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Topological Properties of the Approximate Subdifferential
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
207, 345]360 Ž1997.
AY975264
Topological Properties of the Approximate Subdifferential* Rene ´ Henrion Institute of Applied Mathematics, Humboldt Uni¨ ersity Berlin, D-10099 Berlin, Germany Submitted by Helene ´` Frankowska Received November 30, 1994
The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be obtained as the Kuratowski]Painleve ´ limit of approximate subdifferentials from a one-parametric family of Lipschitzian functions. Sharper characterizations are possible for strongly compact subsets. As a consequence, in any Hilbert space the approximate subdifferential of a suitable Lipschitzian function may be homeomorphic Žboth in the strong and weak topology. to the Cantor set. Further results relate the approximate subdifferential to specific topological types, to the one-dimensional case Žwhich is extraordinary in some sense., and to the value function of a C 1-optimization problem. Q 1997 Academic Press
1. INTRODUCTION The approximate subdifferential was first introduced in finite dimensions by Mordukhovich w21x via normal cones to epigraphs. An equivalent definition based on Dini subdifferentials was found by Ioffe w10x. The same authors gave several approaches for generalizing this concept to the infinite-dimensional case w19, 20, 11, 12x. Being nonconvex in general, the approximate subdifferential enjoys a minimality property among a family of ‘‘reasonable’’ subdifferentials Žcompare w10x, Theorem 9.. On the other hand, it offers a rich calculus. We merely point out the chain rules by Ioffe w12x or Jourani and Thibault w15x or the expression for the subdifferential * Research supported by grants from the Deutsche Forschungsgemeinschaft and the Graduiertenkolleg Geometrie und Nichtlineare Analysis, Humboldt University Berlin. 345 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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of marginal functions provided by Mordukhovich w22x, which all work without the need of convexification. As a consequence, many promising applications in nonsmooth optimization have been reported. For instance, Glover, Craven, and Flam ˚ w5x and Glover and Craven w6x considered first order optimality conditions in Banach spaces. More general optimality conditions were established before by El Abdouni and Thibault w4x. Other papers are concerned with the metric regularity of feasible sets w14, 16, 17x, w23x. It is interesting to note that, in finite dimensions, Mordukhovich w23x could give a complete characterization of the metric regularity of multivalued mappings, whereas in the Banach space setting Jourani and Thibault w16, 17x developed an approach via so-called compactly Lipschitzian mappings, thereby exploiting the chain rule mentioned above. Potential advantages of the approximate subdifferential might be related to the fact that it is not restricted to be a convex set. This motivates the question of which topological properties can be expected in general. It will turn out that, even for Lipschitzian functions, there is a rich variety of topological types that can occur. Starting with the basic definitions, denote by X, X * a Banach space with its dual and let f : X ª R j y`, `4 be an extended-valued function. First, recall the Dini subdifferential w9x of f at some point x g X:
y f Ž x . s
½
x* g X * < x* Ž h . F dy f Ž x ; h . ;h g X 4 , B,
f Ž x . - `, else,
where dy f Ž x; h. s lim inf u ª h, t x 0 ty1 Ž f Ž x q tu. y f Ž x .. refers to the lower Dini directional derivative of f at x in direction h. For the following denote by F the collection of finite-dimensional subspaces of X and for any subset S : X put fS Ž x . [ f Ž x . , s `,
if x g S, else.
DEFINITION 1.1 Žapproximate subdifferential .. For z g X define
¡F f Ž z . s~ ¢B, a
Lg F
lim sup y f xqLŽ x . ,
if f Ž z . - `,
xªz f Ž x .ªf Ž z .
else.
In the definition, lim sup has to be understood in the Kuratowski]Painleve ´ sense with respect to the strong topology in X and the weak*-topology in X *. More explicitly, for some multifunction K: X i X * one has x* g lim sup K Ž x . xªz f Ž x .ªf Ž z .
m there are nets xa ª z, xUa ©w* xU : f Ž xa . ª f Ž z . , xUa g K Ž xa . .
APPROXIMATE SUBDIFFERENTIAL
347
In a similar way the lim inf of such a multifunction is defined by x* g lim inf K Ž x . xªz f Ž x .ªf Ž z .
m for all nets xa ª z with f Ž xa . ª f Ž z . there is a net xUa ©w* x*, xUa g K Ž xa . .
If both kinds of limits coincide Žthe second being always contained in the first. then one simply speaks of the limit lim of K at z. Certainly, one always has the inclusion y f Ž x . : a f Ž x .. For locally Lipschitzian functions the main relation between the approximate and Clarke’s subdifferential is C f Ž x . s clco a f Ž x ., where clco refers to the convex, weak*-closure, and furthermore a f Ž z . is weak*-compact. The following lemma Žsee Proposition 2.1 and Corollary 5.1.1 in w11x. gives expressions of a f Ž z . which may be easier to handle than the original involved definition: LEMMA 1.1. One always has
a f Ž z . s x* g X * < there are nets xa ª z, xUa ©w* x*, La g F : f Ž xa . ª f Ž z . , xUa g yf x aqLa Ž xa . , La cofinal with F 4 Ž here ‘‘La cofinal with F ’’ means ;L g F ;a 'a 0 G a , L : La .. If X is a 0 Banach space which is separable or admits an equi¨ alent Gateaux-differentiable norm, then for each lower semicontinuous function f : X ª R j y`, `4 it holds that a f Ž z . s lim sup «yf Ž x . , xªz f Ž x .ªf Ž z . « x0
where «y f Ž x . is defined as
¡ x* g X < x*Ž h . F d ~ f Ž x. s . - `, ¢B,if f Ž xelse. U
«y
y
f Ž x ; h . q « 5 h 5 ;h g X 4 ,
2. RESULTS As one can already see from the simple example f Ž x . s y< x <, where a f Ž0. s y1, 14 , the approximate subdifferential need not be convex, not even connected. In a first step for characterizing its topological properties, one could look for certain conditions yielding specific topological types. In the following lemma such conditions are imposed onto the local behavior of the Dini subdifferential, which is the main ingredient of the approximate subdifferential.
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348 LEMMA 2.1. conditions:
For f : X ª R, z g X Ž Banach space. consider the following
ŽA1. y f zqLŽ z . s
lim sup xªz , f Ž x .ªf Ž z .
y f xqLŽ x . ;L g F.
ŽA2. lim inf x ª z, f Ž x .ª f Ž z . y f Ž x . / B. ŽA3. y f Ž z . is weak*-compact and there exists « ) 0 such that the relation y f Ž x . l y f Ž z . / B holds for all x with 5 x y z 5 and < f Ž x . y f Ž z .< - « . Then one has the following implications. 1. ŽA1. « a f Ž z . is con¨ ex. 2. ŽA2. « a f Ž z . is star-shaped. 3. ŽA3. « a f Ž z . is connected. 4. If dim X - ` and f is locally Lipschitzian, then ŽA3. « ŽA1. and ŽA2. « ŽA1. or a f Ž z . is a singleton. Proof. For verifying implication 1, note that the definition of a f Ž z . together with ŽA1. means
a f Ž z . s
F yf zqLŽ z . ,
Lg F
which is convex by convexity of y f zqLŽ z .. Next, let ŽA2. be fulfilled. Then there exists some y* g lim inf x ª z, f Ž x .ª f Ž z . y f Ž x .. In order to show star-shapedness of a f Ž z . it is sufficient to prove that the line segments w x*, y*x are contained in a f Ž z . for all x* g a f Ž z .. Now, let any such x* be given. By Lemma 1.1, there are nets xa ª z, xUa ©w* x*, and La : F such that f Ž xa . ª f Ž z ., xUa g y f x aqLaŽ xa ., and La is cofinal with F. According to the definition of lim inf there exists a net yaU ©w* y* with yaU g y f Ž xa . : y f x aqLaŽ xa .. Then, by convexity of the Dini subdifferential, one gets
yf x aqLa Ž xa . 2 txUa q Ž 1 y t . yaU ©w* tx* q Ž 1 y t . y* for all t g w0, 1x, but this means w x*, y*x : a f Ž z .. For proving implication 3, it is sufficient to show that for each x* g a f Ž z . there is some y* g y f Ž z . such that w x*, y*x : a f Ž z .. Then, indeed, two arbitrary points xU1 , xU2 g a f Ž z . may be joined by a path w xU1 , yU1 x j w yU1 , yU2 x j w yU2 , xU2 x which is completely contained in a f Ž z . because w yU1 , yU2 x : y f Ž z . : a f Ž z . by convexity of the Dini subdifferential. This means Žpath-. connectedness of a f Ž z .. In order to verify the afore-mentioned assumption, let any x* g a f Ž z . be given. Then the same kind of nets xa , xUa , La as in the proof of implication 2 exist. Moreover, after passing to subnets if necessary, one may assume that La is not only cofinal with F , but even
349
APPROXIMATE SUBDIFFERENTIAL
residual, i.e., for all L g F there exists an a 0 such that L : La for all a G a 0 . Now ŽA3. means existence of a net yaU g y f Ž xa . l y f Ž z . : y f x aqLaŽ xa . l y f Ž z .. By the assumed weak*-compactness of y f Ž z . we have yaUX ©w* y* for some subnet and some y* g y f Ž z .. Again convexity of the Dini subdifferential implies
y f x aXqLa X Ž xa 9 . 2 txUa 9 q Ž 1 y t . yaU9 ©w* tx* q Ž 1 y t . y*
; t g w 0, 1 x .
La being residual w.r.t. F , the subnet La 9 is easily shown to be cofinal with F , whence w x*, y*x : a f Ž z ., as desired. The last implication Ž4., was shown in w7x. By this lemma, conditions ŽA1., ŽA2., and ŽA3. imply successively weaker topological types. There are examples of lower semicontinuous functions in two variables such that ŽA2. is fulfilled but convexity is violated or ŽA3. is fulfilled but star-shapedness of the approximate subdifferential fails to hold Žsee w7x.. In this sense, the indicated conditions are specific. On the other hand, the last statement of the lemma shows that in the Žfinite-dimensional. Lipschitzian case, ŽA2. and ŽA3. imply nothing but convexity. In this situation it seems hard to find reasonable conditions similar to Lemma 2.1 which are specific for a certain topological type. This motivates a more detailed study of the locally Lipschitzian case. Before proving the main result, we shall need the following lemma which facilitates the computation of a f Ž z . according to Definition 1.1. The third assertion of the lemma could be deduced from more general results in w13x, but to make the presentation self-contained we include its proof, which in the given specific setting is quite simple. LEMMA 2.2. Let X be a Hilbert space, C : X a weakly compact subset, and f : X ª R defined by f Ž z . s min² z, x :< x g C 4 . Denote EŽ z . s x g C <² z, x : s f Ž z .4 . Then the following equalities hold: dy f Ž z ; h . s min ² x, h:< x g E Ž z . 4 , aE Ž z . s 1
«
E Ž z . s yf Ž z . ;
aE Ž z . G 2
lim sup «yf Ž z . s lim sup yf Ž z . . zª0 « x0
«
Ž 1. yf Ž z . s B, Ž 2.
Ž 3.
zª0
Proof. Since f is a concave function, its directional derivative exists and coincides with dy f Ž z; h.. On the other hand, the value of the directional derivative computes from Ž1.; see, e.g., w2, Proposition 4.4x.
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Concerning the first relation of Ž2. one concludes from EŽ z . s x 4 and from Ž1. that dy f Ž z; h. s ² x, h: ;h g X; hence y f Ž z . s x 4 . For the second relation, note that the nonemptiness of y f Ž z . implies Žby definition. dy f Ž z; h. G ydy f Ž z; yh. ;h g X. Now, choosing x 1, x 2 g EŽ z ., x 1 / x 2 , and assuming y f Ž z . / B we obtain from Ž1. for arbitrary h g X, min ² x 1 , h: , ² x 2 , h: 4 G min ² x, h: s dy f Ž z ; h . G ydy f Ž z ; yh . xgE Ž z .
s y min ² x, yh: s max ² x, h: xgE Ž z .
xgE Ž z .
G max ² x 1 , h: , ² x 2 , h: 4 . This inequality holding for all h g X, one gets the contradiction x 1 s x 2 . Finally, for proving Ž3., first observe that the inclusion = is trivially fulfilled because of «y = y. For the reverse inclusion let x g lim sup «yf Ž z . . zª0 « x0
This means existence of nets zl ª 0, xl ©w x, «l ª 0, xl g «yl f Ž zl ., and «l ) 0. Let U be a Žstrong. open neighborhood of 0 and V be a weak open neighborhood of x. Then V contains a base neighborhood of the type W s y g X < ² y y x, y i : - « , y i g X , i s 1, . . . , n4 and for l G l0 one has <² xl y x, y i :< - «r2 Ž i s 1, . . . , n.. This implies y g W : V whenever 5 y y xl 5 - «rŽ2Ž1 q max 5 y i 5... Summarizing, the existence of an index l* such that zl* g U and xl* q B«l* : V follows. The last relation yields EŽ zl* . : V. Indeed, since xl* g «yl* f Ž zl* ., one has for any ¨ g EŽ zl* . and h g X that wcompare the definition of «y f and Ž1.x: ² xl* , h: F min ² ¨ 9, h:< ¨ 9 g E Ž zl* . 4 q «l* 5 h 5 F ² ¨ , h: q «l* 5 h 5 . From this it follows that 5 xl* y ¨ 5 F «l* ; hence EŽ zl* . : xl* q B«l* : V. Since C _ V is weakly compact, this last inclusion together with continuity of f implies existence of some Žstrong. open neighborhood of U9 of zl* such that EŽ y . : V ; y g U9. As a concave function f is Gateaux-differentiable on a dense subset of X w1x. Now, U l U9 is a nonempty set which consequently must contain some point y such that f possesses some Gateaux derivative y* at y. Then, by Ž2.,
y f Ž y . s y* 4 s E Ž y . : V .
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APPROXIMATE SUBDIFFERENTIAL
Summarizing, to each pair U, V of Žstrong, weak. open neighborhoods of 0, x we can assign y wU, V x g U, yUwU, V x g V such that yUwU, V x g y f Ž y wU, V x .. In this way one obtains converging nets ym ª 0, ymU ©w x, and ymU g y f Ž ym . Ž m s wU, V x.; hence x g lim sup yf Ž z . zª0
as desired. Combining Ž3. in Lemma 2.2 with Lemma 1.1 one gets COROLLARY 2.1. Under the assumption of Lemma 2.2 the approximate subdifferential of f Ž defined in the lemma. computes as
a f Ž 0 . s lim sup yf Ž z . . zª0
As was stated in the introductory section, the approximate subdifferential of a locally Lipschitzian function defined on a Banach space is weak*-compact. In turn, the following theorem shows that in a Hilbert space setting each weakly compact set may be approximated by subdifferentials of Lipschitzian functions. THEOREM 2.1. For any weakly compact subset K of any Hilbert space Ž H , ² :. there exists a one-parametric family f u : R = H ª R of Lipschitzian functions with some common modulus L, such that lim ux 0 a f uŽ0, 0. s 04 = K holds in the sense of Kuratowski]Painle¨ ´ e con¨ ergence. Proof. Let c be a constant with 5 x 5 2 - c ; x g K Žwhich exists by weak compactness of K . and fix a parameter u with 0 - u F 1rc. In R = H Žwhich is a Hilbert space by canonical extension of the inner product. consider the ellipsoidal surface Eu s Ž t , x . g R = H < u 5 x 5 2 q Ž t y 1 . s 1 4 2
as well as the subset Ku s Ž t, x. g R = H < x g K , t s 1 y
½
'1 y u 5 x 5 5 . 2
Note that K u is correctly defined and K u : Eu . Furthermore, the function f u : R = X ª R with f u Ž a , z . s min ² Ž a , z . , Ž t , x . :< Ž t , x . g clco K u 4
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is correctly defined as well Žby weak compactness of clco K u .. Since K u : w 0, 1 x = B Ž 0; 'c . : B Ž Ž 0, 0 . ; 'c q 1 . , one has clco K u : B ŽŽ0, 0.; 'c q 1 .; hence 'c q 1 is a modulus of Lipschitz continuity for f u Žnot depending on u.. Next we show the validity of the following relations which are essential for proving the assertion of the theorem: pr Ž a f u Ž 0, 0 . < 0 4 = H . s 0 4 = K ,
Ž 4.
pr Ž a f u Ž 0, 0 .
Ž 5.
Here, prŽ?< M . denotes projection onto a closed subspace of M of R = H . To show first the inclusion 04 = K : prŽ a f uŽ0, 0.< 04 = H . it is sufficient to verify the relation K u : a f uŽ0, 0. because of prŽ K u < 04 = H . s 04 = K. So let Ž t, x . g K u . For n g N put a n s Ž1 y t .rn and z n s yŽ urn. x. Then, the Kuhn]Tucker conditions Žrecall that for u ) 0, Eu is a regular surface . yield that the linear function ²Ž a n , z n ., ? : restricted to Eu attains its minimum at the unique point Ž t, x .. Since Ž t, x . g K u : Eu and by a convexity argument one has
Ž t , x . 4 s arg min ² Ž a n , z n . , Ž t9, x9 . :< Ž t9, x9 . g K u 4 s arg min ² Ž a n , z n . , Ž t9, x9 . :< Ž t9, x9 . g clco K u 4 . Therefore, with the notation introduced in Lemma 2.2, EŽ a n , z n . s Ž t, x .4 and Ž2. implies y f uŽ a n , z n . s Ž t, x .4 . Together with Ž a n , z n . ª Ž0, 0. and using the trivial sequence Ž t n , x n . ' Ž t, x ., Corollary 2.1 provides
Ž t , x . g lim sup yf u Ž a , z . s a f u Ž 0, 0 . . Ž a , z .ª Ž0, 0 .
Before proving the reverse inclusion of Ž4. and Ž5. we show first the relation
Ž t*, x* . g y f u Ž a , z .
«
x* g K ,
t* g 0, 1 y '1 y uc . Ž 6 .
Indeed, from Ž2. in Lemma 2.2 one has
Ž t*, x*. 4 s arg min ² Ž a , z . , Ž t9, x9 . :< Ž t9, x9 . g clco K u 4
Ž 7.
and by convexity of clco K u there exists a sequence Ž t n , x n .4 : K u such that ²Ž a , z ., Ž t n , x n .: ªn ²Ž a , z ., Ž t*, x*.:. Weak compactness of clco K u then yields the existence of a weakly convergent subsequence Ž t n l , x n l . ©l
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APPROXIMATE SUBDIFFERENTIAL
Žt , y . g clco K u . Consequently, ²Ž a , z ., Žt , y .: s ²Ž a , z ., Ž t*, x*.:; hence Žt , y . s Ž t*, x*. because of Ž7.. It follows that x n ©l x* with x n g K l l wsince Ž t n , x n . g K u and by definition of K u x. Weak compactness of K l l yields x* g K, as desired. On the other hand, from the definition of K u it follows that 0 F t n l F 1 y '1 y uc , which proves the second implication of Ž6.. Now, let
Ž t , x . g a f u Ž 0, 0 . s lim sup yf u Ž a , z . Ž a , z .ª Ž0, 0 .
Žcompare Corollary 2.1., i.e., there exist nets Ž al, zl . ª Ž0, 0. and Ž tl , xl . © Ž t, x . with Ž tl, xl . g y f uŽ al , zl .. Equation Ž6. yields xl g K and tl g w0, 1 y '1 y uc x, whence x g K by weak compactness of K and t g w0, 1 y '1 y uc x. Thus, Ž4. and Ž5. are completely proved. In order to verify the assertion of the theorem it suffices to show the relation lim sup a f u Ž 0, 0 . : 0 4 = K : lim inf a f u Ž 0, 0 . , ux0
ux0
Ž 8.
which directly implies equality between all three sets. Considering first some element Ž t, x . g lim sup u x 0 a f uŽ0, 0. one has converging nets ul ª 0, Ž tl, xl . © Ž t, x ., and Ž tl , xl . g a f ulŽ0, 0.. Then Ž4. implies xl g K which ensures x g K because of weak compactness of K. Moreover, Ž5. provides tl g w0, 1 y 1 y ul c x. Therefore tl ª 0 and t s 0, as was to be proved. Concerning the second inclusion let x g K. We have to show that Ž0, x . g lim inf ux 0 a f uŽ0, 0.. To this aim consider an arbitrary net ul ª 0. Put xl ' x and choose}by virtue of Ž4. and Ž5. }some tl with Ž tl , x . g a f ulŽ0, 0. and 0 F tl F 1 y 1 y ul c . Obviously, Ž tl , xl . © Ž0, x ., as desired.
'
'
It is clear that a similar approximation result as in Theorem 2.1 does not hold, for instance, for Clarke’s subdifferential because of the restriction to convexity. This will become even clearer from the following topological derivations of the theorem. LEMMA 2.3. For any strongly compact subset K of any Hilbert space H there exists a Lipschitzian function f : R = H ª R such that a f Ž0, 0. is homeomorphic to K in the strong topology. If moreo¨ er K is contained in some finite-dimensional subspace of H , then a f Ž0, 0. is homeomorphic to K in the weak topology too. Proof. Fix any admissible parameter u in Theorem 2.1 Žsee beginning of the proof. and consider the corresponding set K u . Via the functions f : K ª K u and c : K u ª K defined by f Ž x . s Ž1 y 1 y u 5 x 5 2 , x . and
'
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c Ž t, x . s x we see that f defines a bijection from K to K u with inverse fy1 s c . Moreover, f is strongly continuous while c is both strongly and weakly continuous. Hence f defines a homeomorphism between K and K u in the strong topology. In particular K u inherits strong compactness from K. Now we show that a f Ž0, 0. s K u . Define the Lipschitzian function f : R = H ª R by f Ž a , z . s min²Ž a , z ., Ž t, x .:<Ž t, x . g K u4 which is justified by strong Žhence weak. compactness of K u . Repeating the lines of argumentation in the proof of Theorem 2.1 Žbut restricting considerations to K u itself instead of its convex closure. one derives the inclusion K u : a f Ž0, 0.. The reverse inclusion would follows from the relation y f Ž a , z . : K u ;Ž a , z . due to weak compactness of K u and to the definition of the approximate subdifferential. To verify the mentioned relation, note that
Ž t*, x* . g yf Ž a , z . « Ž t*, x* . 4 s arg min ² Ž a , z . , Ž t9, x9 . :< Ž t9, x9 . g K u 4 as a consequence of Ž2. in Lemma 2.2. In particular, Ž t*, x*. g K u , as was to be shown. Concerning the second part of the lemma, note that in this case the function f , introduced in the beginning of this proof, becomes weakly continuous as was also true for its inverse c . Therefore f is a homeomorphism between K and K u in the weak topology too. This lemma extends a corresponding result in w7x ŽTheorem 3.2. for finite-dimensional H . It allows the following topological characterization of the approximate subdifferential: COROLLARY 2.2. For any Hilbert space H there exists a Lipschitzian function f : R = H ª R such that a f Ž0, 0. is homeomorphic to the Cantor set in the weak as well as the strong topology. In particular, a f Ž0, 0. is totally disconnected in both topologies. Finally we note that in w7x it was proved that for each compact subset K : R n there exists a locally Lipschitzian function f : R nq 2 ª R such that the so-called partial approximate subdifferential Žintroduced by Jourani and Thibault w14x in the context of functions depending on a parameter. of f coincides Žat the origin. with 04 = K : R nq 1. Here, any given compact subset may be even realized itself Žup to imbedding into a space with one extra dimension.. Revisiting Lemma 2.3 and Corollary 2.2 one notices that for the definition of the desired Lipschitzian function one additional dimension is needed. In particular, the construction Žaccording to Corollary 2.2. of an approximate subdifferential being totally disconnected requires the domain of the Lipschitzian function f to have at least dimension 2. This suggests a question about possible topological types of the approximate
APPROXIMATE SUBDIFFERENTIAL
355
subdifferential in the one-dimensional case. Theorem 2.2 below indicates, that for real continuous functions f : R ª R the approximate subdifferential is either an interval or the disjoint union of two intervals, which is obviously a topological restriction. For the proof of this one-dimensional characterization two auxiliary results are needed. To this aim, for functions f : R ª R we introduce the following notations Žwith z g R.: d l Ž z . s lim inf ty1 Ž f Ž z q tu . y f Ž z . . ; t x0 uªy1
d r Ž z . s lim inf ty1 Ž f Ž z q tu . y f Ž z . . , t x0 uª1
where improper values "` are allowed. Obviously,
y f Ž z . s yd l Ž z . , d r Ž z .
Ž 9.
holds, with the interval to be interpreted appropriately for improper values. PROPOSITION 2.1. Then
Let f : R ª R be continuous, and t 1 , t 2 g R, t 1 - t 2 .
d r Ž t 1 . - yd l Ž t 2 .
« ;c g Ž d r Ž t 1 . , yd l Ž t 2 . . ' t g Ž t 1 , t 2 . : c g yd l Ž t . , d r Ž t . s y f Ž t . ,
d r Ž t 1 . ) yd l Ž t 2 .
Ž 10 .
« ;c g Ž yd l Ž t 2 . , d r Ž t 1 . . ' t g Ž t 1 , t 2 . : c g d r Ž t . , yd l Ž t . .
Ž 11 .
Proof. Concerning Ž10. let c g Ž d r Ž t 1 ., yd l Ž t 2 ... Then the function x ¬ f Ž x . y cx, restricted to x g w t 1 , t 2 x, attains its minimum at a point of the open interval Ž t 1 , t 2 .. To see this, assume that t 1 is a minimizer. It follows that f Ž t q t 1 . y f Ž t 1 . G ct ; t g w 0, t 2 y t 1 x . This however yields lim inf ty1 Ž f Ž t 1 q ut . y f Ž t 1 . . G c, t x0 uª1
leading to the contradiction c F d r Ž t 1 .. In a similar way t 2 may be excluded as a minimizer via the contradiction c G yd l Ž t 2 .. As a consequence, there exists t g Ž t 1 , t 2 . such that f Ž x . y cx G f Ž t . y ct ; x g w t 1 , t 2 x. Choose d ) 0 with Ž t y d , t q d . : Ž t 1 , t 2 . to get f Ž t q t . y f Ž t . G ct ; t g Ž yd , d . , Ž . Ž . which immediately implies 10 . For 11 consider the maximum Žrather than minimum. of f Ž x . y cx and repeat the same arguments.
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LEMMA 2.4. Let f : R ª R be continuous and y f Ž z . / B for some z g R. Then a f Ž z . is a closed Ž possibly unbounded. inter¨ al. Proof. By assumption, there is some e g y f Ž z . : a f Ž z .. It will be sufficient to show that a g a f Ž z ., a - e implies Ž a, e . : a f Ž z . Žthe proof is running along similar lines for a ) e .. The general definition, Definition 1.1, of the approximate subdifferential reduces in the present constellation Žfinite dimensions, continuous function. to a f Ž z . s lim sup x ª z y f Ž x ., where the lim sup may be generated by sequences instead of nets. Accordingly, there exist sequences t k ª 0,
a k ª a,
a k g yf Ž z q t k . .
Ž 12 .
Let c g Ž a, e . be arbitrarily given. We have to show that c g a f Ž z .. For k G k 0 one has c ) a k and we may suppose without loss of generality that c f y f Ž z q t k . j y f Ž z . wsince otherwise the assertion c g a f Ž z . follows immediatelyx. Combining Ž12. with Ž9. one gets yd l Ž z q t k . F a k F d r Ž z q t k . - c - yd l Ž z . F e F d r Ž z .
;k G k 0 .
Ž 13 . As an obvious consequence of this relation we have t k / 0 ;k. One of the following two cases must hold true: Case 1. There is a negative subsequence t k p - 0. Application of Ž10. to the relation d r Ž z q t k p . - c - yd l Ž z . wsee Ž13.x yields existence of tp g Ž t k , 0. with c g y f Ž z q tp .. Then c g a f Ž z . due to t p ª 0. p
Case 2. There is a positive subsequence t k p ) 0. Fix an arbitrary c g Ž c, e . and apply Ž11. to the relation d r Ž z . ) c ) yd l Ž z q t k p . wcompare Ž13.x. Hence, there exists tp g Ž0, t k p . with c F yd l Ž z q tp .. Next choose an index q Ž p . to fulfill t k qŽ p. - tp . Application of Ž10. to the relation d r Ž z q t k qŽ p. . - c - c F yd l Ž z q tp . finally provides c g y f Ž z q m p . for some m p g Ž t k qŽ p., tp .. Now c g a f Ž z . due to m p ª 0. According to Lemma 2.4 nonemptiness of the Dini subdifferential implies convexity of the approximate subdifferential in one dimension. Unfortunately, a generalization of this fact to the multidimensional case is not possible even in the Lipschitzian case. This can be seen from the function f : R 2 ª R defined by f Ž x, y . s max 0, min x, y44 . Here y f Ž0, 0. s 0, 04 / B, while a f Ž0, 0. s wŽ0, 0., Ž0, 1.x j wŽ0, 0., Ž1, 0.x Žwith brackets referring to line segments., which is obviously not convex. THEOREM 2.2. Let f : R ª R be continuous. Then a f Ž z . contains at most two connected components Ž for all z g R..
APPROXIMATE SUBDIFFERENTIAL
357
Proof. Negating the assertion means the existence of elements a b - c with a, b, c g a f Ž z . and which are contained in three pairwise different connected components. By definition there are sequences t k , t k , m k ª 0, a k ª a, bk ª b, c k ª c, a k g yf Ž z q t k . ,
bk g yf Ž z q t k . ,
c i g yf Ž z q m k . . Ž 14 .
As a consequence of Lemma 2.4 one has t k , t k , m k / 0 ;k; otherwise a k Žor bk or c k . g y f Ž z . for some k, so a f Ž z . would be an interval in contradiction to our assumption of at least three connected components. Passing to appropriate subsequences one may assume, without loss of generality, that each of the sequences t k 4 , t k 4 , m k 4 in Ž14. has a constant Ždefinite. sign. For two of them this sign must be equal, so we have t k - 0, t k - 0 ;k or
t k ) 0, t k ) 0 ;k
Ž 15 .
Žthe subsequent proof being identical for the remaining possible pairs t k 4 , m k 4 , and t k 4 , m k 4.. We are done if we can show that Ž a, b . : a f Ž z ., because then w a, b x : a f Ž z . in contradiction to the assumption that a and b come from different connected components of a f Ž z .. So choose any e g Ž a, b .. For some k 0 it holds that a k - e - bk and e f y f Ž z q t k . j y f Ž z q t k . ;k G k 0 Žnegating the second relation would immediately provide the desired result e g a f Ž z ... Consequently one gets wcompare Ž9.x yd l Ž z q t k . F a k F d r Ž z q t k . - e - yd l Ž z q t k . F bk F d r Ž z q t k . .
Ž 16 . From this chain it is evident that t k / t k ;k G k 0 , so one of the following two cases holds true: Case 1. There exists a subsequence such that t k p - t k p ; p. Then, Ž10. applied to the relation d r Ž z q t k p . - e - yd l Ž z q t k p . yields the existence of some a p g Ž t k p, t k p . such that e g y f Ž z q a p .. It follows e g a f Ž z . since a p ª 0. Case 2. There exists a subsequence such that t k p ) t k p; p. First we consider the situation t k - 0, t k - 0 ;k wsee Ž15.x. Fix an arbitrary e g Ž a, e .. Again we may assume that a k - e. Then, Ž11. applied to the p relation d r Ž z q t k p . ) e ) yd l Ž z q t k p . yields the existence of some b p g Žt k , t k . such that e G d r Ž z q b p .. Obviously, b p - 0 and b p ª 0. In p p particular we may select an index q Ž p . with t k qŽ p. ) b p . Then, Ž10. applied to the relation d r Ž z q b p . F e - e - yd l Ž z q t k qŽ p. . yields the existence of some gp g Ž b p , t k qŽ p. . such that e g y f Ž z q gp .. Since gp ª 0 we arrive at e g a f Ž z .. The second situation of Ž15. is treated in a similar manner, but now by choosing e g Ž e, b ., deriving the relation e F yd l Ž z q b p ., and considering a sequence t k qŽ p. - b p .
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We note that, in the case of locally Lipschitzian functions, the result of Theorem 2.2 can also be derived from w3x. From an example in the same reference one can see that on any countable subset of the line, which in particular could be dense, the approximate subdifferential of some suitably defined Lipschitz function is disconnected. The following proposition demonstrates that such behavior is possible even for a very nice nonsmooth function like the value function of a parametric optimization problem of class C 1. PROPOSITION 2.2. There exists a function f g C 1 ŽR 2 , R., such that for the corresponding minimum function g Ž z . s min f Ž x, z .< x g w0, 1x4 the approximate subdifferential a g Ž z . is disconnected Ž noncon¨ ex in particular . on a dense set of points z g R. Proof. Denote by Z the set of all z g R representable as z s "i2yj Ž i, j g N.. Then Z is a dense subset of R and it is possible to construct a C 1-function f : R 2 ª R with the properties: ; z g Z,
EŽ z . s x1 Ž z . , x 2 Ž z . 4
f
Ž x1 Ž z . , z . / z
f z
Ž x2 Ž z. , z., Ž 17 .
where EŽ z . s x g w0, 1x< f Ž y, z . G f Ž x, z . ; y g w0, 1x4 . The details of this construction are described in w8x in the context of maximum functions with a dense subset of nondifferentiability. Now consider the minimum function g Ž z . s min f Ž x, z . < x g w 0, 1 x 4 , which has the one-sided directional derivative dg Ž z ; h . s min
½
f z
Ž x, z . ? h < x g E Ž z .
5
; z, h g R.
Therefore, the lower Dini directional derivative dy g Ž z; h. Žsee the Introduction. coincides with dg Ž z; h. and the Dini subdifferential simply becomes the ‘‘interval’’ w f l Ž z ., f r Ž z .x, where
f l Ž z . s max f r Ž z . s min
½ ½
f z f z
Ž x, z . < x g E Ž z . ,
5 Ž .5
Ž x, z . < x g E z
.
Since f l Ž z . G f r Ž z . it holds that y g Ž z . s f l Ž z .4 l f r Ž z .4 for all z g R. Now, for some fixed z g Z we have by Ž17. Žwithout loss of generality., EŽ z . s x1 , x 2 4 ,
a1 ) a2 ,
where a i s
f z
Ž x i , z . Ž i s 1, 2 . .
359
APPROXIMATE SUBDIFFERENTIAL
It follows that y g Ž z . s B. Moreover, continuity of the function fr z implies that ;« ) 0 'd ) 0,
EŽ z . :
½
x 1 q « w y1, 1 x ,
if z g Ž z y d , z . ,
x q « w y1, 1 x ,
if z g Ž z, z q d . .
2
Therefore, lim z z f l Ž z . s f l Ž z . s a 1 and lim z x z f r Ž z . s f r Ž z . s a 2 . Summarizing we arrive at
a g Ž z . s lim sup f l Ž z . 4 l f r Ž z . 4 : lim sup f l Ž z . , f r Ž z . 4 s a 1 , a 2 4 . zªz
zªz
On the other hand, g being locally Lipschitzian, there are sequences z n x z and z n z, such that g 9Ž z n . exists; hence y g Ž z n . s g 9Ž z n .4 s f l Ž z n .4 s f r Ž z n .4 . It results that lim y g Ž z n . s lim f l Ž z n . 4 s a 1 4 ,
zn z
zn z
y
lim g Ž z n . s lim f r Ž z n . 4 s a 2 4 ,
zn x z
znx z
whence a g Ž z . s a 1 , a 2 4 . Since z g Z was arbitrarily chosen, the approximate subdifferential is disconnected on a dense subset of R. ACKNOWLEDGMENTS The author would like to thank Professor Thibault ŽUniversity of Montpellier, France. and ŽHumboldt University, Berlin, Germany. for helpful discussions on the Professor Romisch ¨ subject of this paper.
REFERENCES 1. E. Asplund, Frechet differentiability of convex functions, Acta Math. 121 Ž1968., 31]47. ´ 2. J.-P. Aubin, ‘‘Optima and Equilibria,’’ Springer-Verlag, New York, 1993. 3. J. M. Borwein and S. Fitzpatrick, Characterization of Clarke subgradients among onedimensional multifunctions, CECM Preprint 94:006 Ž1994.. 4. B. El Abdouni and L. Thibault, Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces, Optimization 26 Ž1992., 277]285. 5. B. M. Glover, B. D. Craven, and S. D. Flam, ˚ A generalized Karush]Kuhn]Tucker optimality condition without constraint qualification using the approximate subdifferential, Numer. Funct. Anal. Optim. 14, Ž1993., 333]353. 6. B. M. Glover and B. D. Craven, A Fritz John optimality condition using the approximate subdifferential, J. Optim. Theory Appl. 82 Ž1994., 253]265. 7. R. Henrion, Topological Characterization of the Approximate Subdifferential in the Finite-Dimensional Case, ZOR}Math. Methods Oper. Res. 41, Ž1995., 161]173. 8. R. Henrion, On maximum functions with a dense set of points of nondifferentiability, in ‘‘Approximation and Optimization,’’ Vol. 3, Peter Lang, Frankfurt am Main, 1993. 9. A. D. Ioffe, Calculus on Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal. 8 Ž1984., 517]539.
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10. A. D. Ioffe, Approximate subdifferentials and applications. I: The finite-dimensional theory, Trans. Amer. Math. Soc. 281 Ž1984., 389]416. 11. A. D. Ioffe, Approximate subdifferentials and applications. II: Functions on locally convex spaces, Mathematika 33 Ž1986., 111]128. 12. A. D. Ioffe, Approximate subdifferentials and applications. 3: The metric theory, Mathematika 36 Ž1989., 1]38. 13. A. D. Ioffe, Proximal analysis and approximate subdifferential, J. London Math. Soc. 41 Ž1990., 175]192. 14. A. Jourani and L. Thibault, Approximate subdifferential and metric regularity: The finite-dimensional case, Math. Programming 47 Ž1990., 203]218. 15. A. Jourani and L. Thibault, The approximate subdifferential of composite functions, Bull. Austral. Math. Soc. 47 Ž1993., 443]455. 16. A. Jourani and L. Thibault, Approximations and metric regularity in mathematical programming in Banach space, Math. Oper. Res. 18, Ž1993., 390]401. 17. A. Jourani and L. Thibault, Metric regularity for strongly compactly Lipschitzian mappings, Nonlinear Anal. 24 Ž1995., 229]240. 18. A. Jourani and L. Thibault, Extensions of subdifferential calculus rules in Banach spaces and applications, submitted for publication. 19. A. Y. Kruger and B. S. Mordukhovich, Extreme points and Euler equations in nondifferentiable optimization problems, Dokl. Akad. Nauk BSSR 24 Ž1980., 684]687. 20. A. Y. Kruger, Properties of generalized differentials, Siberian Math. J. 26 Ž1985., 822]832. 21. B. S. Mordukhovich, The maximum principle in the problem of time-optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 Ž1976., 960]969. 22. B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization, SIAM J. Appl. Math. 58 Ž1992.. 23. B. S. Mordukhovich, Complete characterization of openness, metric regularity and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 Ž1993., 1]35.
207, 345]360 Ž1997.
AY975264
Topological Properties of the Approximate Subdifferential* Rene ´ Henrion Institute of Applied Mathematics, Humboldt Uni¨ ersity Berlin, D-10099 Berlin, Germany Submitted by Helene ´` Frankowska Received November 30, 1994
The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be obtained as the Kuratowski]Painleve ´ limit of approximate subdifferentials from a one-parametric family of Lipschitzian functions. Sharper characterizations are possible for strongly compact subsets. As a consequence, in any Hilbert space the approximate subdifferential of a suitable Lipschitzian function may be homeomorphic Žboth in the strong and weak topology. to the Cantor set. Further results relate the approximate subdifferential to specific topological types, to the one-dimensional case Žwhich is extraordinary in some sense., and to the value function of a C 1-optimization problem. Q 1997 Academic Press
1. INTRODUCTION The approximate subdifferential was first introduced in finite dimensions by Mordukhovich w21x via normal cones to epigraphs. An equivalent definition based on Dini subdifferentials was found by Ioffe w10x. The same authors gave several approaches for generalizing this concept to the infinite-dimensional case w19, 20, 11, 12x. Being nonconvex in general, the approximate subdifferential enjoys a minimality property among a family of ‘‘reasonable’’ subdifferentials Žcompare w10x, Theorem 9.. On the other hand, it offers a rich calculus. We merely point out the chain rules by Ioffe w12x or Jourani and Thibault w15x or the expression for the subdifferential * Research supported by grants from the Deutsche Forschungsgemeinschaft and the Graduiertenkolleg Geometrie und Nichtlineare Analysis, Humboldt University Berlin. 345 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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of marginal functions provided by Mordukhovich w22x, which all work without the need of convexification. As a consequence, many promising applications in nonsmooth optimization have been reported. For instance, Glover, Craven, and Flam ˚ w5x and Glover and Craven w6x considered first order optimality conditions in Banach spaces. More general optimality conditions were established before by El Abdouni and Thibault w4x. Other papers are concerned with the metric regularity of feasible sets w14, 16, 17x, w23x. It is interesting to note that, in finite dimensions, Mordukhovich w23x could give a complete characterization of the metric regularity of multivalued mappings, whereas in the Banach space setting Jourani and Thibault w16, 17x developed an approach via so-called compactly Lipschitzian mappings, thereby exploiting the chain rule mentioned above. Potential advantages of the approximate subdifferential might be related to the fact that it is not restricted to be a convex set. This motivates the question of which topological properties can be expected in general. It will turn out that, even for Lipschitzian functions, there is a rich variety of topological types that can occur. Starting with the basic definitions, denote by X, X * a Banach space with its dual and let f : X ª R j y`, `4 be an extended-valued function. First, recall the Dini subdifferential w9x of f at some point x g X:
y f Ž x . s
½
x* g X * < x* Ž h . F dy f Ž x ; h . ;h g X 4 , B,
f Ž x . - `, else,
where dy f Ž x; h. s lim inf u ª h, t x 0 ty1 Ž f Ž x q tu. y f Ž x .. refers to the lower Dini directional derivative of f at x in direction h. For the following denote by F the collection of finite-dimensional subspaces of X and for any subset S : X put fS Ž x . [ f Ž x . , s `,
if x g S, else.
DEFINITION 1.1 Žapproximate subdifferential .. For z g X define
¡F f Ž z . s~ ¢B, a
Lg F
lim sup y f xqLŽ x . ,
if f Ž z . - `,
xªz f Ž x .ªf Ž z .
else.
In the definition, lim sup has to be understood in the Kuratowski]Painleve ´ sense with respect to the strong topology in X and the weak*-topology in X *. More explicitly, for some multifunction K: X i X * one has x* g lim sup K Ž x . xªz f Ž x .ªf Ž z .
m there are nets xa ª z, xUa ©w* xU : f Ž xa . ª f Ž z . , xUa g K Ž xa . .
APPROXIMATE SUBDIFFERENTIAL
347
In a similar way the lim inf of such a multifunction is defined by x* g lim inf K Ž x . xªz f Ž x .ªf Ž z .
m for all nets xa ª z with f Ž xa . ª f Ž z . there is a net xUa ©w* x*, xUa g K Ž xa . .
If both kinds of limits coincide Žthe second being always contained in the first. then one simply speaks of the limit lim of K at z. Certainly, one always has the inclusion y f Ž x . : a f Ž x .. For locally Lipschitzian functions the main relation between the approximate and Clarke’s subdifferential is C f Ž x . s clco a f Ž x ., where clco refers to the convex, weak*-closure, and furthermore a f Ž z . is weak*-compact. The following lemma Žsee Proposition 2.1 and Corollary 5.1.1 in w11x. gives expressions of a f Ž z . which may be easier to handle than the original involved definition: LEMMA 1.1. One always has
a f Ž z . s x* g X * < there are nets xa ª z, xUa ©w* x*, La g F : f Ž xa . ª f Ž z . , xUa g yf x aqLa Ž xa . , La cofinal with F 4 Ž here ‘‘La cofinal with F ’’ means ;L g F ;a 'a 0 G a , L : La .. If X is a 0 Banach space which is separable or admits an equi¨ alent Gateaux-differentiable norm, then for each lower semicontinuous function f : X ª R j y`, `4 it holds that a f Ž z . s lim sup «yf Ž x . , xªz f Ž x .ªf Ž z . « x0
where «y f Ž x . is defined as
¡ x* g X < x*Ž h . F d ~ f Ž x. s . - `, ¢B,if f Ž xelse. U
«y
y
f Ž x ; h . q « 5 h 5 ;h g X 4 ,
2. RESULTS As one can already see from the simple example f Ž x . s y< x <, where a f Ž0. s y1, 14 , the approximate subdifferential need not be convex, not even connected. In a first step for characterizing its topological properties, one could look for certain conditions yielding specific topological types. In the following lemma such conditions are imposed onto the local behavior of the Dini subdifferential, which is the main ingredient of the approximate subdifferential.
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348 LEMMA 2.1. conditions:
For f : X ª R, z g X Ž Banach space. consider the following
ŽA1. y f zqLŽ z . s
lim sup xªz , f Ž x .ªf Ž z .
y f xqLŽ x . ;L g F.
ŽA2. lim inf x ª z, f Ž x .ª f Ž z . y f Ž x . / B. ŽA3. y f Ž z . is weak*-compact and there exists « ) 0 such that the relation y f Ž x . l y f Ž z . / B holds for all x with 5 x y z 5 and < f Ž x . y f Ž z .< - « . Then one has the following implications. 1. ŽA1. « a f Ž z . is con¨ ex. 2. ŽA2. « a f Ž z . is star-shaped. 3. ŽA3. « a f Ž z . is connected. 4. If dim X - ` and f is locally Lipschitzian, then ŽA3. « ŽA1. and ŽA2. « ŽA1. or a f Ž z . is a singleton. Proof. For verifying implication 1, note that the definition of a f Ž z . together with ŽA1. means
a f Ž z . s
F yf zqLŽ z . ,
Lg F
which is convex by convexity of y f zqLŽ z .. Next, let ŽA2. be fulfilled. Then there exists some y* g lim inf x ª z, f Ž x .ª f Ž z . y f Ž x .. In order to show star-shapedness of a f Ž z . it is sufficient to prove that the line segments w x*, y*x are contained in a f Ž z . for all x* g a f Ž z .. Now, let any such x* be given. By Lemma 1.1, there are nets xa ª z, xUa ©w* x*, and La : F such that f Ž xa . ª f Ž z ., xUa g y f x aqLaŽ xa ., and La is cofinal with F. According to the definition of lim inf there exists a net yaU ©w* y* with yaU g y f Ž xa . : y f x aqLaŽ xa .. Then, by convexity of the Dini subdifferential, one gets
yf x aqLa Ž xa . 2 txUa q Ž 1 y t . yaU ©w* tx* q Ž 1 y t . y* for all t g w0, 1x, but this means w x*, y*x : a f Ž z .. For proving implication 3, it is sufficient to show that for each x* g a f Ž z . there is some y* g y f Ž z . such that w x*, y*x : a f Ž z .. Then, indeed, two arbitrary points xU1 , xU2 g a f Ž z . may be joined by a path w xU1 , yU1 x j w yU1 , yU2 x j w yU2 , xU2 x which is completely contained in a f Ž z . because w yU1 , yU2 x : y f Ž z . : a f Ž z . by convexity of the Dini subdifferential. This means Žpath-. connectedness of a f Ž z .. In order to verify the afore-mentioned assumption, let any x* g a f Ž z . be given. Then the same kind of nets xa , xUa , La as in the proof of implication 2 exist. Moreover, after passing to subnets if necessary, one may assume that La is not only cofinal with F , but even
349
APPROXIMATE SUBDIFFERENTIAL
residual, i.e., for all L g F there exists an a 0 such that L : La for all a G a 0 . Now ŽA3. means existence of a net yaU g y f Ž xa . l y f Ž z . : y f x aqLaŽ xa . l y f Ž z .. By the assumed weak*-compactness of y f Ž z . we have yaUX ©w* y* for some subnet and some y* g y f Ž z .. Again convexity of the Dini subdifferential implies
y f x aXqLa X Ž xa 9 . 2 txUa 9 q Ž 1 y t . yaU9 ©w* tx* q Ž 1 y t . y*
; t g w 0, 1 x .
La being residual w.r.t. F , the subnet La 9 is easily shown to be cofinal with F , whence w x*, y*x : a f Ž z ., as desired. The last implication Ž4., was shown in w7x. By this lemma, conditions ŽA1., ŽA2., and ŽA3. imply successively weaker topological types. There are examples of lower semicontinuous functions in two variables such that ŽA2. is fulfilled but convexity is violated or ŽA3. is fulfilled but star-shapedness of the approximate subdifferential fails to hold Žsee w7x.. In this sense, the indicated conditions are specific. On the other hand, the last statement of the lemma shows that in the Žfinite-dimensional. Lipschitzian case, ŽA2. and ŽA3. imply nothing but convexity. In this situation it seems hard to find reasonable conditions similar to Lemma 2.1 which are specific for a certain topological type. This motivates a more detailed study of the locally Lipschitzian case. Before proving the main result, we shall need the following lemma which facilitates the computation of a f Ž z . according to Definition 1.1. The third assertion of the lemma could be deduced from more general results in w13x, but to make the presentation self-contained we include its proof, which in the given specific setting is quite simple. LEMMA 2.2. Let X be a Hilbert space, C : X a weakly compact subset, and f : X ª R defined by f Ž z . s min² z, x :< x g C 4 . Denote EŽ z . s x g C <² z, x : s f Ž z .4 . Then the following equalities hold: dy f Ž z ; h . s min ² x, h:< x g E Ž z . 4 , aE Ž z . s 1
«
E Ž z . s yf Ž z . ;
aE Ž z . G 2
lim sup «yf Ž z . s lim sup yf Ž z . . zª0 « x0
«
Ž 1. yf Ž z . s B, Ž 2.
Ž 3.
zª0
Proof. Since f is a concave function, its directional derivative exists and coincides with dy f Ž z; h.. On the other hand, the value of the directional derivative computes from Ž1.; see, e.g., w2, Proposition 4.4x.
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Concerning the first relation of Ž2. one concludes from EŽ z . s x 4 and from Ž1. that dy f Ž z; h. s ² x, h: ;h g X; hence y f Ž z . s x 4 . For the second relation, note that the nonemptiness of y f Ž z . implies Žby definition. dy f Ž z; h. G ydy f Ž z; yh. ;h g X. Now, choosing x 1, x 2 g EŽ z ., x 1 / x 2 , and assuming y f Ž z . / B we obtain from Ž1. for arbitrary h g X, min ² x 1 , h: , ² x 2 , h: 4 G min ² x, h: s dy f Ž z ; h . G ydy f Ž z ; yh . xgE Ž z .
s y min ² x, yh: s max ² x, h: xgE Ž z .
xgE Ž z .
G max ² x 1 , h: , ² x 2 , h: 4 . This inequality holding for all h g X, one gets the contradiction x 1 s x 2 . Finally, for proving Ž3., first observe that the inclusion = is trivially fulfilled because of «y = y. For the reverse inclusion let x g lim sup «yf Ž z . . zª0 « x0
This means existence of nets zl ª 0, xl ©w x, «l ª 0, xl g «yl f Ž zl ., and «l ) 0. Let U be a Žstrong. open neighborhood of 0 and V be a weak open neighborhood of x. Then V contains a base neighborhood of the type W s y g X < ² y y x, y i : - « , y i g X , i s 1, . . . , n4 and for l G l0 one has <² xl y x, y i :< - «r2 Ž i s 1, . . . , n.. This implies y g W : V whenever 5 y y xl 5 - «rŽ2Ž1 q max 5 y i 5... Summarizing, the existence of an index l* such that zl* g U and xl* q B«l* : V follows. The last relation yields EŽ zl* . : V. Indeed, since xl* g «yl* f Ž zl* ., one has for any ¨ g EŽ zl* . and h g X that wcompare the definition of «y f and Ž1.x: ² xl* , h: F min ² ¨ 9, h:< ¨ 9 g E Ž zl* . 4 q «l* 5 h 5 F ² ¨ , h: q «l* 5 h 5 . From this it follows that 5 xl* y ¨ 5 F «l* ; hence EŽ zl* . : xl* q B«l* : V. Since C _ V is weakly compact, this last inclusion together with continuity of f implies existence of some Žstrong. open neighborhood of U9 of zl* such that EŽ y . : V ; y g U9. As a concave function f is Gateaux-differentiable on a dense subset of X w1x. Now, U l U9 is a nonempty set which consequently must contain some point y such that f possesses some Gateaux derivative y* at y. Then, by Ž2.,
y f Ž y . s y* 4 s E Ž y . : V .
351
APPROXIMATE SUBDIFFERENTIAL
Summarizing, to each pair U, V of Žstrong, weak. open neighborhoods of 0, x we can assign y wU, V x g U, yUwU, V x g V such that yUwU, V x g y f Ž y wU, V x .. In this way one obtains converging nets ym ª 0, ymU ©w x, and ymU g y f Ž ym . Ž m s wU, V x.; hence x g lim sup yf Ž z . zª0
as desired. Combining Ž3. in Lemma 2.2 with Lemma 1.1 one gets COROLLARY 2.1. Under the assumption of Lemma 2.2 the approximate subdifferential of f Ž defined in the lemma. computes as
a f Ž 0 . s lim sup yf Ž z . . zª0
As was stated in the introductory section, the approximate subdifferential of a locally Lipschitzian function defined on a Banach space is weak*-compact. In turn, the following theorem shows that in a Hilbert space setting each weakly compact set may be approximated by subdifferentials of Lipschitzian functions. THEOREM 2.1. For any weakly compact subset K of any Hilbert space Ž H , ² :. there exists a one-parametric family f u : R = H ª R of Lipschitzian functions with some common modulus L, such that lim ux 0 a f uŽ0, 0. s 04 = K holds in the sense of Kuratowski]Painle¨ ´ e con¨ ergence. Proof. Let c be a constant with 5 x 5 2 - c ; x g K Žwhich exists by weak compactness of K . and fix a parameter u with 0 - u F 1rc. In R = H Žwhich is a Hilbert space by canonical extension of the inner product. consider the ellipsoidal surface Eu s Ž t , x . g R = H < u 5 x 5 2 q Ž t y 1 . s 1 4 2
as well as the subset Ku s Ž t, x. g R = H < x g K , t s 1 y
½
'1 y u 5 x 5 5 . 2
Note that K u is correctly defined and K u : Eu . Furthermore, the function f u : R = X ª R with f u Ž a , z . s min ² Ž a , z . , Ž t , x . :< Ž t , x . g clco K u 4
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is correctly defined as well Žby weak compactness of clco K u .. Since K u : w 0, 1 x = B Ž 0; 'c . : B Ž Ž 0, 0 . ; 'c q 1 . , one has clco K u : B ŽŽ0, 0.; 'c q 1 .; hence 'c q 1 is a modulus of Lipschitz continuity for f u Žnot depending on u.. Next we show the validity of the following relations which are essential for proving the assertion of the theorem: pr Ž a f u Ž 0, 0 . < 0 4 = H . s 0 4 = K ,
Ž 4.
pr Ž a f u Ž 0, 0 .
Ž 5.
Here, prŽ?< M . denotes projection onto a closed subspace of M of R = H . To show first the inclusion 04 = K : prŽ a f uŽ0, 0.< 04 = H . it is sufficient to verify the relation K u : a f uŽ0, 0. because of prŽ K u < 04 = H . s 04 = K. So let Ž t, x . g K u . For n g N put a n s Ž1 y t .rn and z n s yŽ urn. x. Then, the Kuhn]Tucker conditions Žrecall that for u ) 0, Eu is a regular surface . yield that the linear function ²Ž a n , z n ., ? : restricted to Eu attains its minimum at the unique point Ž t, x .. Since Ž t, x . g K u : Eu and by a convexity argument one has
Ž t , x . 4 s arg min ² Ž a n , z n . , Ž t9, x9 . :< Ž t9, x9 . g K u 4 s arg min ² Ž a n , z n . , Ž t9, x9 . :< Ž t9, x9 . g clco K u 4 . Therefore, with the notation introduced in Lemma 2.2, EŽ a n , z n . s Ž t, x .4 and Ž2. implies y f uŽ a n , z n . s Ž t, x .4 . Together with Ž a n , z n . ª Ž0, 0. and using the trivial sequence Ž t n , x n . ' Ž t, x ., Corollary 2.1 provides
Ž t , x . g lim sup yf u Ž a , z . s a f u Ž 0, 0 . . Ž a , z .ª Ž0, 0 .
Before proving the reverse inclusion of Ž4. and Ž5. we show first the relation
Ž t*, x* . g y f u Ž a , z .
«
x* g K ,
t* g 0, 1 y '1 y uc . Ž 6 .
Indeed, from Ž2. in Lemma 2.2 one has
Ž t*, x*. 4 s arg min ² Ž a , z . , Ž t9, x9 . :< Ž t9, x9 . g clco K u 4
Ž 7.
and by convexity of clco K u there exists a sequence Ž t n , x n .4 : K u such that ²Ž a , z ., Ž t n , x n .: ªn ²Ž a , z ., Ž t*, x*.:. Weak compactness of clco K u then yields the existence of a weakly convergent subsequence Ž t n l , x n l . ©l
353
APPROXIMATE SUBDIFFERENTIAL
Žt , y . g clco K u . Consequently, ²Ž a , z ., Žt , y .: s ²Ž a , z ., Ž t*, x*.:; hence Žt , y . s Ž t*, x*. because of Ž7.. It follows that x n ©l x* with x n g K l l wsince Ž t n , x n . g K u and by definition of K u x. Weak compactness of K l l yields x* g K, as desired. On the other hand, from the definition of K u it follows that 0 F t n l F 1 y '1 y uc , which proves the second implication of Ž6.. Now, let
Ž t , x . g a f u Ž 0, 0 . s lim sup yf u Ž a , z . Ž a , z .ª Ž0, 0 .
Žcompare Corollary 2.1., i.e., there exist nets Ž al, zl . ª Ž0, 0. and Ž tl , xl . © Ž t, x . with Ž tl, xl . g y f uŽ al , zl .. Equation Ž6. yields xl g K and tl g w0, 1 y '1 y uc x, whence x g K by weak compactness of K and t g w0, 1 y '1 y uc x. Thus, Ž4. and Ž5. are completely proved. In order to verify the assertion of the theorem it suffices to show the relation lim sup a f u Ž 0, 0 . : 0 4 = K : lim inf a f u Ž 0, 0 . , ux0
ux0
Ž 8.
which directly implies equality between all three sets. Considering first some element Ž t, x . g lim sup u x 0 a f uŽ0, 0. one has converging nets ul ª 0, Ž tl, xl . © Ž t, x ., and Ž tl , xl . g a f ulŽ0, 0.. Then Ž4. implies xl g K which ensures x g K because of weak compactness of K. Moreover, Ž5. provides tl g w0, 1 y 1 y ul c x. Therefore tl ª 0 and t s 0, as was to be proved. Concerning the second inclusion let x g K. We have to show that Ž0, x . g lim inf ux 0 a f uŽ0, 0.. To this aim consider an arbitrary net ul ª 0. Put xl ' x and choose}by virtue of Ž4. and Ž5. }some tl with Ž tl , x . g a f ulŽ0, 0. and 0 F tl F 1 y 1 y ul c . Obviously, Ž tl , xl . © Ž0, x ., as desired.
'
'
It is clear that a similar approximation result as in Theorem 2.1 does not hold, for instance, for Clarke’s subdifferential because of the restriction to convexity. This will become even clearer from the following topological derivations of the theorem. LEMMA 2.3. For any strongly compact subset K of any Hilbert space H there exists a Lipschitzian function f : R = H ª R such that a f Ž0, 0. is homeomorphic to K in the strong topology. If moreo¨ er K is contained in some finite-dimensional subspace of H , then a f Ž0, 0. is homeomorphic to K in the weak topology too. Proof. Fix any admissible parameter u in Theorem 2.1 Žsee beginning of the proof. and consider the corresponding set K u . Via the functions f : K ª K u and c : K u ª K defined by f Ž x . s Ž1 y 1 y u 5 x 5 2 , x . and
'
RENE ´ HENRION
354
c Ž t, x . s x we see that f defines a bijection from K to K u with inverse fy1 s c . Moreover, f is strongly continuous while c is both strongly and weakly continuous. Hence f defines a homeomorphism between K and K u in the strong topology. In particular K u inherits strong compactness from K. Now we show that a f Ž0, 0. s K u . Define the Lipschitzian function f : R = H ª R by f Ž a , z . s min²Ž a , z ., Ž t, x .:<Ž t, x . g K u4 which is justified by strong Žhence weak. compactness of K u . Repeating the lines of argumentation in the proof of Theorem 2.1 Žbut restricting considerations to K u itself instead of its convex closure. one derives the inclusion K u : a f Ž0, 0.. The reverse inclusion would follows from the relation y f Ž a , z . : K u ;Ž a , z . due to weak compactness of K u and to the definition of the approximate subdifferential. To verify the mentioned relation, note that
Ž t*, x* . g yf Ž a , z . « Ž t*, x* . 4 s arg min ² Ž a , z . , Ž t9, x9 . :< Ž t9, x9 . g K u 4 as a consequence of Ž2. in Lemma 2.2. In particular, Ž t*, x*. g K u , as was to be shown. Concerning the second part of the lemma, note that in this case the function f , introduced in the beginning of this proof, becomes weakly continuous as was also true for its inverse c . Therefore f is a homeomorphism between K and K u in the weak topology too. This lemma extends a corresponding result in w7x ŽTheorem 3.2. for finite-dimensional H . It allows the following topological characterization of the approximate subdifferential: COROLLARY 2.2. For any Hilbert space H there exists a Lipschitzian function f : R = H ª R such that a f Ž0, 0. is homeomorphic to the Cantor set in the weak as well as the strong topology. In particular, a f Ž0, 0. is totally disconnected in both topologies. Finally we note that in w7x it was proved that for each compact subset K : R n there exists a locally Lipschitzian function f : R nq 2 ª R such that the so-called partial approximate subdifferential Žintroduced by Jourani and Thibault w14x in the context of functions depending on a parameter. of f coincides Žat the origin. with 04 = K : R nq 1. Here, any given compact subset may be even realized itself Žup to imbedding into a space with one extra dimension.. Revisiting Lemma 2.3 and Corollary 2.2 one notices that for the definition of the desired Lipschitzian function one additional dimension is needed. In particular, the construction Žaccording to Corollary 2.2. of an approximate subdifferential being totally disconnected requires the domain of the Lipschitzian function f to have at least dimension 2. This suggests a question about possible topological types of the approximate
APPROXIMATE SUBDIFFERENTIAL
355
subdifferential in the one-dimensional case. Theorem 2.2 below indicates, that for real continuous functions f : R ª R the approximate subdifferential is either an interval or the disjoint union of two intervals, which is obviously a topological restriction. For the proof of this one-dimensional characterization two auxiliary results are needed. To this aim, for functions f : R ª R we introduce the following notations Žwith z g R.: d l Ž z . s lim inf ty1 Ž f Ž z q tu . y f Ž z . . ; t x0 uªy1
d r Ž z . s lim inf ty1 Ž f Ž z q tu . y f Ž z . . , t x0 uª1
where improper values "` are allowed. Obviously,
y f Ž z . s yd l Ž z . , d r Ž z .
Ž 9.
holds, with the interval to be interpreted appropriately for improper values. PROPOSITION 2.1. Then
Let f : R ª R be continuous, and t 1 , t 2 g R, t 1 - t 2 .
d r Ž t 1 . - yd l Ž t 2 .
« ;c g Ž d r Ž t 1 . , yd l Ž t 2 . . ' t g Ž t 1 , t 2 . : c g yd l Ž t . , d r Ž t . s y f Ž t . ,
d r Ž t 1 . ) yd l Ž t 2 .
Ž 10 .
« ;c g Ž yd l Ž t 2 . , d r Ž t 1 . . ' t g Ž t 1 , t 2 . : c g d r Ž t . , yd l Ž t . .
Ž 11 .
Proof. Concerning Ž10. let c g Ž d r Ž t 1 ., yd l Ž t 2 ... Then the function x ¬ f Ž x . y cx, restricted to x g w t 1 , t 2 x, attains its minimum at a point of the open interval Ž t 1 , t 2 .. To see this, assume that t 1 is a minimizer. It follows that f Ž t q t 1 . y f Ž t 1 . G ct ; t g w 0, t 2 y t 1 x . This however yields lim inf ty1 Ž f Ž t 1 q ut . y f Ž t 1 . . G c, t x0 uª1
leading to the contradiction c F d r Ž t 1 .. In a similar way t 2 may be excluded as a minimizer via the contradiction c G yd l Ž t 2 .. As a consequence, there exists t g Ž t 1 , t 2 . such that f Ž x . y cx G f Ž t . y ct ; x g w t 1 , t 2 x. Choose d ) 0 with Ž t y d , t q d . : Ž t 1 , t 2 . to get f Ž t q t . y f Ž t . G ct ; t g Ž yd , d . , Ž . Ž . which immediately implies 10 . For 11 consider the maximum Žrather than minimum. of f Ž x . y cx and repeat the same arguments.
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356
LEMMA 2.4. Let f : R ª R be continuous and y f Ž z . / B for some z g R. Then a f Ž z . is a closed Ž possibly unbounded. inter¨ al. Proof. By assumption, there is some e g y f Ž z . : a f Ž z .. It will be sufficient to show that a g a f Ž z ., a - e implies Ž a, e . : a f Ž z . Žthe proof is running along similar lines for a ) e .. The general definition, Definition 1.1, of the approximate subdifferential reduces in the present constellation Žfinite dimensions, continuous function. to a f Ž z . s lim sup x ª z y f Ž x ., where the lim sup may be generated by sequences instead of nets. Accordingly, there exist sequences t k ª 0,
a k ª a,
a k g yf Ž z q t k . .
Ž 12 .
Let c g Ž a, e . be arbitrarily given. We have to show that c g a f Ž z .. For k G k 0 one has c ) a k and we may suppose without loss of generality that c f y f Ž z q t k . j y f Ž z . wsince otherwise the assertion c g a f Ž z . follows immediatelyx. Combining Ž12. with Ž9. one gets yd l Ž z q t k . F a k F d r Ž z q t k . - c - yd l Ž z . F e F d r Ž z .
;k G k 0 .
Ž 13 . As an obvious consequence of this relation we have t k / 0 ;k. One of the following two cases must hold true: Case 1. There is a negative subsequence t k p - 0. Application of Ž10. to the relation d r Ž z q t k p . - c - yd l Ž z . wsee Ž13.x yields existence of tp g Ž t k , 0. with c g y f Ž z q tp .. Then c g a f Ž z . due to t p ª 0. p
Case 2. There is a positive subsequence t k p ) 0. Fix an arbitrary c g Ž c, e . and apply Ž11. to the relation d r Ž z . ) c ) yd l Ž z q t k p . wcompare Ž13.x. Hence, there exists tp g Ž0, t k p . with c F yd l Ž z q tp .. Next choose an index q Ž p . to fulfill t k qŽ p. - tp . Application of Ž10. to the relation d r Ž z q t k qŽ p. . - c - c F yd l Ž z q tp . finally provides c g y f Ž z q m p . for some m p g Ž t k qŽ p., tp .. Now c g a f Ž z . due to m p ª 0. According to Lemma 2.4 nonemptiness of the Dini subdifferential implies convexity of the approximate subdifferential in one dimension. Unfortunately, a generalization of this fact to the multidimensional case is not possible even in the Lipschitzian case. This can be seen from the function f : R 2 ª R defined by f Ž x, y . s max 0, min x, y44 . Here y f Ž0, 0. s 0, 04 / B, while a f Ž0, 0. s wŽ0, 0., Ž0, 1.x j wŽ0, 0., Ž1, 0.x Žwith brackets referring to line segments., which is obviously not convex. THEOREM 2.2. Let f : R ª R be continuous. Then a f Ž z . contains at most two connected components Ž for all z g R..
APPROXIMATE SUBDIFFERENTIAL
357
Proof. Negating the assertion means the existence of elements a b - c with a, b, c g a f Ž z . and which are contained in three pairwise different connected components. By definition there are sequences t k , t k , m k ª 0, a k ª a, bk ª b, c k ª c, a k g yf Ž z q t k . ,
bk g yf Ž z q t k . ,
c i g yf Ž z q m k . . Ž 14 .
As a consequence of Lemma 2.4 one has t k , t k , m k / 0 ;k; otherwise a k Žor bk or c k . g y f Ž z . for some k, so a f Ž z . would be an interval in contradiction to our assumption of at least three connected components. Passing to appropriate subsequences one may assume, without loss of generality, that each of the sequences t k 4 , t k 4 , m k 4 in Ž14. has a constant Ždefinite. sign. For two of them this sign must be equal, so we have t k - 0, t k - 0 ;k or
t k ) 0, t k ) 0 ;k
Ž 15 .
Žthe subsequent proof being identical for the remaining possible pairs t k 4 , m k 4 , and t k 4 , m k 4.. We are done if we can show that Ž a, b . : a f Ž z ., because then w a, b x : a f Ž z . in contradiction to the assumption that a and b come from different connected components of a f Ž z .. So choose any e g Ž a, b .. For some k 0 it holds that a k - e - bk and e f y f Ž z q t k . j y f Ž z q t k . ;k G k 0 Žnegating the second relation would immediately provide the desired result e g a f Ž z ... Consequently one gets wcompare Ž9.x yd l Ž z q t k . F a k F d r Ž z q t k . - e - yd l Ž z q t k . F bk F d r Ž z q t k . .
Ž 16 . From this chain it is evident that t k / t k ;k G k 0 , so one of the following two cases holds true: Case 1. There exists a subsequence such that t k p - t k p ; p. Then, Ž10. applied to the relation d r Ž z q t k p . - e - yd l Ž z q t k p . yields the existence of some a p g Ž t k p, t k p . such that e g y f Ž z q a p .. It follows e g a f Ž z . since a p ª 0. Case 2. There exists a subsequence such that t k p ) t k p; p. First we consider the situation t k - 0, t k - 0 ;k wsee Ž15.x. Fix an arbitrary e g Ž a, e .. Again we may assume that a k - e. Then, Ž11. applied to the p relation d r Ž z q t k p . ) e ) yd l Ž z q t k p . yields the existence of some b p g Žt k , t k . such that e G d r Ž z q b p .. Obviously, b p - 0 and b p ª 0. In p p particular we may select an index q Ž p . with t k qŽ p. ) b p . Then, Ž10. applied to the relation d r Ž z q b p . F e - e - yd l Ž z q t k qŽ p. . yields the existence of some gp g Ž b p , t k qŽ p. . such that e g y f Ž z q gp .. Since gp ª 0 we arrive at e g a f Ž z .. The second situation of Ž15. is treated in a similar manner, but now by choosing e g Ž e, b ., deriving the relation e F yd l Ž z q b p ., and considering a sequence t k qŽ p. - b p .
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We note that, in the case of locally Lipschitzian functions, the result of Theorem 2.2 can also be derived from w3x. From an example in the same reference one can see that on any countable subset of the line, which in particular could be dense, the approximate subdifferential of some suitably defined Lipschitz function is disconnected. The following proposition demonstrates that such behavior is possible even for a very nice nonsmooth function like the value function of a parametric optimization problem of class C 1. PROPOSITION 2.2. There exists a function f g C 1 ŽR 2 , R., such that for the corresponding minimum function g Ž z . s min f Ž x, z .< x g w0, 1x4 the approximate subdifferential a g Ž z . is disconnected Ž noncon¨ ex in particular . on a dense set of points z g R. Proof. Denote by Z the set of all z g R representable as z s "i2yj Ž i, j g N.. Then Z is a dense subset of R and it is possible to construct a C 1-function f : R 2 ª R with the properties: ; z g Z,
EŽ z . s x1 Ž z . , x 2 Ž z . 4
f
Ž x1 Ž z . , z . / z
f z
Ž x2 Ž z. , z., Ž 17 .
where EŽ z . s x g w0, 1x< f Ž y, z . G f Ž x, z . ; y g w0, 1x4 . The details of this construction are described in w8x in the context of maximum functions with a dense subset of nondifferentiability. Now consider the minimum function g Ž z . s min f Ž x, z . < x g w 0, 1 x 4 , which has the one-sided directional derivative dg Ž z ; h . s min
½
f z
Ž x, z . ? h < x g E Ž z .
5
; z, h g R.
Therefore, the lower Dini directional derivative dy g Ž z; h. Žsee the Introduction. coincides with dg Ž z; h. and the Dini subdifferential simply becomes the ‘‘interval’’ w f l Ž z ., f r Ž z .x, where
f l Ž z . s max f r Ž z . s min
½ ½
f z f z
Ž x, z . < x g E Ž z . ,
5 Ž .5
Ž x, z . < x g E z
.
Since f l Ž z . G f r Ž z . it holds that y g Ž z . s f l Ž z .4 l f r Ž z .4 for all z g R. Now, for some fixed z g Z we have by Ž17. Žwithout loss of generality., EŽ z . s x1 , x 2 4 ,
a1 ) a2 ,
where a i s
f z
Ž x i , z . Ž i s 1, 2 . .
359
APPROXIMATE SUBDIFFERENTIAL
It follows that y g Ž z . s B. Moreover, continuity of the function fr z implies that ;« ) 0 'd ) 0,
EŽ z . :
½
x 1 q « w y1, 1 x ,
if z g Ž z y d , z . ,
x q « w y1, 1 x ,
if z g Ž z, z q d . .
2
Therefore, lim z z f l Ž z . s f l Ž z . s a 1 and lim z x z f r Ž z . s f r Ž z . s a 2 . Summarizing we arrive at
a g Ž z . s lim sup f l Ž z . 4 l f r Ž z . 4 : lim sup f l Ž z . , f r Ž z . 4 s a 1 , a 2 4 . zªz
zªz
On the other hand, g being locally Lipschitzian, there are sequences z n x z and z n z, such that g 9Ž z n . exists; hence y g Ž z n . s g 9Ž z n .4 s f l Ž z n .4 s f r Ž z n .4 . It results that lim y g Ž z n . s lim f l Ž z n . 4 s a 1 4 ,
zn z
zn z
y
lim g Ž z n . s lim f r Ž z n . 4 s a 2 4 ,
zn x z
znx z
whence a g Ž z . s a 1 , a 2 4 . Since z g Z was arbitrarily chosen, the approximate subdifferential is disconnected on a dense subset of R. ACKNOWLEDGMENTS The author would like to thank Professor Thibault ŽUniversity of Montpellier, France. and ŽHumboldt University, Berlin, Germany. for helpful discussions on the Professor Romisch ¨ subject of this paper.
REFERENCES 1. E. Asplund, Frechet differentiability of convex functions, Acta Math. 121 Ž1968., 31]47. ´ 2. J.-P. Aubin, ‘‘Optima and Equilibria,’’ Springer-Verlag, New York, 1993. 3. J. M. Borwein and S. Fitzpatrick, Characterization of Clarke subgradients among onedimensional multifunctions, CECM Preprint 94:006 Ž1994.. 4. B. El Abdouni and L. Thibault, Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces, Optimization 26 Ž1992., 277]285. 5. B. M. Glover, B. D. Craven, and S. D. Flam, ˚ A generalized Karush]Kuhn]Tucker optimality condition without constraint qualification using the approximate subdifferential, Numer. Funct. Anal. Optim. 14, Ž1993., 333]353. 6. B. M. Glover and B. D. Craven, A Fritz John optimality condition using the approximate subdifferential, J. Optim. Theory Appl. 82 Ž1994., 253]265. 7. R. Henrion, Topological Characterization of the Approximate Subdifferential in the Finite-Dimensional Case, ZOR}Math. Methods Oper. Res. 41, Ž1995., 161]173. 8. R. Henrion, On maximum functions with a dense set of points of nondifferentiability, in ‘‘Approximation and Optimization,’’ Vol. 3, Peter Lang, Frankfurt am Main, 1993. 9. A. D. Ioffe, Calculus on Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal. 8 Ž1984., 517]539.
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