Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

Transcription

1 Nonlinear Analysis 68 (2008) Dini derivaive and a characerizaion for Lipschiz and convex funcions on Riemannian manifolds O.P. Ferreira Universidade Federal de Goiás, Insiuo de Maemáica e Esaísica, Campus Samambaia, Caixa Posal 131, Goiânia, GO, , Brazil Received 14 December 2006; acceped 19 December 2006 Absrac Dini derivaives in Riemannian manifold seings are sudied in his paper. In addiion, a characerizaion for Lipschiz and convex funcions defined on Riemannian manifolds and sufficien opimaliy condiions for consrain opimizaion problems in erms of he Dini derivaive are given. c 2007 Elsevier Ld. All righs reserved. Keywords: Dini derivaive; Convex funcions; Lipschiz funcions; Riemannian manifolds 1. Inroducion In he las few years, severals imporan conceps of nonsmooh analysis have been exended from Euclidean space o a Riemannian manifold seing, in order o go furher in he sudy of opimizaion problems and relaed opics. Works dealing wih his subjec include hose by Azagra, Ferrera and López-Mesas [1], Azagra, Ferrera [2], Ferreira [6] and Ledyaev and Zhu [9 11]. I is worhwhile o menion ha exensions of conceps and echniques of opimizaion from Euclidean spaces o Riemannian manifolds have been exensively sudied in several papers including [7,13 15,18]. Lipschiz and convex funcions play an imporan role in nonsmooh analysis on linear spaces and, as is well known, he Dini derivaive is a very useful ool in he analysis of hese funcions. Our aim in his paper is o sudy some properies of Dini derivaives in a Riemannian manifolds conex and provide a characerizaion for Lipschiz and convex funcions in erms of his derivaive. In addiion, we obain sufficien opimaliy condiions for consrain opimizaion problems in ha seing. A couple of papers have deal wih he issue of characerizaion for Lipschiz and convex funcions in he conex of linear spaces. Clarke, Sern and Wolenski [4] have given a characerizaion for Lipschiz funcions in erms of boh he Dini derivaive and proximal subgradiens in he conex of Hilber space. Poliquin [16] provided a characerizaion for convex funcions in erms of proximal subgradiens on he Euclidean space R n and Clarke, Sern and Wolenski [4] using a novel approach exended his resul o Hilber space. Correa, Jofré and Thibaul [5] have given a characerizaion for convex funcions in erms of Clarke subdifferenials in he conex of Banach space. Also, Tel.: ; fax: address: orizon@ma.ufg.br X/$ - see fron maer c 2007 Elsevier Ld. All righs reserved. doi: /j.na

2 1518 O.P. Ferreira / Nonlinear Analysis 68 (2008) Luc and Swaminahan [12] have obained a characerizaion for convex funcions in erms of Dini derivaives on real opological vecor spaces. The organizaion of he paper is as follows: In Secion 1.1, we lis some basic noaions and auxiliary resuls used in his presenaion. In Secion 2 we inroduce he concep of Dini derivaives of funcions defined on Riemannian manifolds and obain some resuls for i. In Secion 3 we provide a characerizaion for Lipschiz funcions in erms of Dini derivaives in a Riemannian manifold seing. In Secion 4, we obain a characerizaion for convex funcions defined on Riemannian manifolds in erms of he Dini derivaive. In Secion 5 we give sufficien opimaliy condiions for consrain opimizaion problems in erms of he Dini derivaive. We conclude his paper by making some remarks abou exensions of our resuls Noaion and auxiliary resuls In his secion we recall some noaions, definiions and basic properies of Riemannian manifolds used hroughou he paper. They can be found in many inroducory books on Riemannian Geomery, for example in [17]. Throughou he paper, M is a smooh manifold and C 1 is he class of all coninuously differeniable funcions on M. The space of vecor fields on M is denoed by X (M), by T p M he angen space of M a p and by T M = x M T x M he angen bundle of M. Le M be endowed wih a Riemannian meric,, wih corresponding norm denoed by, so ha M is now a Riemannian manifold. Le us recall ha he meric can be used o define he lengh of a piecewise C 1 curve c : [a, b] M joining p o q, i.e., such ha c(a) = p and c(b) = q, by l(c) = b a c () d. Minimizing his lengh funcional over he se of all such curves we obain a disance d(p, q), which induces he original opology on M. Also, he meric induces a map f C 1 (M) grad f X (M), which associaes o each f is gradien via he rule grad f, X = d f (X), for all X X (M). The chain rule generalizes o his seing in he usual way: ( f c) () = grad f (c()), c (), for all curves c C 1. Le c be a curve joining poins p and q in M and le be a Levi-Civia connecion associaed o (M,, ). For each [a, b], induces an isomery, relaive o,, P(c) a : T c(a) M T c() M, he so-called parallel ranslaion along c from c(a) o c(). A vecor field V along c is said o be parallel if c V = 0. If c iself is parallel, hen we say ha c is a geodesic. The geodesic equaion γ γ = 0 is a second order nonlinear ordinary differenial equaion, so he geodesic γ is deermined by is posiion and velociy a one poin. I is easy o check ha γ is consan. We say ha γ is normalized if γ = 1. A geodesic γ : [a, b] M is said o be minimal if is lengh is equal o he disance beween is end poins, i.e. l(γ ) = d(γ (a), γ (b)). A finie dimensional Riemannian manifold is complee if is geodesics are defined for any value of. The Hopf Rinow heorem assers ha if he Riemannian manifold M is complee, hen any pair of poins in M can be joined by a (no necessarily unique) minimal geodesic. Moreover, (M, d) is a complee meric space and is closed and bounded subses are compac. In his paper, we assume ha all manifolds are complee and finie dimensional. The exponenial map exp p : T p M M is defined by exp p v = γ v (1), where γ v is he geodesic defined by is posiion p and velociy v a p. We can prove ha, γ v () = exp p v for any value of. For p M, le { i p := sup r > 0 : exp p Br (op) } is diffeomorphism, where o p denoes he origin of T p M and B r (o p ) := {v T p M : v o p < r}. Noe ha if 0 < δ < i p hen exp p B δ (o p ) = B δ (p), where B δ (p) := {q M : d(p, q) < δ}. The number i p is called he injeiviy radius of M a p. For any p M, he map d 2 (p,.) C (B i p (p)), where B i p (p) = {q M : d(p, q) < i p }, and grad d 2 (p, q) = 2 expq 1 p, for all q B i p (p). Throughou he paper, M denoes a finie dimensional Riemannian manifold which is complee and for any p M, for δ > 0 we denoe by B δ (p) he open ball and by B δ [p] he closed ball cenered an p. The se C M is said o be convex, if for any p, q C he minimal geodesics joining p o q are conained in C. Le U be an open subse of M. From now on, we denoe by F(U) he class of all funcion f : M (, + ] which are lower semiconinuous on U and dom( f ) U, where dom( f ) = {x M : f (x) < + }. If U = M we denoe F for F(U).

3 O.P. Ferreira / Nonlinear Analysis 68 (2008) Dini derivaive In his secion we sudy some properies of he Dini derivaive of locally Lipschiz funcions. Our main resul shows ha he Dini derivaive does no depend on he curve, namely, i jus depends on he direcion. Definiion 1. A funcion f F(U) is said o be Lipschiz on V, of rank L 0, if V dom( f ) and here holds f (p) f (q) Ld(p, q), p, q V, where d is he Riemannian disance on M. We denoe he se of all Lipschiz funcion on V, of rank L, by Lip L (V ). Definiion 2. A funcion f F(U) is said o be Lipschiz a p, of rank L p, if here exiss δ > 0 such ha f Lip L p (B δ (p)) and locally Lipschiz on V if i is Lipschiz a every p V. Now, we are going o presen hree imporan examples of Lipschiz funcions ha emerge in he sudy of Riemannian manifolds, see [17,8]. Example 1. Le M be a non-compac Riemannian manifold. A geodesic γ : [0, + ) M parameerized by arclengh and emanaing from p is called a ray emanaing from p if d(γ (), γ (s)) = s, for all, s > 0. For a ray γ, he Busemann funcion b γ : M (, + ], defined by b γ (q) = lim ( d (q, γ ())), + is Lipschiz of rank 1, see [6]. Example 2. Le M be a Riemannian manifold and S a closed subse of M. The disance funcion d S (p) = inf{d(p, s) : s S} is Lipschiz of rank 1, see [6]. Example 3. Le f F and λ > 0. Suppose ha f is bounded below by he consan k. Then, i easy o show ha he funcion f λ : M (, + ] defined by f λ (p) = inf { f (q) + λd 2 (p, q) }, is also bounded below by k. Moreover, following he same paern used o prove he firs par of he Theorem 5.1 on page 44 of [3], we can prove ha f λ is locally Lipschiz on is domain. We remark ha he Lipschiz properies depend on he Riemannian meric defined on M. In oher words, if he meric on M is changed hen he se of Lipschiz funcions on M becomes differen from he previous one, see Example 4.4 in [6]. Definiion 3. The lower Dini derivaive of f F a p dom( f ) in he direcion of v T p M is defined by f (p, v) = lim inf 0 + f (γ ()) f (p), where γ : R M is he geodesic such ha γ (0) = p and γ (0) = v. If p dom( f ) define f (p, v) = for all v T p M. Noe ha if f F is a locally Lipschiz funcion on dom( f ), hen dom( f ) is an open se and for all p dom( f ) and v T p M here exiss he direcional derivaive and f (p, v) < +. Now, we are going o obain some properies for lower Dini derivaives. We begin wih some preliminaries. Le p M and c 1, c 2 : ( ε, ε) M be a wo C 1 curves, such ha c 1 (0) = c 2 (0) = p, c 1 (0) = v and c 2 (0) = w. Assume ha he image ses c 1 (( ε, ε)), c 2 (( ε, ε)) are in B i p (p). Le α : [0, 1] ( ε, ε) M be a variaion of geodesics given by α(, s) = exp c1 (s) (exp c 1 (s) 1 c 2 (s)). Noe ha, for each s ( ε, ε) we have α(0, s) = c 1 (s), α(1, s) = c 2 (s) and he curve α s : [0, 1] M given by α s () = α(, s) is a geodesic. In paricular, α 0 () = α(, 0) = p is a consan geodesic. Now, consider he vecor fields T (, s) := α α (, s), and J(, s) = (, s). (1)

4 1520 O.P. Ferreira / Nonlinear Analysis 68 (2008) Above definiions imply ha T (, s) is angen o geodesic α s and J(, s), he Jacobi vecor field hrough α s, saisfies he following differenial equaion J(, s) + R(J(, s), T (, s))t (, s) = 0, (2) where R is he curvaure ensor, see [17]. Lemma 1. Le p M, c 1 and c 2 be wo C 1 curves in M such ha c 1 (0) = c 2 (0) = p, c 1 (0) = v and c 2 (0) = w. Then d(c 1 (s), c 2 (s)) lim = w v. s 0 + s Proof. To simplify noaions, define ψ(s) = d(c 1 (s), c 2 (s)). Consider α, he variaion of geodesics defined by (1), hen ψ(s) = α s () = T (, s). Since T (, 0) = α 0() = 0 he firs derivaive of ψ 2 a s = 0 is given by d ds (ψ2 (s)) s=0 = 2 T (, 0), T (, 0) = 0, (3) and he second derivaive by d 2 ds 2 (ψ2 (s)) s=0 = 2 = 2 In addiion, Eq. (2) becomes J(, 0) = 0. T (, 0), T (, 0) T (, 0), T (, 0) + 2 T (, 0), T (, 0). (4) Now, he laer equaliy ogeher wih he condiions J(0, 0) = v and J(1, 0) = w implies ha J(, 0) = v +(w v). Using Symmery s Lemma (see Lemma 2.2, pp. 35 in [17]) and his equaliy we have T (, 0) = α (, 0) = So, subsiuing he las equaliy in (4), we obain d 2 ds 2 (ψ2 (s)) s=0 = 2 w v 2. α (, 0) = J(, 0) = w v. Thus, as ψ(0) = 0 i follows from he laer equaliy and (3) ha ψ 2 (s) = w v 2 s 2 + O(s 2 ), wih lim s 0 + O(s 2 )/s 2 = 0. Therefore, he resul follows from he definiion of ψ. Corollary 1. If f F is Lipschiz in p wih consan L p, hen f (p,.) Lip L p (T p M). Proof. Le v and w be in T p M. Le γ, η be he geodesics wih γ (0) = η(0) = p, γ (0) = v and η (0) = w. Firs noe ha f (γ ()) f (p) f (η()) f (p) f (γ ()) f (η()) =. Since f is Lipschiz in p and γ (0) = η(0) = p we have f (γ ()) f (η()) L p d(γ (), η()) for all [0, ε) and some ε > 0. This inequaliy ogeher wih he above equaliy imply ha f (γ ()) f (p) L p d(γ (), η()) f (η()) f (p) f (γ ()) f (p) for all [0, ε). Now, as γ (0) = v and η (0) = w we obain from Lemma 1 ha d(γ (), η()) lim = v w. 0 + d(γ (), η()) + L p,

5 O.P. Ferreira / Nonlinear Analysis 68 (2008) Therefore, aking lim inf in he above inequaliy and considering he laer equaliy, we obain f (p, v) L p v w f (p, w) f (p, v) + L p v w, which implies ha f (x,.) is Lipschiz wih consan L p. Corollary 2. Le f F be a Lipschiz funcion in p dom( f ). For all C 1 curves c : [0, ε) M saisfying c(0) = p and c (0) = v here holds f (p, v) = lim inf 0 + f (c()) f (p). Proof. Le γ be a geodesic wih γ (0) = p and γ (0) = v. Since f is Lipschiz in p, a similar argumen used in he proof of he laer corollary implies f (γ ()) f (p) L p d(γ (), c()) f (c()) f (p) f (γ ()) f (p) d(γ (), c()) + L p, for all [0, ε) and some ε ε, where L p is a Lipschiz consan of f in p. Now, as c (0) = γ (0) = v, Lemma 1 implies ha lim 0 + d(γ (), c())/ = 0. So, aking lim inf in he laer inequaliy we obain he saemen. Corollary 3. Le f F(U) be a locally Lipschiz funcion on U and I R an open inerval. If c : I U is a C 1 curve, hen ( f c) (, 1) = f ( c(), c () ), I. Proof. This follows from Corollary Characerizaion for Lipschiz funcions In his secion we presen a characerizaion for Lipschiz funcions defined on Riemannian manifolds. I is worh poining ou ha our resuls in his secion were obained by adaping, for our conex, he echniques inroduced by Clarke, Sern and Wolenski [4] for characerizing Lipschiz funcions in Hilber spaces. Proposiion 1. Le U be an open convex subse of M and le f F(U). If f is locally Lipschiz on dom f wih consan L everywhere, hen he following saemens hold: (i) dom f = U; (ii) f Lip L (U). Proof. For (i). Firs noe ha dom( f ) U. I remains o show ha U dom( f ). For ha, ake q U. Now, ake p dom( f ) and a minimal geodesic γ : [0, 1] M such ha γ (0) = p and γ (1) = q. Since U is convex i follows ha γ ([0, 1]) U. We claim ha q dom( f ). Suppose no. Thus we have ha := sup{ (0, 1] : f (γ ()) < + } < 1, and since p dom( f ) and f is locally Lipschiz on dom f we also have 0 <. So, aking (0, ) we have γ ([0, ]) dom f. Since γ ([0, ]) is compac and f is locally Lipschiz on dom f, we can ake 0 = 0 < 1 < < n = and posiive numbers δ 0 δ n 1 saisfying γ ([ i, i+1 ]) B δi (γ ( i )) U and f Lip L (B δi (γ ( i ))), for all i = 0,..., n 1. As he geodesic γ is minimal, using he local Lipschiz propery we obain f (γ ( n 1 )) = f (p) + ( f (γ ( i+1 )) f (γ ( i ))) i=0 n 1 f (p) + L d (γ ( i ), γ ( i+1 )) i=0 f (p) + L d ( p, γ ( ) ). (5)

6 1522 O.P. Ferreira / Nonlinear Analysis 68 (2008) Since f is lower semiconinuous, leing goes o in he laer equaion we conclude ha f (γ ( )) < +, which implies ha γ ( ) dom( f ). According o f being Lipschiz a γ ( ), he definiion of is violaed. Therefore, we obain ha f is finie on he enire γ ([0, 1]). In paricular, q dom( f ), so U dom( f ) and he firs saemen follows. For (ii). Take p, q U and a minimal geodesic γ : [0, 1] M such ha γ (0) = p and γ (1) = q. Since U is convex i follows from iem (i) ha γ ([0, 1]) dom( f ). Wih an analogous argumen used o obain (5) we can show ha f (q) f (p) + Ld(p, q). Now, by reversing he roles of p and q, i easy o conclude ha f Lip L (U), and he second saemen is proved. Theorem 1. Le f F and U be an open convex subse of M. Then f is Lipschiz on U of rank L 0 if, and only if, f (p, v) L v, p U, v T p M. Proof. Firs, suppose ha f is Lipschiz on U of rank L 0. Take p U, v T p M. Le γ be he geodesic such ha γ (0) = p and γ (0) = v. Since γ (0) = p and f is Lipschiz on U of rank L 0, here exiss δ > 0 such ha f (γ ()) f (p) Ld(γ (), p) L v, [0, δ). So, using he definiion of he lower Dini derivaive, he above inequaliy implies ha f (p, v) = lim inf 0 + f (γ ()) f (p) d(γ (), p) L lim inf = L v. 0 + Reciprocally, suppose ha f (p, v) L v, for all p U and v T p M. Le p 0 dom( f ). Since f F(U), ake 0 < δ < i p such ha f is bounded below on B 4δ (p 0 ) U. Le K > L and q B δ (p 0 ). Define g : U [0, + ] as K d(p, q) if p B 2δ [q]; (d(p, q) 2δ)2 g(p) = K d(p, q) + if p B 3δ (q) \ B δ (q); 3δ d(p, q) + oherwise. I is easy o see ha g F(U). Moreover, as g C 1 (B 3δ (q) \ {q}) and 0 < δ < i p here holds [ grad g(p) = K + 2(d(p, q) 2δ)(3δ d(p, q)) + d(p, q)(d(p, q) 2δ) (3δ d(p, q)) 2 ] ( exp p 1 q/d(p, q)), for all p B 3δ (q) \ B 2δ (q). Now, noe ha f + g is in F(U), goes o + as p goes o he boundary of B 3δ (q) and is bounded below on B 3δ (q). Therefore, as M is a complee Riemannian manifold of finie dimension, here exiss p B 3δ (q) a minimizer for f + g. Firs we assume ha p q. Since p B 3δ (q) \ {q}, g C 1 (B 3δ (q) \ {q}) and f + g F(U) we have f (p, v) + grad g(p ), v 0 v T p M. (6) For simplifying he noaions se v := exp 1 p q/d(p, q) and l := 2(d(p, q) 2δ)(3δ d(p, q)) + d(p, q)(d(p, q) 2δ) (3δ d(p, q)) 2. So, grad g(p ) = K v if p B 2δ [q] and grad g(p ) = [K + l]v if p B 3δ (q) \ B 2δ (q). Thus, leing v = v in (6) and aking ino accoun ha l 0 for p B 3δ (q) \ B 2δ (q) we obain f (p, v ) K v > L v,

7 O.P. Ferreira / Nonlinear Analysis 68 (2008) which conradics our assumpion. Consequenly we mus have p = q. Due o he fac ha g(q) = 0, he poin q is a minimizer of f + g and B δ (p 0 ) B 2δ [q] we have f (q) = ( f + g)(q) ( f + g)(p) f (p) + K d(p, q), p B δ (p 0 ). Since we can change he roles of q and p in he above argumen he following inequaliy holds f (q) f (p) K d(p, q), p, q B δ (p 0 ). Leing K go o L in he laer inequaliy, we conclude ha for any p 0 dom f here exiss δ > 0 such ha f Lip L B δ (p 0 ). Thus we have shown ha f is locally Lipschiz on dom( f ) wih he same rank L everywhere. Therefore, for finishing he proof use he Proposiion 1. Example 4. Le SR n be he se of symmeric marices endowed wih he Frobenius meric defined by U, V = r (V U) and le SR n ++ be he se of posiive definie symmeric marices. Le he funcion f : SRn ++ R be defined by f (X) = ln de X. I easy o see ha he funcion f is no Lipschiz on SR n ++. For each X SRn ++ define a new inner produc in SR n as U, V = X 1 U X 1, V U, V SR n. Endowing SR n ++ wih he Riemannian meric,. we obain a complee Riemannian manifold. We denoe by M his Riemannian manifold. Noe ha f C 1 on M. So, f (X, V ) = grad f (X), V V T X M. Because he usual gradien of f on SR n ++ is f (X) = X 1, we have ha he gradien of f on M is given by grad f (X) = X f (X)X = X. So, grad f (X) 2 = grad f (X), grad f (X) = 1. Thus, f (X, V ) = grad f (X), V grad f (X) V = V, V T X M. From Theorem 1 i follows ha f is Lipschiz on M of rank L = 1. Example 5. Le Ω = {p = (p 1, p 2 ) R 2 : p 2 > 0} and le f : Ω R be given by f (p) = ln(p 2 ). I easy o see ha f is no a Lipschiz funcion on Ω wih respec o he Euclidean meric,. Le G be a 2 2 marix defined by G(p) = (g i j (p)), where g 11 (p) = g 22 (p) = 1 p2 2, g 12 (p) = g 21 (p) = 0. Endowing Ω wih he Riemannian meric, defined by u, v = G(p)v, u, we obain a complee Riemannian manifold H 2, namely, he upper half-plane model of Hyperbolic space. Noe ha f C 1 and he gradien of f in H 2 is given by grad f (p) = G(p) 1 f (p) = (0, p 2 ), where f is he usual gradien of f in Ω. I is simple o show ha grad f (p) = 1 and so, f (p, v) = grad f (p), v v, v T p H 2. Therefore, from Theorem 1 i follows ha f is Lipschiz on H 2 of rank L = Characerizaion for convex funcions In his secion we obain a characerizaion for convex funcions defined on Riemannian manifolds. As usual, in he ype of characerizaion we shall presen, firs we obain a resul like he mean value heorem adaped o our seing. Definiion 4. A funcion f : M (, + ] is said o be convex (respecively, sricly convex) if for all minimal geodesics γ : [a, b] M, he composiion f γ : [a, b] (, + ] is convex (respecively, sricly convex).

8 1524 O.P. Ferreira / Nonlinear Analysis 68 (2008) I follows from he above definiion ha if f : M (, + ] is a convex funcion hen dom( f ) and he sub-level ses {p M : f (p) k} are convex ses, for all k R. Example 6. Le S n = {p R n+1 : p = 1} be a uniary sphere. Fix p = (0,..., 0, 1) R n+1. Seing p = (p 1..., p n, p n+1 ), define ϕ : S n (, + ] as { d( p, p), if pn+1 0; ϕ(p) = + if p n+1 < 0. Noe ha ϕ F, bu ϕ is no a convex funcion and is dom(ϕ) = {p S n : p n+1 0} is no a convex se. Now, le C {p S n : p n+1 > 0} be a closed and convex se and define ρ : S n (, + ] as { d( p, p), if p C; ρ(p) = + if p C. Noe ha ρ is lower semiconinuous, convex and is domain C is closed. In general, for all p M and C {p M : d( p, p) < π/2} a closed convex se, he funcion η : S n (, + ] defined as η(p) = d( p, p), if p C and η(p) = + if p C, is lower semiconinuous and convex. In he above example dom(ϕ) = {p R n : p n+1 0} is closed and is inerior is convex. As dom(ϕ) is no convex we conclude ha, in general, he closure of a convex se is no convex. Noe ha ϕ is Lipschiz of rank L = 1 in in(dom(ϕ)), bu no in dom(ϕ). Definiion 5. A funcion f F is locally bounded in p if here exis δ > 0 and r > 0 such ha f (q) r for all p B δ (p), and f F is locally bounded in dom( f ) if i is locally bounded a all poins p dom( f ). Noe ha if f F is locally bounded in dom( f ) hen dom( f ) mus be open. Proposiion 2. Le f F be locally bounded in dom( f ). If f is convex, hen f is locally Lipschiz on dom( f ). Proof. See Proposiion 5.2 in [1]. We remark ha in n-dimensional Euclidean spaces for proving ha a convex funcion f is locally bounded in dom( f ) wo imporan resuls are used, namely, for each p dom( f ) here exiss a n-dimensional simplex dom( f ) such ha p in( ) and Jensen s inequaliy. However, as far as we know resuls like hese in he Riemannian conex have no been sudied ye. From now on, we assume ha f F and is locally Lipschiz on dom( f ) wihou explicily menioning hem in he saemens of our resuls. Noe ha in his case f is locally bounded in dom( f ) and dom( f ) is open. Example 7. Le S n = {p R n+1 : p = 1} be a uniary sphere. Fix p = (0,..., 0, 1) R n+1. Seing p = (p 1..., p n, p n+1 ), define ζ : S n (, + ] as { ln(π/2 d( p, p)), if pn+1 > 0; ζ(p) = + if p n+1 < 0. ζ is lower semiconinuous, convex and is domain is open. Moreover, ζ locally Lipschiz in dom(ζ ). Example 8. Le M = {p R n : p = 1} be a uniary sphere. Seing p = (p 1..., p n ), define ψ : M (, + ] as n ln(p i ) if p 1 > 0,..., p n > 0, ψ(p) = i=1 + oherwise. ψ is lower semiconinuous, convex and is domain is open. Moreover, ψ is locally Lipschiz in dom(ψ).

9 O.P. Ferreira / Nonlinear Analysis 68 (2008) Proposiion 3. If f is convex, hen for all p dom( f ) and v T p M here holds f (p, v) = lim 0 + f (γ ()) f (p) = inf >0 f (γ ()) f (p), where γ is a minimal geodesic such ha γ (0) = p and γ (0) = v. Proof. I follows from convexiy of f γ. Corollary 4. If f is convex and γ : [a, b] dom( f ) is a minimal geodesic, hen here holds ( ) f (γ ( ), γ ( )) + f (γ ( )) f (γ ()),, [a, b]. Proof. Is an immediae consequence of Proposiion 3. Definiion 6. Le f F be a locally Lipschiz funcion on dom( f ). The lower Dini derivaive f : T M [, + ] is said o be monoone (respecively, sricly monoone) if, for any minimal geodesic γ : [a, b] M wih is end poins in dom( f ), he map ϕ γ : [a, b] [, + ] defined by ϕ γ () = f (γ (), γ ()) is monoone non-decreasing (respecively, increasing). Noe ha in he above definiion we do no assume ha dom( f ) is a convex se. Lemma 2. If γ is a geodesic such ha γ (0) = p, γ (1) = q and γ ([0, 1]) dom( f ), hen here exis, ˆ (0, 1) such ha f ( γ ( ), γ ( ) ) f (q) f (p) f ( γ (ˆ), γ (ˆ) ). Proof. Le ψ : [0, 1] R be defined by ψ(s) = f (γ (s)) s( f (q) f (p)). So, from Corollary 3 is sufficien o show ha here exis, ˆ (0, 1) such ha ψ (, 1) 0 ψ (ˆ, 1). Firs we will prove ha here exiss (0, 1) such ha ψ (, 1) 0. Since ψ is coninuous in [0, 1] and ψ(0) = ψ(1) = f (p), here exiss s 1 (0, 1] a global minimum of ψ. Now le s 0 (0, s 1 ). We have wo possibiliies: (i) ψ (s 0, 1) 0 or (ii) ψ (s 0, 1) > 0. If (i) occurs ake = s 0, oherwise ψ (s 0, 1) > 0 implies ha here exiss s (s 0, s 1 ) such ha ψ(s 0 ) < ψ( s). Hence, by he coninuiy of ψ here exiss (s 0, s 1 ) such ha ψ(s) ψ( ) for all s [s 0, s 1 ], which implies ψ (, 1) 0. The second inequaliy can be shown by a similar argumen o he firs. Corollary 5. If γ is a geodesic in M such ha 2 > 1, γ ([ 1, 2 ]) dom( f ), hen here exis, ˆ ( 1, 2 ) such ha f ( γ ( ), γ ( ) ) f (γ ( 2)) f (γ ( 1 )) 2 1 f ( γ (ˆ), γ (ˆ) ). Proof. Defining he geodesic α(s) = γ (s 2 + (1 s) 1 ) we have α(0) = γ ( 1 ) and α(1) = γ ( 2 ). Now, noe ha v f (p, v) is posiively homogeneous, for all p dom( f ). So, applying Lemma 2 he saemen follows. Lemma 3. If f is monoone, hen dom( f ) is convex. Proof. Take p, q dom( f ) and γ a minimal geodesic joining p o q. Assume wihou lose of generaliy ha γ (0) = p and γ (1) = q. Define ˆ := inf{ [0, 1] : [γ (), q] dom( f )}, := sup{ [0, 1] : [p, γ ()] dom( f )}. Since f is locally Lipschiz on dom( f ) we have ha dom( f ) is open, so 0 ˆ < 1 and 0 < 1. Assume by conracion ha γ ([0, 1]) dom( f ) or equivalenly ha 0 < ˆ and < 1. Now, ake 0 < λ < ˆ < τ < 1. The definiions of ˆ and imply ha he minimal geodesic segmens joining γ (τ) o q and p o γ (λ) are in dom( f ). Thus, from Corollary 5 here exis 1 (0, λ) and 2 (τ, 1) such ha f (γ (λ)) f (p) λf (γ ( 1 ), γ ( 1 )), (1 τ) f (γ ( 2 ), γ ( 2 )) f (q) f (γ (τ)).

10 1526 O.P. Ferreira / Nonlinear Analysis 68 (2008) The above inequaliies ogeher wih monooniciy of he derivaive f imply ha f (γ (λ)) λ + f (γ (τ)) (1 τ) f (p) λ f (p) λ + f (γ ( 1 ), γ ( 1 )) + f (q) (1 τ) f (γ ( 2 ), γ ( 2 )) + f (q) (1 τ). Since f F, 0 < and ˆ < 1, leing λ go o and τ go o ˆ in he above equaion we obain f (γ ( )) + f (γ (ˆ)) (1 ˆ) f (p) + f (q) (1 ˆ). So, f (γ (ˆ)) < + and f (γ ( )) < +, i.e., γ (ˆ), γ ( ) dom( f ). As dom( f ) is open we obain a conradicion wih he definiions of ˆ and. Therefore, γ ([0, 1]) dom( f ) and he saemen follows. Theorem 2. f is convex if, and only if, f is monoone. Proof. Assume ha f is convex. Firs noe ha dom( f ) is convex. Le γ : [a, b] M be a minimal geodesic wih is end poins in dom( f ) and ake 1, 2 [a, b] such ha 1 < 2. Since dom( f ) is convex we have ha γ ([ 1, 2 ]) dom( f ) and using Corollary 4 we obain ha ( 2 1 ) f ( γ ( 1 ), γ ( 1 ) ) + f (γ ( 1 )) f (γ ( 2 )), ( 1 2 ) f ( γ ( 2 ), γ ( 2 ) ) + f (γ ( 2 )) f (γ ( 1 )). Therefore, as 1 < 2 i follows from las wo inequaliies ha f ( γ ( 1 ), γ ( 1 ) ) f ( γ ( 2 ), γ ( 2 ) ), hence f is monoone. Now, assume ha f is monoone. Le γ : [a, b] M be a geodesic and le 1, 2 [a, b] be such ha 1 < 2. If γ ( 1 ) dom( f ) or γ ( 2 ) dom( f ) we have f γ ((1 λ)( 1 + λ 2 )) (1 λ) f (γ ( 1 )) + λf (γ ( 2 )), for all λ [0, 1]. Now, suppose ha γ ( 1 ), γ ( 2 ) dom( f ). So, Lemma 3 implies ha γ ([ 1, 2 ]) dom( f ). Thus, for all ( 1, 2 ) we obain from Corollary 5 ha here exis s 1 ( 1, ) and s 2 (, 2 ) such ha f (γ ()) f (γ ( 1 )) ( 1 ) f (γ (s 1 ), γ (s 1 )), ( 2 ) f (γ (s 2 ), γ (s 2 )) f (γ ( 2 )) f (γ ()). Thus, leing = (1 λ) 1 + λ 2 in he above equaion we obain from monooniciy of f ha (1 λ) f (γ ( 1 )) + λf (γ ( 2 )) f (γ ()) = (1 λ)[ f (γ ()) f (γ ( 1 ))] + λ[ f (γ ( 2 )) f (γ ())] (1 λ)λ( 2 1 )[ f (γ (s 1 ), γ (s 1 )) + f (γ (s 2 ), γ (s 2 ))] 0, for all λ [0, 1]. Which implies ha f γ ((1 λ) 1 + λ 2 ) (1 λ) f (γ ( 1 )) + λf (γ ( 2 )), for all λ [0, 1]. Therefore f is convex. 5. Sufficien opimaliy condiions for opimizaion problems In his secion we are going o obain sufficien opimaliy condiions for consrain opimizaion problems in erms of he Dini derivaive. Proposiion 4. Le C M be a convex se. Assume ha f : M R is convex in C. Consider he following nonlinear programming problem { min f (p) (P) s.. p C. Le p C. If for all p C we have ha f (p, γ p p (0)) 0, where γ p p is a minimal geodesic from p o p wih γ p p(0) = p and γ p p(1) = p. Then p C is a soluion o (P).

11 O.P. Ferreira / Nonlinear Analysis 68 (2008) Proof. Given p, p C and γ p p a minimal geodesic from p o p wih γ p p(0) = p and γ p p(1) = p. Since C is convex we have ha γ p p([0, 1]) C. Now, as f is convex we conclude from Corollary 4 ha f (p) f (p ) + f (p, γ p p (0)). Because f (p, γ p p (0)) 0, he laer inequaliy implies ha f (p) f (p ). So, p is a soluion o (P). Corollary 6. Le f, g i : M R be convex, for i = 1,..., m. Consider he following nonlinear programming problem { min f (p) ( P) s.. g i (p) 0, i = 1,..., m. Le p be a feasible poin o ( P). If for all p, a feasible poin o ( P), here exiss a vecor µ = (µ 1,..., µ m ) R m such ha m m f (p, γ p p ( )) + µ i g i (p, γ p p ( )) 0, µ 0, and µ i g i (p ) = 0, (7) i=1 where γ p p is a minimal geodesic from p o p wih γ p p (0) = p and γ p p (1) = p. Then p is a soluion o ( P). Proof. Since f, g i : M R are convex, for i = 1,..., m, and µ 0 we conclude ha C := {p M : g i (p) 0, i = 1,..., m} is convex and h : M R defined by h(p) = f (p) + m i=1 µ i g i (p) is also convex. Moreover, f (p) h(p), for all p C. Take p C and γ p p a minimal geodesic from p o p wih γ p p(0) = p and γ p p(1) = p. From he firs inequaliy in (7) we obain ha h (p, γ p p (0)) 0 and as p C i follows from Proposiion 4 ha p saisfies h(p) h(p ), for all p C. Thus, from (8) and he equaliy in (7) we obain ha f (p) h(p) h(p ) = f (p ), for all p C, and he proposiion is proved. 6. Final remarks A complee, simply conneced Riemannian manifold of nonposiive secional curvaure is called a Hadamard manifold, see Examples 4 and 5 above. The Hadamard Caran Theorem [17] assers ha he opological and differenial srucure of a Hadamard manifold coincide wih hose of a Euclidean space of he same dimension. More precisely, a any poin p M, he exponenial map exp p : T p M M is a diffeomorphism. Furhermore, for any wo poins p, q M here exiss a unique geodesic joining p o q which is minimal. So, Definiion 6 becomes Definiion 7. Le f F be a locally Lipschiz funcion on dom( f ). The lower Dini derivaive f : T M [, + ] is said o be monoone if f (p, exp p 1 q) + f (q, exp q 1 p) 0, for any wo poins p, q M. Since we have obained resuls only for locally Lipschiz funcions defined on finie dimensional Riemannian manifolds, we expec ha he resuls in his paper will be one more sep oward a characerizaion for Lipschiz and convex funcions in more general seings. Acknowledgemen The auhor was suppored in par by CNPq Gran /2005-8, FUNAPE/UFG, PADCT-CNPq, PRONEX Opimizaion (FAPERJ/CNPq) and PROCAD (CAPES). i=1 (8)

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