c 2000 Society for Industrial and Applied Mathematics

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1 SIAM J. CONTROL OPTIM. Vol. 38, No. 2, pp c 2000 Society fo Industial and Applied Mathematics NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS A. S. LEWIS AND R. E. LUCCHETTI Abstact. Given a nonlinea function h sepaating a convex and a concave function, we povide vaious conditions unde which thee exists an affine sepaating function whose gaph is somewhee almost paallel to the gaph of h. Such esults blend Fenchel duality with a vaiational pinciple and ae closely elated to the Clake Ledyaev mean value inequality. Key wods. sandwich theoem, squeeze theoem, Fenchel duality, vaiational pinciple, mean value inequality, Clake subdiffeential AMS subject classifications. 49J52, 90C46, 26D07 PII. S Intoduction. The cental theoems in this pape blend two completely distinct types of esult, both fundamental in optimization theoy: Fenchel duality and vaiational pinciples. The simplest vesion of Fenchel duality states that fo any convex functions f and g on R n satisfying f g, a egulaity condition implies the set L def = y R n : f (y)+g ( y) 0} is nonempty (whee f is the Fenchel conjugate of f). Geometically, this means thee exists an affine function sandwiched between f and g. On the othe hand, one of the easiest examples of a vaiational pinciple states that if h is a locally Lipschitz function bounded below on R n, then h has abitaily small Clake subgadients: 0 cl (Im h). Geometically, thee ae points whee the gaph of h is almost hoizontal (in a cetain nonsmooth sense). The theoems we discuss hee combine the featues of both esults above. We conside functions f, g, and h as befoe, now satisfying f h g, and unde vaious egulaity conditions we pove L cl (Im h). Geometically, thee ae affine functions between f and g whose gaphs ae somewhee almost paallel to the gaph of h. As we show by means of vaious examples, the existence of a suitable affine sepaating function depends on both local and asymptotic popeties of the thee functions. Hence the egulaity conditions we need to impose combine assumptions on the domains of the pimal functions, f and g, and of thei conjugates, f and g, as well as local and global gowth conditions on h. Received by the editos Febuay 18, 1998; accepted fo publication (in evised fom) May 27, 1999; published electonically Febuay 9, This eseach was patially suppoted by the Natual Sciences and Engineeing Reseach Council of Canada. Depatment of Combinatoics and Optimization, Univesity of Wateloo, Wateloo, Ontaio, Canada N2L 3G1 (aslewis@oion.uwateloo.ca; aslewis). Politecnico di Milano, Facoltá di Ingegneia di Como, P.le Gebetto 6, Como, Italy (el@komodo.ing.unico.it). 613

2 614 A. S. LEWIS AND R. E. LUCCHETTI The key tool fo ou esults is a ecent, somewhat supising mean value inequality of Clake Ledyaev [3], ephased as a hybid sandwich theoem in [6]. We illustate the application of this type of esult with two appaently simple but athe emakable consequences. Fist, any convex function f and locally Lipschitz function h f satisfy domf cl (Im h). Second (a squeeze theoem ), any locally Lipschitz functions p h q with p(0) = h(0) = q(0) satisfy p(0) h(0) q(0). We have not been able to find simple poofs o efeences fo eithe of these two esults. 1 With the exception of this last squeeze theoem, ou esults do not appea to be substantially easie with the assumption that h is smooth (in which case h educes to the singleton h). We believe they povide futhe evidence of the depth, applicability, and fundamental natue of the Clake Ledyaev inequality in optimization theoy. 2. Notation and peliminay esults. We begin by eviewing some basic ideas fom convex analysis (see [7]). Given a convex set A R n, we denote by aff A the smallest affine space containing A and by i A the set of the intenal points of A aff A (with the induced topology). Obseve that i A is a nonempty convex set. Given a function f : R n [, ], we denote its effective domain by and by epi f its epigaph, the set dom f def = x R n : f(x) < } epi f def = (x, ) R n R : f(x)}, a convex set if and only if f is convex. The hypogaph of the function g is instead hyp( g) def = (x, ) R n R : g(x)}, again a convex set if and only if g is convex. The epigaph of f is closed if and only if f is lowe semicontinuous in the usual sense. We shall wite f Γ 0 to mean that epi f is nonempty, closed, and convex and does not contain vetical lines. Fo a set A, let I A be the indicato function of the set A, 0 if x A, I A (x) = othewise. In paticula, I kb denotes the indicato function of the ball centeed at 0 R n and with adius k. 1 Subsequent investigations evealed altenative appoaches to the last theoem independent of the Clake Ledyaev esult [1]. Nonetheless, the oiginal appoach we pesent hee emains attactive fo its tanspaency.

3 NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS 615 The Fenchel conjugate of a function f : R n [, ] is the function f : R n [, ] defined by f (y) def = sup y, x f(x)}, x R n a convex lowe semicontinuous function (even if f is not) which belongs to Γ 0 if f does. Futhemoe, f = f, poviding f Γ 0. The function f is said to be cofinite if its conjugate f satisfies dom f = R n. f(x) It is easy to see this is equivalent to saying that lim x x = [5, Chapte 10, Poposition 1.3.8]. The subdiffeential of a convex function f at a point x dom f is the closed convex set f(x) def = y R n : f(z) f(x)+ y, z x fo all z R n }. The fundamental connection between the subdiffeentials of a function and of its Fenchel conjugate is shown by the following Fenchel identity: y f(x) f (x)+f (y)= y, x. It follows, in paticula, that y f(x) if and only if x f (y), poviding f Γ 0. Given two functions p, q : R n [, ], we define the infimal convolution between them by (p q)(x) = inf y R np(y)+q(x y)}, a convex function if p and q ae, possibly assuming the value, and that may fail to be lowe semicontinuous. Finally, fo a function p Γ 0, we denote by p k def = p k, the infimal convolution between p and k : this function is the lagest k-lipschitz function minoizing p. Fo moe about convex functions, the inteested eade is invited to consult [4], [5], [7]. Next we biefly conside the subdiffeential of a locally Lipschitz function h : U R, whee U is an open subset of R n. This notion is not uniquely defined in the liteatue, and hee we make the choice of the Clake subdiffeential (see [2]) which is moe suited fo ou scopes, as an example in the final section will show. To define it, fist let us intoduce the notion of genealized diectional deivative of h at the point x in the diection v: h (x, v) def = lim sup z x t 0 h(z + tv) h(z). t The function v h (x, v) is eveywhee finite, subadditive, and positively homogeneous; hence, in paticula, it is continuous and convex. Then the subdiffeential of h at x is defined as h(x) def = y R n : y, v h (x, v) fo all v R n }, a nonempty closed convex set. Moeove, if k is a Lipschitz constant fo h, the subdiffeential is nom-bounded by k. In paticula, the multifunction h is bounded on

4 616 A. S. LEWIS AND R. E. LUCCHETTI bounded sets. Obseve that the Clake subdiffeential of h at x is the same set as the (convex) subdiffeential of the function v h (x, v) atv= 0, a simple but useful popety we shall use thoughout this pape. Fo moe about nonsmooth analysis fo locally Lipschitz functions, the inteested eade is invited to consult [2]. In this pape we shall deal with two convex functions f,g Γ 0 and a locally Lipschitz function h such that f h g. Fo a moment, let us focus on the poblem of nonemptyness of the set L def = y R n : f (y)+g ( y) 0}. This set can be chaacteized in a moe geometic way, as the following easy poposition states. Poposition 2.1. Let f,g Γ 0. Then, fo y R n, if and only if thee exists a R such that f (y)+g ( y) 0 f(x) a + y, x g(x) fo all x R n. Thus the poblem of nonemptyness of L is equivalent to finding an affine sepaato lying below f and above g. This can be stated in tems of a sepaation poblem fo the sets epi f and hyp( g). The assumption f gensues that i epif i hyp( g) =, (see [4, Chapte 4, Poposition 1.1.9]) and this in tun implies that epi f and hyp( g) can be sepaated by a hypeplane [4, Chapte 3, Theoem 4.1.4]. Howeve, it can happen that the only sepaating hypeplane is vetical, which unfotunately says nothing about nonemptyness of L. The fist esult stating that L is nonempty is the following well-known Fenchel duality theoem [7, Theoem 31.1], which in ou setting can be ephased in the following way. Theoem 2.1. Let f,g Γ 0 be such that f gand suppose Then thee exists y R n such that i (dom f) i (dom g). f (y)+g ( y) 0. We illustate the ole of the assumption on the domains of f and g with the help of the following fou examples, whee the set L is always empty. Example 2.1. x if x 0, f(x) = othewise, 0 if x =0, g(x) = othewise. Hee i (dom f) i (dom g) =.

5 NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS 617 Example if uv 1,u 0, f(u, v) = othewise, 0 if u 0,v =0, g(u, v) = othewise. Hee we have dom f dom g =. Example 2.3. u if v = 1, f(u, v) = othewise, 0 if v =0, g(u, v) = othewise. Hee the distance between dom f and dom g is 1. In the last two examples the domains of f and g do not intesect, while in the fist example a cucial ole is played by the fact that inf(f + g) = 0. In the following example inf(f + g) > 0, and yet thee is no affine sepaato. Obseve that such an example could not be povided in one dimension [4, Chapte 1, Remak 3.3.4]. Example uv if u, v 0, f(u, v) = othewise, 1 2 uv if u 0,v 0, g(u, v) = othewise. A staightfowad calculation shows f (u,v 1 if u )= 0,u v 1, othewise, Thus the set L is empty. g (u,v 1 if u )= 0,u v 1, othewise. 3. Sandwich theoems. We tun now to the case of thee functions f, g, and h, such that f and g ae convex, h is locally o globally Lipschitz, and f h g. Examples 2.2, 2.3, and 2.4 show that the existence of a locally Lipschitz function h between f and g (take h(x, y) = xy in all cases) does not change the situation: thee is no affine sepaato. Fist let us ecall now some known esults. Theoem 3.1 (see [6, Theoem 2]). Let C be a nonempty convex compact set in R n.letf,g Γ 0 and with domains contained in C. Leth:R n Rbe Lipschitz on a neighbohood of C. Suppose moeove f h gon C.

6 618 A. S. LEWIS AND R. E. LUCCHETTI Then thee exist c C and y h(c) such that f (y)+g ( y) 0. Boundedness of C can be elaxed at the expense of equiing moe about the functions f and g and/o about the function h. Specifically, we have the following two esults. Theoem 3.2 (see [6, Theoem 7]). Let C be a nonempty closed convex set in R n.letf,g Γ 0 be cofinite, with domains contained in C. Moeove suppose int (dom f) int (dom g). Let h : R n R be locally Lipschitz on a neighbohood of C and suppose f h g on C. Then thee exist c C and y h(c) such that f (y)+g ( y) 0. Theoem 3.3 (see [6, Theoem 8]). Let C be a nonempty closed convex set in R n. Let f,g Γ 0 be cofinite, with domains contained in C. Let h : R n R be globally Lipschitz on a neighbohood of C and suppose f h gon C. Then thee exist c C and y h(c) such that f (y)+g ( y) 0. Obseve one does not need a qualification condition on the domains of f and g if h is globally Lipschitz. In the last two theoems, howeve, cofiniteness is equied, which can be egaded as a (stong) qualification condition on the domains of the conjugates. The fist esult we want to pove deals simply with the existence of the affine sepaato. To pove it, we need the following poposition about egulaizing Fenchel poblems. Poposition 3.1. Suppose p, q Γ 0 and Then, fo all lage k, we have and ( ) i (dom p) i (dom q). inf(p + q) = inf(p k + q k ) agmin(p + q) = agmin(p k + q k ). Poof. To pove the fist equality, we need to pove only inf(p + q) inf(p k + q k ). Thee is nothing to pove if inf(p + q) =. Theefoe, let us assume it is finite. (It cannot be because of ( ).) By Fenchel duality, thee is y R n such that Take k> y. Then inf(p + q) =p (y)+q ( y).

7 NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS 619 inf(p + q) =p (y)+q ( y)=(p +I kb )(y)+(q +I kb )( y) =(p k ) (y)+(q k ) ( y) inf z R n((p k) (z)+(q k ) ( z)) = inf(p k + q k ) inf(p + q). This shows the fist equality and also that y as above is optimal fo the poblem of minimizing, on R n,(p k ) ( )+(q k ) ( ). Now, witing down optimality conditions, we obtain, using k> y, x agmin(p + q) p(x)+q(x)= p (y) q ( y), x p (y) q ( y), x (p + I kb )(y) (q + I kb )( y), x (p k ) (y) (q k ) ( y), x agmin(p k + q k ). We begin ou sequence of main esults by poving some vaiants of Fenchel duality, whee the usual egulaity condition is eplaced by the existence of a Lipschitz sepaato. Theoem 3.4. Fo f,g Γ 0, suppose i (dom f ) i ( dom g ). Suppose futhe thee exists a locally Lipschitz function h such that f h g. Then thee is y R n such that f (y)+g ( y) 0. (Moeove, if inf(f + g) =0, then such a y can be found in the ange of h.) Poof. Fom Poposition 3.1, applied to p = f and q( ) =g ( ), we have and inf((f ) k ( )+(g ) k ( )) = inf(f ( )+g ( )) agmin((f ) k ( )+(g ) k ( )) = agmin(f ( )+g ( )) fo all lage k. Apply Theoem 3.1 to the functions f + I kb, h, and g + I kb, fo lage k, and the set C = kb, tofindy k Im h such that If (f ) k (y k )+(g ) k ( y k ) 0. y k agmin((f ) k ( )+(g ) k ( )) (as in the case when inf(f + g) = 0), then we deduce f (y k )+g ( y k ) 0 and we conclude. Othewise, fo all lage k, 0 > inf((f ) k ( )+(g ) k ( )) = inf(f ( )+g ( )).

8 620 A. S. LEWIS AND R. E. LUCCHETTI Thus thee is y R n such that f (y)+g ( y) 0, as equied. We povided hee the esult when inf(f + g) = 0 fo the sake of completeness. Howeve obseve that unde the assumptions of Theoem 3.4 inf f + g is attained. In this cicumstance the squeeze theoem in the next section will povide a moe pecise esult. With espect to the ole of the assumptions in Theoem 3.4, Example 2.1 shows the set L can be empty if we do not assume the existence of a locally Lipschitz function sandwiched between f and g, while Example 2.2 shows the necessity of the qualification condition i (dom f ) i ( dom g ). We tun now to the poblem of poviding conditions unde which the slope of an affine sepaato can be found in the closue of the ange of the Clake subdiffeential of the sepaating function h. To do this, we pove fist the following poposition. Poposition 3.2. Suppose f,g Γ 0 satisfy f g.fok=1,2,..., define and L k def = y R n :(f ) k (y)+(g ) k ( y) 0} L def = y R n : f (y)+g ( y) 0}. Then L k is a deceasing collection of closed convex sets containing L, and If moeove the condition y k L k, y k y implies y L. 0 int (dom f dom g) holds, then the sets L k fo lage k ae all contained in a compact set. Poof. Since (f ) k (f ) k+1 f, clealy L L k+1 L k, fo all k>0. Let us pove that, if y k is such that y k y and then (f ) k (y k )+(g ) k ( y k ) 0 fo all k, f (y)+g ( y) 0. Fom Poposition 2.1 thee exists a k R such that f(x) a k + y k,x g(x) fo all x kb. It is easy to show the sequence a k } is bounded, so it has some cluste a R. (Use the boundedness of y k } and the existence of an element x dom f dom g.) It follows that f(x) a + y, x g(x) fo all x R n, so y L. We have poved the fist pat of the claim. Now define a function v(w) = inf x Rn(f(x + w)+g(x))

9 NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS 621 and a sequence of functions deceasing pointwise to v, v k (w) = inf x R n((f + I kb)(x + w)+(g+i kb )(x)). Obseve that (v k ) (y) =(f ) k (y)+(g ) k ( y) and v (y) =f (y)+g ( y) and that dom v = dom f dom g, so that 0 int (dom v). Since v is continuous at 0, thee exist eals >0 and α and a cube C such that B C int(dom v) and v α 1onC. Hence, fo lage k we have v k α on each vetex of C and hence on B, so(v k ) (w) w α fo all points w in R n, and theefoe L k (α/)b. We ae now eady fo a new esult. Theoem 3.5. Fo f,g Γ 0, and locally Lipschitz h : R n R satisfying f h g, suppose Then 0 int (dom f dom g). y cl (Im h):f (y)+g ( y) 0. Poof. Apply Theoem 3.1 to the functions f + I kb h (g+i kb ), fo lage k. Then thee exists y k Im ( h):(f ) k (y k )+(g ) k ( y k ) 0. By Poposition 3.2 the sequence (y k ) clustes and any cluste point satisfies the equied popety. We intend now to pove that the constaint qualification in Theoem 3.5 can be eplaced by an assumption involving the gowth of f and h at infinity. To do this, we need the following poposition. Poposition 3.3. Fo f Γ 0 and locally Lipschitz h satisfying f h, suppose f(x) lim inf x x > max Then, fo all lage k, f k h. Poof. Let 0 <a<band c be such that lim sup x h(x) x, 0 }. f(x) x b, h(x) x a, fo all x such that x c. Then thee exists bf R such that f(x) + b x fo all x R n, and f has bounded level sets. Fo the sake of contadiction, suppose thee exists, fo each k N, x k such that f k (x k ) <h(x k ). Two cases can occu. (i) (x k ) is unbounded. Taking a subsequence, we can suppose x k. Fo k>b,wehavef k (x k ) +b x k. It follows that a x k h(x k )>f k (x k ) +b x k,

10 622 A. S. LEWIS AND R. E. LUCCHETTI a contadiction. (ii) (x k ) is bounded. Again taking a subsequence, we can suppose x k x. Pick m> x and so that h is -Lipschitz on mb. Since f has compact level sets, fo each k thee is y k such that h(x k ) >f k (x k )=f(y k )+k x k y k.ash(x k ) h(x), fo lage k one has f(y k ) f(y k )+k x k y k h(x)+1. Thus (y k ) is bounded and, taking anothe subsequence, we can suppose y k y. Since k x k y k h(x)+1 inf f fo all k, we deduce y = x. Thus, fo lage k, x k,y k mb, so h(x k ) h(y k )+ x k y k f(y k )+k x k y k <h(x k ), a contadiction. Theoem 3.6. Fo f,g Γ 0 and locally Lipschitz h : R n R satisfying f h g, suppose Then lim inf x } f(x) x > max h(x) lim sup x x, 0. y cl (Im h):f (y)+g ( y) 0. Poof. Fom Poposition 3.3, f k h gfo lage k. Since we know dom f k dom g = R n, Theoem 3.5 implies thee exists y cl Im h with f (y)+g ( y) (f k ) (y)+g ( y) 0. The poof of the theoem above elies on the fact that we ae able to constuct a function p Γ 0 such that f p h and whose domain contains intenal points. Howeve, to do this is not always possible, as the following example shows. Example 3.1. Let 0 if v =0, f (u, v) = othewise and h(u, v) = uv. Suppose the convex function p satisfies f p h. Then clealy p(u, 0)=0fo all u. Fo any eal u and v and positive intege, 1 p(u, v) =1 p(u, v)+ 1p(u, 0) ( p u, v ) ( = p u, v ) + p(, 0) ( u + 2p, v ) 2 2 (u + )v 2 v 2

11 NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS 623 as. Hence p(u, v) =+ wheneve v 0,sop=f. The next esult deals with the case of h being globally Lipschitz. Theoem 3.7. Fo f,g Γ 0 and globally Lipschitz h : R n R, suppose f h g. Then y cl (Im h):f (y)+g ( y) 0. Poof. Apply Theoem 3.3 to the functions f + I kb, h, and g + I kb to obtain the existence of y k Im h such that (f + I kb ) (y k )+(g+i kb ) ( y k ) 0. Since h is globally Lipschitz, (y k ) has a cluste point y. Then we conclude with the help of Poposition 3.2. The next example shows that in geneal no y Im h satisfies f (y)+g ( y) 0. Example 3.2. Let f (x) = x, 1+x 2 if x 0, g(x)= othewise, x exp( x) if x 0, h(x) = 2x 1 othewise. Then all the assumptions of Theoem 3.5 ae fulfilled, and moeove h is globally Lipschitz. We end the section by poving a unilateal esult which can be egaded as a genealized vaiational pinciple. Theoem 3.8. Fo f Γ 0 and locally Lipschitz h : R n R, suppose f h. Then cl (Im h) dom f. Poof. Choose any point z dom f and eal k> z. Define g( ) =I kb ( ) inf kb h and apply Theoem 3.5. The special case f = 0 gives the well-known vaiational esult that a locally Lipschitz function h which is bounded above satisfies 0 cl (Im h). 4. Squeeze theoems. In this section we specialize the situation studied befoe. We shall make the futhe assumption that thee is a point whee the thee functions ae equal. In this case, as we shall see, we ae able to povide moe pecise esults. We shall stat with the following easy poposition, that we state without poof. Poposition 4.1. Fo f,g Γ 0 satisfying f g, suppose thee exists x such that f(x) = g(x). Then y : f (y)+g ( y) 0}= f(x) g(x). We pove now a convex squeeze theoem. Theoem 4.1. Fo f,g Γ 0 and locally Lipschitz h : R n R satisfying f h g, suppose thee exists x R n such that f(x) = g(x). Then f(x) h(x) g(x).

12 624 A. S. LEWIS AND R. E. LUCCHETTI Poof. Without loss of geneality, we can suppose x = 0. Fo each positive intege, asf h gon 1 B, we can apply Theoem 3.1 to find x 1 B and y h(x ) with By Poposition 4.1 (f + I 1 B ) (y )+(g+i1 B ) ( y ) 0. y (f + I 1 B )(0) (g+i1 B )(0) = f(0) g(0). Since h is locally bounded, thee exists a subsequence (y k )of(y ) conveging to some y, and since x k 0 and h is closed, y h(0). The next squeeze theoem deals instead with thee locally Lipschitz functions. To pove it, we need the following poposition. Poposition 4.2. Let f : R n R be locally Lipschitz and suppose δ>0. Then f(0) + f (0,x)+δ x >f(x) fo all small nonzeo x. Poof. Suppose that f(0) = 0, that f is k-lipschitz nea 0, and that, fo the sake of contadiction, thee is a sequence (x ) such that x 0 fo all and x 0, with Thus f (0,x )+δ x f(x ). f (0, Suppose, without loss of geneality, It follows that lim sup f (0, ) x + δ f(x ) x x. x x d. Then ) x f(x ) + δ lim sup x x f( x d)+f(x ) f( x d) = lim sup x f( x d)+k x x d ) lim sup x f (0,d). f (0,d)+δ f (0,d), which is impossible. Theoem 4.2. Suppose f, h, g : R n R ae thee locally Lipschitz functions such that f h g. Moeove, suppose f(x) =g(x)fo some x. Then f(x) h(x) g(x). Poof. Suppose, without loss of geneality, f(0) = h(0) = g(0) = 0. By Poposition 4.2, fo each N, thee exists ε > 0 such that f (0,x)+ x ( h(x) ( g) (0,x)+ x )

13 NONSMOOTH DUALITY, SANDWICH, AND SQUEEZE THEOREMS 625 fo all x such that x ε. Now, take ε<ε. Then f (0,x)+ x +I εb (x) h(x) ( ( g) (0,x)+ x ) +I εb (x) fo all x R n. We can then apply Theoem 4.1 to get an element y such that ( y f (0, )+ ) +I εb ( ) (0) h(0) ( ( g) (0, )+ ) +I εb ( ) (0) = ( f(0) + 1 ) B h(0) ( g(0) + 1 ) B. Since y h(0), the sequence (y ) is bounded and any of its cluste points does the job. Coollay 4.1. Let f 1 f 2 f k : R n R be locally Lipschitz. Suppose f 1 (0) = =f k (0). Then k f i (0), i=1 povided at least one of the following conditions holds: k =1,2,3; at least one f i is smooth; n =1,2. Poof. The cases k = 2 and the case when f i is smooth follow fom the sum ule applied to f 1 f 2 and f j f i, espectively. The case k = 3 is Theoem 4.2 and the cases n = 1,2 ae consequences of Theoem 4.2 and Helly s theoem. 5. Final emaks. We have seen some sandwich and squeeze theoems, dealing with convex and locally Lipschitz functions. While the convex subdiffeential is standad, thee ae seveal notions of subdiffeential fo locally Lipschitz functions. Hee we use the Clake subdiffeential athe than, fo instance, the appoximate subdiffeential, because the latte is not suitable fo the esults we seek. Conside the following simple example. Example 5.1. Let f (x) =I [ 1,1] (x), and g(x) = x +I [ 1,1] (x), h(x) = x. Then f h gand h is (globally) Lipschitz. Howeve L = y : f (y)+g (y) 0}=0}, while the appoximate subdiffeential of h is the set 1, 1}. Finally, hee is a list of questions we leave to the inteested eade.

14 626 A. S. LEWIS AND R. E. LUCCHETTI imply imply Question 1. Does Question 2. Does i (dom f) i (dom g) L cl (Im h)? i (dom f ) i ( dom g ) L cl (Im h)? Question 3. 2 Does the nonsmooth squeeze esult of Coollay 4.1 hold moe geneally fo any n, k? REFERENCES [1] J. M. Bowein and S. P. Fitzpatick, Duality Inequalities and Sandwiched Functions, pepint, [2] F. H. Clake, Optimization and Nonsmooth Analysis, John Wiley and Sons, New Yok, [3] F. H. Clake and Yu. S. Ledyaev, Mean value inequalities, Poc. Ame. Math. Soc., 122 (1994), pp [4] J. B. Hiiat-Uuty and C. Lemaéchal, Convex Analysis and Minimization Algoithms I, Gundlehen Math. Wiss., 305, Spinge Velag, Belin, [5] J. B. Hiiat-Uuty and C. Lemaéchal, Convex Analysis and Minimization Algoithms II, Gundlehen Math. Wiss., 306, Spinge Velag, Belin, [6] A. S. Lewis and D. Ralph, A nonlinea duality esult equivalent to the Clake Ldyaev mean value inequality, Nonlinea Anal., 26 (1996), pp [7] R. T. Rockafella, Convex Analysis, Pinceton Univesity Pess, Pinceton, NJ, Afte this pape was submitted, this question was esolved in the affimative in [1].