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). ESAIM: Cocv, September 1998, Vol. 3, 329{343 332 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 () ESAIM: Cocv, September 1998, Vol. 3, 329{343 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): ESAIM: Cocv, September 1998, Vol. 3, 329{343 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) ESAIM: Cocv, September 1998, Vol. 3, 329{343 336 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 ESAIM: Cocv, September 1998, Vol. 3, 329{343 LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS 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 ESAIM: Cocv, September 1998, Vol. 3, 329{343 338 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. ESAIM: Cocv, September 1998, Vol. 3, 329{343 LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS 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). ESAIM: Cocv, September 1998, Vol. 3, 329{343 340 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 ESAIM: Cocv, September 1998, Vol. 3, 329{343 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) ESAIM: Cocv, September 1998, Vol. 3, 329{343 342 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. References 1] C. Amara: Directions de majoration d'une fonction quasiconvexe et applications, Serdica Math. J., 23, 1997, 1001{1018. 2] H. Attouch, R.Wets: Epigraphical analysis, in Analyse non lineaire, H. Attouch, J.-P. Aubin, F. Clarke, I. Ekeland eds., Gauthier-Villars, Paris, 1989, 73{100. 3] H. Attouch, M. Volle: Cutting and scanning methods in set-valued analysis, SetValued Analysis, 4, 1996, 135{155. 4] M. Avriel, W. Diewert, S. Schaible, I. Zang: Generalized concavity, Plenum Press, New York, 1987. 5] F. Baccelli, G. Cohen, G.J. Olsder, J.-P. Quadrat: Synchronization and linearity: an algebra for discrete event systems, John Wiley and Sons, New York, 1992. 6] G. Barles: Solutions de viscosite des equations de Hamilton-Jacobi, Math. et Appl. 17, Springer-Verlag, 1994. 7] E.N. Barron, R. Jensen, W. Liu: Hopf-Lax-Type Formula for ut + H (u Du ) = 0, J. Di. Eq., 126, 1996, 48{61. 8] E.N. Barron, W. Liu: Calculus of variations in L1 , Appl. Math. Optim., 35, 1997, 237{263. 9] G. Birkho: Lattice Theory, Providence, A.M.S., 1966. 10] M.G. Crandall, P.-L. Lions: Viscosity solutions of Hamilton Jacobi equations, Trans. Amer. Math. Soc., 277, 1983, 1{42. 11] J.-P. Crouzeix: Polaires quasiconvexes et dualite, C. R. Acad. Sci. Paris, 297, Serie A, 1974, 955{958. 12] J.-P. Crouzeix: Deux notions de polarite pour les fonctions quasiconvexes, C. R. Acad. Sci. Paris, 282, Serie A, 1976, 107{109. ESAIM: Cocv, September 1998, Vol. 3, 329{343 LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS 343 13] J.-P. Crouzeix: Contributions a l'etude des fonctions quasiconvexes, Thesis, University of Clermont-Ferrand II, France, 1977. 14] L.C. Evans: Some min-max methods for the Hamilton-Jacobi equation, Ind. Math. J., 33{1, 1984, 31{50. 15] W. Fenchel: Convex cones, sets and functions, University of Princeton, 1953. 16] M. Gondran: Analyse minimax, C. R. Acad. Sci. Paris, 323, Serie I, 1996, 1249{1252. 17] L. Hormander: Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Ark. Mat., 3, 1954, 181{186. 18] D.T. Luc, M. Volle: Level set under inmal convolution and level addition, J.O.T.A., 94{3, 1997, 695{714. 19] J.E. Martinez-Legaz: Quasiconvex duality theory by generalized conjugations methods, Optimization, 19, 1988, 603{652. 20] J.E. Martinez-Legaz, I. Singer: _-Dualities and ?-dualities, Optimization, 22{4, 1991, 483{511. 21] J.-J. Moreau: Inf-convolution, sous-additivite, convexite des fonctions numeriques, J. Math. Pures Appl., 49, 1970, 109{154. 22] J.-P. Penot: Starshaped functions and conjugacy, Preprint, University of Pau, France, 1996. 23] J.-P. Penot, M. Volle: Another duality scheme for quasiconvex problems, International Series of Numerical Math., 84, 1988, Birkauser Verlag Basel, 259{275. 24] J.-P. Penot, M. Volle: Inversion of real-valued functions and applications, ZOR Methods and Models of Operations Research, 34, 1990, 117{141. 25] J.-P. Penot, M. Volle: On quasiconvex duality, Math. Oper. Res., 15{4, 1990, 597{ 625. 26] P. Plazanet: Contributions a l'analyse des fonctions convexes et des dierences de fonctions convexes. Application a l'optimisation et a la theorie des e.d.p., Thesis, University Paul Sabatier, Toulouse, France, 1990. 27] R.T. Rockafellar: Duality theorems for convex functions, Bull. Amer. Math. Soc., 70, 1964, 189{192. 28] R.T. Rockafellar: Convex Analysis, Princeton, 1970. 29] A.M. Rubinov, B.M. Glower: On generalized quasiconvex conjugation, Contemp. Math., 204, 1997, 199{216. 30] A. Seeger, M. Volle: On a convolution operation obtained by adding level sets: classical and new results, Operations Research, 29{2, 1995, 131{154. 31] I. Singer: The lower semicontinuous quasiconvex hull as a normalized second conjugate, Nonlinear Anal. Theory Methods Appl., 7, 1983, 1115{1121. 32] P.T. Thach: A nonconvex duality with a zero duality gap and applications, SIAM J. Optim., 4, 1994, 44{64. 33] S. Traore, M. Volle: On the level sum of two convex functions on Banach spaces, Journal of Convex Analysis, Part II of the Special Issues in Honor of R.T. Rockafellar, 3{1, 1996, 141{151. 34] M. Volle: Conjugaison par tranches, Ann. Mat. Pura ed Appl., 139, 1985, 279{311. 35] M. Volle: Quasiconvex duality for the max of two functions, Lecture Notes in Economics and Math. Systems, 452, 1997, 365{379. 36] M. Volle: Conditions initiales quasiconvexes dans les equations de Hamilton-Jacobi, C. R. Acad. Sci. Paris, 325, Serie I, 1997, 167{170. ESAIM: Cocv, September 1998, Vol. 3, 329{343

© Copyright 2023 Paperzz