b(r) y < x y ) = sup(r R : b(r) = ) = inf
R
2
2
jj jj
h
jj jj
i
h
2
i
;1
thus,
' (x) = sup sup(r R : b(r) < x y = y ) inf
be
R
y=
60
= sup be ( x y = y ):
y=
60
As be is `.s.c. nondecreasing we then have ' (x) = be (sup x y = y ) =
y=
60
be( x ).
2
h
h
i jj jj
_
i jj jj
h
i jj jj
jj jj
nondecreasing. Then
Corollary 4.12. Let f = a jj jj with a : 0 +1! R
f c = a
jj jj
where a is the `.s.c. hull of a.
Proof. By Proposition 4.10, f c (y s) = a~e (s)jjy jj setting b = ~ae be is
nothing but the `.s.c. hull a~ of a~. By Proposition 4.11 we then have f c =
a~ jj jj = a jj jj.
Corollary 4.13. Let ' : Y
0 +1] nondecreasing thus
R
!
R , '(y s) = b(s) y with b : R
jj jj
!
'c (y s) = b(s) y jj jj
for any y 2 Rn, s 2 R.
Proof. By Proposition 4.11, ' = be jj jj now let a(r) := be (r) for r 0 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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339
Application to Hamilton-Jacobi equations
In order to solve the Bellman equation in L1 spaces, an explicit formula
has been recently derived by Barron and al. (7]) for the corresponding
Hamilton-Jacobi equation
ut = ;H (Du u) on ]0 +1 Rn with u(0 x) = g(x):
(5.1)
The hamiltonian H (y r) is here nite-valued, continuous, nondecreasing in
r, sublinear in y, and the function g Lipschitz continuous and bounded.
Thus, the unique viscosity solution (6], 10], ...) is given by
5.
x z
u(t x) = zinf
2Rn(h( t ) g(z))
;
_
where h is the quasiconvex function dened by
h(x) = inf fr 2 R : H (y r) hx yi 8y 2 Rng:
As H (y ) is nondecreasing, observe that the function h above is nothing
but the -conjugate of H , namely:
h(x) = H (x) = supn supfr 2 R : H (y r) < hx yig:
y2R
On the other hand, setting ht] (x) = h( xt ) one can write
u(t x) = (ht]4g)(x):
(5.2)
We are interested in the case when the initial condition g is quasiconvex but
not necessarily Lipschitz continuous or bounded (see 36]). Let us give some
general properties inspired by the convex case (see e.g. 14] 26]).
be an extended real-valued quasiconvex funcLemma 5.1. Let h : Rn ! R
tion then
hr+s] = hr]4hs]
(5.3)
for all positive real numbers r s.
Moreover the function v dened on ]0 +1Rn by v (t x) = ht](x) is jointly
quasiconvex.
x = r ( x1 )+ s ( x2 )
Proof. Let x x1 x2 2 X with x1 + x2 = x one has r+s
r+s r
r+s s
so that hr+s] (x) hr] (x1 ) _ hs] (x2 ). This proves the inequality in (5.3).
Conversely,
(hr]4hs] )(x) hr] ( r +r s x) _ hs] ( r +s s x) = hr+s] (x):
We prove now that v is jointly quasiconvex. Let x0 x1 2 X , t0 t1 2]0 +1
and 2 0 1]. One has
t1
x1 + (1 ; )t0 x0 1 + (1 ; )x0 =
x := x
t1 + (1 ; )t0 t1 + (1 ; )t0 t1 t1 + (1 ; )t0 t0
therefore, h(s) h( xt 1 ) _ ( xt 0 ), that is v ((t1 x1) + (1 ; )(t0 x0))
1
0
v(t1 x1) _ v(t0 x0).
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340
M. VOLLE
Proposition 5.2. Let g , h be two extended real-valued quasiconvex func-
tions on Rn then the function u dened in (5.2) is jointly quasiconvex on
]0 +1Rn. If, moreover, g and h are `.s.c. and
Cg \ ;Ch = f0g
(5.4)
then u is `.s.c. and exact.
Proof. One has u(t x) = infn(v (t z ) _ g (x ; z )) now the function (t x z )
z2R
n
7;! v (t z ) _ g (x ; z ) is jointly quasiconvex. Taking the inmum on z 2 R
we obtain that u is jointly quasiconvex (see e.g. 13]). Assume that (5.4)
holds. Since Cht] = Ch for any t > 0 one has Cg \ ;Cht] = f0g and u is
exact by Lemma 4.1. Let us prove that u is jointly `.s.c.. Let (tp xp)p2N
be a sequence with values in fu rg (r xed in R) converging to (t x). As
u is exact, there exist two sequences (yp)p2N and (zp)p2N such that xp =
tpyp + zp tp > 0 yp 2 fh rg, zp 2 fg rg. Assume (yp)p2N is
unbounded: there exist "k k!1
!
0+ and a subsequence (ypk )k2N such that
k
!1
"k ypk ! y 6= 0 it follows that y 2 Ch and "k zpk k!1
! ;ty 6= 0 therefore,
y 2 ;Cg that is impossible. So, (yp)p2N is bounded and, taking converging
!
x = ty + z with
subsequences if necessary, we have xp = tp yp + zp p!1
y 2 fh rg and z 2 fg rg. Thus u(t x) h(y) _ h(z) r and the proof
is complete.
The description of the normal cone to a lower level set of u is of crucial
importance in the sequel. Recall that the normal cone of a convex subset
A Rn at a point a 2 A is the closed convex cone N (A a) := fy 2 Rn :
suphx y i ha y ig. The following lemma points out that the normal cone of
x2A
a lower level set of a quasiconvex function includes the Frechet subdierential
of the function.
be an extended real-valued quasiconvex funcLemma 5.3. Let f : Rn ! R
;
1
tion, a 2 f (R), and y 2 Rn such that
lim inf f (a + h) ; f (a) ; hh y i 0
h
jjhjj!0
thus y N ( f f (a) a).
Proof. Let f (a + h) f (a) and " > 0 for t > 0 small enough one has
f (a + th) f (a) t( " + h y ) as f is quasiconvex it follows that
0 f (a) f (a + h) f (a) t( " + h y )
therefore, h y " so that, as " > 0 is arbitrary, y N ( f f (a) a).
Now let us describe the normal cone of a lower level set of u.
Lemma 5.4. Let g and h be two extended real-valued quasiconvex functions
on Rn such that (5.4) holds, let (t x) u;1 (R) r = u(t x), and let x1 x2
in Rn be such that x = x1 + x2, r = h( x1 ) g (x2) thus N ( u r (t x))
t
jj
2
f
jj
g
;
h
;
h
i
_
;
;
h
2
i i
f
g
2
_
f
g
is given by:
x1 yi y) : y 2 N (fh rg x1 ) \ N (fg rg x )g:
f(;h
2
t
t
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LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS
341
Proof. Let (s y ) 2 R Rn as the quasiconvex function u is exact one has
(s y ) 2 N (fu rg (t x)) i
st + hx1 yi + hx2 yi st + hx1 yi + hx2 yi
for all t > 0 x1 2 tfh rg x2 2 fg rg.
Taking t = t and x2 = x2 we get y 2 N (fh rg xt1 ).
Taking t = t and x1 = x1 we get y 2 N (fg rg x2 ).
Taking x1 = tx1 and x2 = x2 we get s = ;h x1 y i.
t
t
We then have the inclusion in the announced formula the other inclusion
being obvious, the proof is complete.
We are now in a position to state the main result of this section saying that
if the initial condition g is continuous and quasiconvex then the function u
in (5.2) is still a viscosity solution of (5.1). More precisely:
Theorem 5.5. Let H be as in the beginning of this section, h = H , and g :
Rn ! R a continuous quasiconvex function such that h(0) inf
g. Thus, the
Rn
continuous quasiconvex function u(t x) = (ht] 4g )(x) is a viscosity solution
of (5.1).
Proof. By Theorem 3.5 one has H = hc as H is nite valued the sets
fh < r g are bounded so that Ch = f0g. It then follows from Proposition
5.2 that u is `.s.c. quasiconvex and exact as g is real-valued, it ensues that
(h(y ) _ g (x ; ty )) so that u is
u is real-valued too. Moreover u(t x) = yinf
2
Rn
upper semicontinuous (for each y 2 Rn, the function (t x) 2]0 +1Rn 7;!
h(y) _ g(x ; ty) is continuous). Hence u is continuous with real values. Let
us check that u is a viscosity subsolution ot (5.1). On this point we follow
the proof of 7], Theorem 3.1: let ' :]0 +1 Rn ! R be a dierentiable
function such that u ; ' has a local maximum at (t0 x0). We have to prove
that 't(t0 x0) ;H (Dx'(t0 x0) u(t0 x0)). By Theorem 3.5 this amounts
to saying that, given x such that h(x) < u(t0 x0), we have
't(t0 x0) ;hx Dx'(t0 x0)i:
(5.5)
Now, for > 0 small enough one has
'(t0 ; x0 ; x) ; '(t0 x0) u(t0 ; x0 ; x) ; u(t0 x0) :
Taking Lemma 5.1 into account we get u(t0 x0) u(t0 x0 x) h(x)
so that '(t0 x0 x) '(t0 x0) 0: Finally, sending 0+ one
;
;
;
;
;
;
;
;
_
!
obtains (5.5).
We prove now that u is a supersolution of (5.1). Let (t x) be a local minimum of u ; ' for jj(t ; t x ; x)jj small enough we then have
u(t x) ; u(t x) '(t x) ; '(t x)
= (t ; t)'t(t x) + hx ; x Dx'(t x)i + "(t x)jj(t ; t x ; x)jj
with lim "(t x) = 0:
(tx)!(tx)
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342
M. VOLLE
Thus,
x) ; hx ; x Dx'(t x)i
0:
lim inf u(t x) ; u(t x) ;jj((tt ;; tt)'tx(t;
x )jj
(tx)!(tx)
0
0
Now, u being jointly quasiconvex, Lemma 5.3 says that
('t(t x) Dx'(t x)) 2 N (fu u(t x)g (t x)):
As u is exact (Proposition 5.2) it follows from Lemma 5.4 that
't(t x) = ;h xt1 Dx'(t x)i
with h( x1 ) u(t x). On the other hand the continuity of H entails
t
H (y r) = hc (y r) = sup hx yi = sup hx yi for all y 2 Rn r 2 R, so
h(x)<r
h(x)r
that
't(t x) ;H (Dx'(t x) u(t x)):
To conclude the proof let us verify the initial condition in (5.1). Let x 2 X
g and h is quasiconvex it follows
and f t (x) := ht] (x) _ g (x; x). As h(0) inf
Rn
that (f t )t>0 is a nonincreasing family. Since h is coercive, t!
lim0 ht] (x) =
+
+1 if x 6= 0 h(0) if x = 0, so that (f t )0<t1 is a nonincreasing family of
equicoercive `.s.c. functions with t!
lim0 f t (x) = +1 if x 6= 0, g (x) if x = 0.
+
t = inf lim f t = g (
f
x):
Therefore, u(0 x) = tlim
inf
!0 Rn
Rn t!0
+
+
Remark 5.6. The condition h(0) infn g in Theorem 5.5 is not very strinR
gent it is in particular satised when the lagrangian H is nonnegative: in
such a case one has h(0) = ;1.
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