Error bounds for an inequality system

by Z iliW u

B.Sc., Xiamen University, 1982 M.Sc., University o f Victoria, 1997

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree o f

D O C T O R OF PH ILO SO PH Y

in the Department of M athem atics and Statistics

We accept this dissertation as conforming to the required standard

Dr. J. J, Ye,^Supervisor (Department o f M athem atics and Statistics)

)r. C. J. Bose, Departmental Member (Departm ent o f M athem atics and Statistics)

Dr. R. Illner, Departmental Member (Departm ent o f M athem atics and Statistics)

Dr. W. S. Lu, Outside Member (Departm ent o f Electrical and Computer Engineering)

:---Dr. Q. jTZRu, Exterral Examiner (Departm ent o f M athem atics and Statistics, Western Michigan University)

© Zili W u, 200 1 Uinversity o f Victoria

All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopy or other means, without the permission o f the author.

A B ST R A C T

For an inequality system , an error bound is an estimation for the distance from any point to the solution set o f the inequality. The Ekeland variational principle (EVP) is an important tool in the study of error bounds. We prove that EVP is equivalent to an error bound result and present several sufficient conditions for an inequality system to have error bounds. In a metric space, a condition is similar to that o f Takahashi. In a Banach space we express conditions in term s o f an abstract subdifferential and the lower Dini derivative. We then discuss error bounds with exponents by a relation between the lower Dini derivatives o f a function and its power function. For an l.s.c. convex function on a reflexive Banach space these conditions turn out to be equivalent. Furthermore a global error bound closely relates to the metric regularity.

Examiners:

D ^ /j . J. Ye, Supervisor/D epartm ent o f M athem atics and Statistics)

Dr. C. J. Bose, Departmental Member (Departm ent o f M athem atics and Statistics)

Dr. R. Illner, Departmental Member (Departm ent o f M athem atics and Statistics)

Dr. W. S. Lu, O utsiae Mernber (Departm ent o f Electrical and Computer Engineering)

Dr. Q. J. Zhir, External Examiner (Departm ent o f M athem atics and Statistics, Western Michigan University)

A bstract ü

T a b le o f C o n te n ts iii

A c k n o w le d g e m e n ts v

1 I n tr o d u c tio n 1

2 V ario u s G e n e ra liz e d S u b d iffe re n tia ls 9

2.1 The Clarke Subdifferential . ... 9

2.2 Various D e riv a tiv e s... 24

2.3 The Michel-Penot S u b d iffe re n tia l... 28

2.4 The Proximal Subdifferential... 30

2.5 The Fréchet Subdifferential... 33

2.6 The Dini Subdifferentials... 39

3 E q u iv a le n t F o rm u la tio n s o f E k e la n d ’s V a ria tio n a l P r in c ip le 44 3.1 The Ekeland Variational Principle ... 45

3.2 The e-conditions of Takahashi and Hamel with a ... 47

3.3 Weak Sharp Minima and Error B o u n d s ... 50

3.5 The Completeness of a Metric Space ... 60

4 Error Bounds for Lower Sem icontinuous Functions 65

4.1 Error Bounds for Nonconvex Functions on Metric Spaces . . . . 66

4.2 5o,-subdifferentials... ... ... 72 4.3 Error Bounds for Lower Semicontinuous Functions on Banach

S p a c e s ... 74 4.4 Error Bounds with E x p o n e n ts ... 83

4.5 Error Bounds for Lower Semicontinuous Convex Functions . . . 92

4.6 Error Bounds with A bstract Constraint S e t s ... 99 4.7 Global Error Bounds and Metric R e g u la r ity ... 107

W ith most sincere appreciation, I thank Dr. Jane J. Ye for her patience, helps and supervision in my research work, in particular, in the process of w ritting this dissertation.

Thanks and appreciation are due to Dr. Chris J. Bose, Dr. Reinhard Illner, Dr. Wu-Sheng Lu and Dr. Qiji J. Zhu for their time in examining my disserta­ tion and providing me with enlightening suggestions and remarks.

I gratefully acknowledge the Department of M athematics and Statistics, University of Victoria, which has provided me with the opportunity and support to work on this dissertation.

Finally, I am greatly indebted to my father, my wife, my brother, my sisters and my daughter for their love, support and encouragement. I would like to extend heartful thanks to Mr. Zhanghong Wu for his impressive instruction and encouragement.

In trod u ction

Let C be a nonempty closed subset of a normed linear space X and gj : X ^

(~oo, +00] be extended real-valued functions for i = 1, • • •, r and j = 1, ■ • •, s. Denote the solution set of an inequality system by

5 := {z E C : /i( z ) < 0, -, A M < 0; g iM = 0 , - =

0}-The set S is said to have a global error bound if it is nonempty and there exists a constant fi > 0 such th a t

ds{x) := inf{||a; - c\\ : c e S } < ||-FM +|| 4- ||G(a;)|| ) Vx € C, (1.1)

where F{x)^ = (/i(z )+ , • • •, f r { x ) + ) E IF with a+ := max{a, 0} for a E R, G{x) = {qi(x) , • • ■ ,gs(x)) € i?* and || • || is the usual Euclidean norm. The set

S is said to have a local error bound if there exist constants p > 0 and e > 0 such th at

dg(a;) < /r(||F (z )+ || 4- ||G(a;)||) Vz E C with ||(F (z )+ ,G (a ;))|| < e.

Apparently if the set S has a global (local) error bound, then functions involved provide a global (local) error estim ate for the distance from any point

optimization. Consider the following optimization problem with equality and inequality constraints:

(P) minimize h{x) subject to f { x ) < 0, g{x) — 0, x E P ”,

where h is Lipschitz of rank L on P ", / : P " P and |^| : P " —>■ P are lower semicontinuous. In this situation the feasible set

P := {z E P " : / ( % ) < 0, ^(z) = 0}

is closed and, by the exact penalization, problem (P) is equivalent to the un­ constrained problem:

(Pa) minimize h{x) + ad s{x ) subject to z E P "

for any a > L (see [17, Proposition 1.3]). The objective function of problem (Pa)

involves the distance function which is usually not easy to deal with. However if S has a global error bound then, for some p > 0,

dg(ar) < /^(/(%)+ + l^(z)l) V% E P ".

It is easy to check th a t every solution of (Fa) solves

(Pan) minimize h(x) + aiJ,(f(x)+ + |^ (r)|) subject to z E P".

On the other hand, if xq solves problem (Pan)> then for each z E P we have

a(fa(%o) < h(z) - h(zo) < Z,||z; - zo|| Vz E 5^.

Thus ads(xo) < Lds{xo). This implies ds{xo) = 0, th at is, xq € S. So Xq solves problem (P ). Therefore problems (P), (Pa) and (Pan) have the same solution set. Hence we can solve problem (F) by studying problem {Pap) which maybe simpler since it is unconstrained and does not contain the distance function.

Since Hoffman [32] obtained a result on global error bound for a linear

inequality systems on FP, the study of error bounds has received more and more

attention in the m athem atical programming literature due to many im portant applications in sensitivity analysis, complementarity problems, implicit function theorem, and the convergence analysis of some descent methods. For more details on error bounds and their applications we see [2, 13, 15, 18, 23, 24, 29, 33, 39, 41, 42, 43, 44, 45, 46, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 72, 83, 89, 90] and the references therein.

Hoffman’s result [32] states th a t if f { x ) = A x + a and g{x) — B x + b for some matrices A, B and some vectors a, b of appropriate dimension then there exists some p > 0 depending on A and B only such th at

ds{x) < n{\\[Ax + a]+|| + \\Bx + 6||) M x e B A .

There are no special conditions required for a linear inequality system to have a global bound. However for a nonlinear inequality system, additional conditions are usually needed. The Slater condition is one of most often used conditions, which postulates the existence of an a:o E C such th a t f{xo) < 0 for / : X —>• BP.

trary bounded closed convex set S' in a normed space; Mangasarian [60] proved th a t there exists a global error bound when / : i?” —> is a convex differen­ tiable function and satisfies an asymptotic constraint qualification; Auslender and Crouzeix [2] extended M angasarian’s result to a nondifferentiable function; Luo and Luo [50] established a global error bound for a convex quadratic inequal­ ity system without other conditions; for a continuous convex inequality system on a reflexive Banach space Deng [23] obtained an error bound result under a Slater condition on the associated recession functions; Deng [24] extended his result to the same system on a Banach space under the Slater condition with the Hausdorff distance assumption.

Some weaker conditions have also been found to characterize a global error bound for a convex function / : R” —> Li [46] showed th a t a convex differentiable inequality system f { x ) < 0 has a global error bound if and only if the system satisfies Abadie’s constraint qualification at each x £ S, th a t is, the tangent cone of S' at % can be expressed as follows

Ts{x) = {y G R" : y) < 0 for each i with fi(x) = 0}.

Lewis and Pang [42] studied a proper lower semicontinuous convex function / : j y —> (—oo, -hoo] and proved th a t / has a global error bound if and only if for some /x > 0 and for any x E B^ with f { x ) — 0 and any normal vector d to S' at Æ the directional derivative f'{x-,d) satisfies f '{ x \d ) > /x“ ^||d||.

It is worth pointing out th a t the above conditions are all in one class which closely relates to points inside the solution set S. The other class of conditions

of functions at the points outside 5^ for which we can use the knowledge of nonsmooth analysis to study error bounds more effectively. To the author's knowledge, it is Ioffe [33] who ffrst obtained a global error bound (as well as metric regularity at a point) for a Lipschitz continuous equality system f { x ) — 0 on a Banach space X under the condition th a t for some /i > 0, 0 < e < +oo, some z E X with f ( z ) = 0 there holds

IICII* > e d°\ f\{x ) 'ix E X with f { x ) 0 and \\x - z\\ < e,

where d ° f{ x ) is the Clarke subdifferential of / at x. The Ekeland variational principle and the sum rule of the Clarke subdifferential are the main tools in the proof of this result from which we can see th a t the Clarke snbdifferential in the condition can be replaced with any smaller subdifferentials satisfying the sum rule. Using Ioffe’s method, Ye [89] and Jourani [39] have sharpened the result of Ioffe by replacing the Clarke subdifferential with the limiting subdifferential in i?” and a partial subdifferential in a general Banach space respectively. For a lower semicontinuous system on R^, Wu [83] has used the fuzzy sum rule (instead of the sum rule) to prove th a t the Clarke subdifferential in Ioffe’s condition can be replaced with the proximal subdifferential. In a Hilbert space, Clarke, Ledyaev, Stern and Wolenski [18, Theorem 3.3.1] have weakened Ioffe’s condition using the proximal subdifferential instead of the Clarke subdifferential. They did not use Ioffe’s m ethod to establish their result (since the proximal subdifferential does not satisfy the sum rule) but the decrease principle (see also Ye [90, Claim]). Ledyaev and Zhu [41] have used the decease principle of

to extend their result in terms of Fréchet subdifferential.

Recently Ng and Zheng [66] have proved the existence of a global error bound for a proper lower semicontinuous function / on a complete metric space

X provided th a t for some /r > 0 and each x Ç. X with f { x ) > 0 there exists a unit vector E X such th a t the lower Dini directional derivative of / at x in the direction hx is less than or equal to — t hat is,

/ I (3;; ha:)

Their result is based on an equivalent result of Ekeland’s variational principle, th a t is, the Takahashi theorem, which asserts the existence of minima for a lower semicontinuous function / on a complete metric space {X, d) when / satisfies the condition 0/ Th&ohoaht, th a t is, for each % E X with infx / < / (z) there exists y E X such th a t y x and f { y ) + d{x, y) < f { x ) .

Wu and Ye [88] obtained a related error bound result for a proper lower semicontinuous function / on a complete metric space X which states th a t the solution set S is nonempty and for some /x > 0 and 0 < e < +00 there holds

ds{x) < iJ,f{x)-^ for all a; € Y with f { x ) < e

provided th a t the set { x E X : f { x ) < e} is nonempty and for each x E X with 0 < f { x ) < e there exists y E X such th a t 0 < /(y ) < e and

0 < d(z, %/) < //[/(%) - /(%/)].

As we know, the Ekeland variational principle and the sum rule or the fuzzy sum rule are very im portant in establishing the above results. One of our pur­ poses in this dissertation is to reveal the equivalent relations among the Ekeland

in chapter 3. We will introduce the concepts of the e-condition of Takahashi and Hamel with a > 0. Then by the Ekeland variational principle we will prove th a t the e-condition of Takahashi implies th a t of Hamel. It follows from this relation th a t the e-condition of Takahashi is sufficient for / to have weak sharp minima. From this we can easily derive the above error bound result for which we will further dem onstrate its theoretical application by establishing a new fixed point theorem and proving the Takahashi theorem.

Our second goal is to study sufficient conditions for an inequality system to possess error bounds in chapter 4. For an inequality system on a general metric space we will discuss conditions similar to the condition of Takahashi. In a Banach space we will introduce the concept of ^^-subdffierential (which satisfies the fuzzy sum rule) and use it to present a sufficient condition similar to Ioffe’s for a lower semicontinuous system. The corresponding result unifies and extends several existing results and is applicable to a system with a closed constraint set. Sufficient conditions for error bounds will also be given in terms of the lower Dini derivative. Such conditions are easy to be translated into those for error bounds w ith exponents. These conditions are shown to be sufficient and necessary for a lower semicontinuous convex function on a reflexive Banach space to have (local and global) error bounds. We will briefly discuss relations between global error bound and metric regularity at the end of chapter 4.

For the above purposes we will gather the basic concepts of various general­ ized subdifferentials and their properties in the next chapter.

defined by

ll^ll* := sup{((,u) : u € X , ||n|| < 1}

while B and B* stand for the closures of open unit balls B of X and B* of X* respectively.

Various G eneralized

S ub different iais

In this chapter we study various generalized subdifferentials and their properties th a t will be used in this dissertation. We begin with the Clarke subdifferential and its calculus in section 2.1. The close relations among the Clarke subdiffer­ ential and various derivatives will be revealed in section 2.2. From section 2.3 to section 2.6 we will discuss the Michel-Penot subdifferential, the proximal subd­ ifferential, the Fréchet subdifferential and the Dini subdifferential respectively.

2.1

T he Clarke Subdifferential

It is well known th a t the Clarke subdifferential is a very im portant concept in nonsmooth analysis, especially as it relates to optimization (see [6, 7, 8, 9, 10, 11, 16, 17, 20, 19, 25, 27, 33, 38, 47, 70, 71, 74, 76, 89]). It was ffrst deffned for a locally Lipschitz function then for a very general class of functions.

D eûnîtion and Properties

D eûnition 2.1 Let C be a nonempty subset of a normed linear space A function / : C —>• i? is said to be Lipschitz {of rank L) on C if for some nonnegative scalar L one has

\ f { y i ) ~ 7(2/2) I < L\\Vi — 2/2II Vyi, 2/2 E C.

We shall say th at f is Lipschitz of rank L near x iî C — x + ôB fox some 5 > 0. A function / is said to be locally Lipschitz on a subset 5 of X if it is Lipschitz near every point in 5".

D e fin itio n 2.2 Let f : X R he Lipschitz near x. For any vector v in X , the

(Clarke) generalized directional derivative of f at x in the direction v, denoted

7° (a;; n), is defined by

y^x t

The Clarke subdifferential {the generalized gradient) of f at x is the subset of

X* given by

a ° 7 ( 4 = 6 X* : < 7°(:r; %;) Vu e X }.

W ith the Clarke subdifferential a wonderful system of theory has been de­ veloped (e.g., see [16]). We will use the following basic properties.

P ro p o s itio n 2.3 ([16, Proposition 2.1.1]) Let f be Lipschitz of rank L near x. Then

(a) The function v —> f ° { x ] v ) is finite, positively homogeneous, and subaddi­ tive on X , and satisfies

|/° ( z ;u )| < Z,||u||.

(b) f°{x; v) is upper semicontinuous as a function of {x, v) and, as a function of V alone, is Lipschitz of rank L on X .

(c ) r ( z ; - u ) = ( - / ) ° ( a ; ; t ; ) .

P r o p o s itio n 2.4 ([16, Proposition 2.1.2]) Let f be Lipschitz of rank L near x. Then

(a) d ° f { x ) is a nonempty, convex, weak*-compact subset of X* and ||^||* < L

for every Ç in d° f{ x ) .

(b) For every v in X , one has

/° (z ; u) = m ax{((, i;) : ( E 9°/(z )} .

P r o p o s itio n 2.5 (The mean value theorem) Let x and y be two distinct points in X and f : X ^ R be Lipschitz on an open set containing

[x, y] := {Xx + (1 - A)y : A G [0,1]}.

Then there exists a point u in {x, y) ;= [z, y } such that

/ W - / ( : c ) G ( ^ / ( n ) , - 4 ,

T h e T a n g e n t C o n e a n d th e N o rm a l C o n e

Let C be a nonempty subset of X . The distance function associated with C

is dehned by

dc{x) = inf{||x — c|| ; c e C} Vz € X .

Clearly the function d c is globally Lipschitz of rank 1 on X , th a t is,

|dc(z) - d c W I < I k -2/11 V z,2/ € X.

Thus by Proposition 2.4 we have ||^||* < 1 for each ^ e d°dc{x).

The distance function is a Lipschitz function frequently used in optimization since if a Lipschitz function / of rank L attains a minimum over C a t z then for

any L > L the function f + L d c attains a minimum at z, th a t is, a constrained

optimization problem can be changed to a unconstrained optimization problem which may be easier to study. In addition the distance function can be used to introduce the concepts of tangent cone to C at z E C and the normal cone, the geometric counterparts of the Clarke directional derivative and the Clarke subdifferential.

Recall th at a set D is a cone x î t D Ç D for all t > 0. Suppose th a t z E C. It is easy to check th a t d^(z; u) > 0 for all u E X. We define the tangent cone

to (7 at z to be the set

Tc{x) = {u E X : dc{x; v) = 0}

and the contingent cone to C at z to be the set

The following theorem imphes th a t the above tangent cone Tc(3:) is inde-pendent of the norm of X .

T h e o re m 2.6 ([16, Theorem 2.4.5]) Let x e C. Then v G T c{x) iff, for every

sequence Xi in C converging to x and sequence ti in (0, +oo) decreasing to 0,

there is a sequence V{ in X converging to v such that æ* + Lvi G C for all i.

It follows from Theorem 2.6 th at Tc{x) Ç K { x ) Vx G C. We say th a t the set C is regWor at x G C provided J c (x ) = A"c(x).

D e fin itio n 2.7 Let f : X (—oo,+oo] be an extended real-valued function.

The effective domain of / is the set

d o m f := {x G X : —oo < / ( x ) < -foo}.

We say th at / is proper if d o m f is nonempty. The epigraph of / : X —)• (—oo, 4-oo] is the set

e p i f := { (x , r ) G d o m f x R : / ( x ) < r } .

The function / is said to be regular at x provided epi f is regular at (x, / ( x ) ) .

When / is Lipschitz near x, by [16, Theorem 2.4.9], the function / is regular at X iff / satisfies the following two conditions;

(a) For all v G X ,th e usual one-sided directional derivative of / at x in the direction v given by

/ ( x - t - t n ) - / ( x )

f'( x ; v ) = lim

^ ^ t-^o+

(b) For all v e X , f'{x; v) = f°{x] v).

P r o p o s itio n 2.8 ([16, Proposition 2.3.12]) Let fi be Lipschitz near x G X for

each i £ I = {i — 1, ■ ■ ■, m } . Denote

f { x ) = max{/j(rr} : i e 1 } and I {x ) — { i E I : fi{x) = f { x ) } .

We have

Ç c o { a ° : 2 G /(T )}

(wA ere co (fenofea connea; A W i), o n j ÿ / i ts regW or o f a: / o r eocA ê in 7 (a;), f/ien

equality holds and f is regular at x.

By Proposition 2.3, the tangent cone Tc(x) is closed and convex. This property allows us to define the normal cone to C at x by polarity with Tc{x)

as follows:

7Vc(z) = { ( € % ' : 2;) < 0 VuG 2b(T)}. The normal cone N c { x ) has the following properties.

P r o p o s itio n 2.9 ([16, Proposition 2.4.2])

7 /c(a ;) = c Z { |J d c (a ;)} ,

A >0

where cl denotes weak* closure.

T h e o re m 2.10 ([16, Theorem 2.4.7]) Let f be Lipschitz near x, and suppose

0 0 d°f { x) . If C is defined as { y E X : f { y ) < f { x ) } , then one has

{ u G % : r ( : c ; n ) < 0 } C 7 b ( a ; ) .

C o ro lla ry 2.11 ([16, Corollary 1, p.56]) le f / 6e n m r a; and 0 0

Nc(ar) Ç U A a°/(z). A>0

If f is regular at x, then equality holds.

P r o p o s itio n 2.12 ([16, Proposition 2.4.4]) ^ C is conrez, (Aen, /o r eacA a; €

C, N c (x ) coincides with the cone of normals in the sense of convex analysis, that is,

JVc(T) = { ( € % ' : ( ^ ,c - a ; ) < 0 VcE C}.

P r o p o s itio n 2.13 ([16, Corollary, p.52]) Suppose th a t / is Lipschitz near a;

and attains a minimum over C at x. Then 0 G d ° f ( x ) + N c(x ).

P r o p o s itio n 2.14 ([16, Corollary, p.61]) Let f be Lipschitz near x. Then an element ^ of X* belongs to d° f { x) iff {f, - 1 ) belongs to Ngpi f( x, f {x) ) .

A n E x te n d e d D e fin itio n o f th e C la rk e S u b d ifie re n tia l

According to Proposition 2.14, the concept of the Clarke subdifferential can be extended to an extended real-valued function.

D e fin itio n 2.15 ([16, Definition 2.4.10]) Let f : X [—oo, -Too] be finite at

X E X. The Clarke subdifferential d° f { x) is the set

a ° / W := {f € V : (Ç .-1 ) 6 f { x ) ) } .

Let C be a subset of X . Then the indicator function of C defined by

I / \ f 0 if z G C 00 i f z $ ( C

is an extended real-valued function. When z is a point in C, the Clarke subd- ifferential d°tl^c{x) happens to be the normal cone to C at x.

P r o p o s itio n 2.16 ([16, Proposition 2.4.12]) Let the point x belong to C. then

and ipc{x) M regular at x iff C is regular at x.

From Proposition 2.14, the new definition of d° f { x) in Definition 2.15 is con­ sistent with the previous one for the locally Lipschitz case. Unlike the Lipschitz case, the new set d° f { x ) may be empty for the non-Lipschitz case. However if / attains a local minimum at x then d° f { x ) must be nonempty.

P r o p o s itio n 2.17 ([16, Proposition 2.4.11]) Let f : X [—oo, -Foo] be finite

at x £ X and / attains its local minimum at x. Then 0 € d°f { x) .

The Clarke subdifferential satisfies the following sum rule and chain rule.

P r o p o s itio n 2.18 (Sum Rule [16, Corollary 1, p .105]) Suppose that is finite

at X and /g is Lipschitz near x. Then one has

a°(A + /2 ) ( T ) c a ° A ( z ) - b a ° /2 ( z ) ,

and there is equality if f i and are also regular at x.

where h \ X ET' is a function whose component functions hi {i = 1, ■ • •, n) ore neor z E % onif g : 72" —^ 72 ia 7,(pacA*(z neor A(a;). One Aoa

9°/ ( z ) Ç c ô { ^ a i ( i : E := (« i, - ,0:^) E 9°g(/i(z))}

i=l

{where œ denotes weak* closed convex hull), and equality holds if g is regular at h{x), each hi is regular at x, and every a of d°g{h{x)) has nonnegative components. {In this case it follows that f is regular at x.)

A function / : X —)■ (—oo, + o c ] is said to be lower semicontinuous at x e X

provided th at

If f is lower semicontinuous at each point x E X , then / is called lower semi­ continuous. Given a topological space X , this is equivalent to saying th a t the epigraph e p i f is closed in X x 72 or th at the level set { x E X : f { x) < r} i closed in X for every r E 72 (see [30, Theorem 4, p .103]).

Similarly we say th a t a function / ; X —>■ (—oo, -foo] is weakly lower

continuous at z E X provided th a t for each sequence {xj} weakly converging to

x we have

/(z;) < lim in f/( z i) . Η»-fCX3

Clearly if / is weakly lower semicontinuous then it is lower semicontinuous. The converse may not be true, but a convex function on a locally convex space is lower semicontinuous if and only if it is weakly lower semicontinuous (see [30, Corollary 2, p .105]). The norm of X* is weak* lower semicontinuous (see also [21, Exercise 9, p .128]).

IS

semi-Let X be a Banach space and / : X —>■ (—00, +00] be lower semicontinuous

at a: G dom f. The Clarke-Rockafellar generalized directional derivative of / at

X in the direction u G X is defined as follows:

r ( x ; 0 : = lim lim sup inf +

e-> 0 + 1 w e v + e B t

y-^x

t-> 0 +

where y-Ux signifies th a t y and f { y ) converge to x and f { x) , respectively.

Clarke [16] proved th a t the extended f ° and d ° f have the same relation as they were in the Lipschitz case.

P r o p o s itio n 2.20 ([16, Corollary, p.97]) One has d° f { x) = 0 iff f°{x; 0) =

—00. Otherwise, one has

= {^ € X* : ((, n> < u) Vu G X },

and

u) = sup{((, u) : ( G /(a;)}.

R e m a r k 2.21 Proposition 2.20 implies th a t if f°{x; 0) ^ —00 then / ° (z; 0) = 0

and f°{x] ■) is sublinear and hence convex.

It is easy for us to obtain the following relation from the above concepts.

P r o p o s itio n 2.22 Let X be a normed linear space and / : X —^ (—00, +00] be

lower semicontinuous. For x G dom f, consider the following:

{i) There exist hg G X and p > 0 such that ||hx|| — 1 and f°{x; hx) < —

Then (i) implies {ii).

P ro o f. For x G d o m f if there exist hx £ X with \\hx\\ = 1 and p > 0 such th at

f°{x; hx) < — t hen for each f G d° f ( x) we have

This implies ||Ç||* > |(^, hx) \ > pT^. This proves th a t (i) implies {ii). ■

T h e S u b d iffe re n tia l o f a C o n v e x F u n c tio n

Recall th at a function f : X (—oo, +oo] is said to be c o n v e x if for every x , y E X and each A G (0,1) we have

f { Xx + (1 — X)y) < Xf { x) + (1 — X)f{y).

The following proposition implies th a t the Clarke subdifferential generalizes the subdifferential in the sense of convex analysis.

P r o p o s itio n 2.23 Let / : AT —> (—oo, +oo] be a convex function on X and

X G d o mf . Then d° f { x) coincides with the subdifferential of f at x in the sense

of convex analysis, that is,

= a / ( z ) : = { ( € % * : ( ( , ! / - z ) < /(%/) - /( a ; ) V{/ G X } .

P ro o f. By definition f G d° f { x) if and only if {f, - 1 ) G Ngp, f{x, f{^))- Since / is convex, epi f is convex. By Proposition 2.12, ({, —1) G Nepi f { x , f { x) ) if and only if

This inequality is in turn equivalent to th at

< / W - / W V^/ e X,

th a t is, f E ^ /( z ) . II

Exam ple 2.24 ([70, Example 2.26]) Let X be a Banach space and / : X —^ (—00, +oo) be a function given by

/ W = Vz E X.

Then

a ° /( z ) = a / ( z ) = {^ E X ' : ( f ,z ) = 11(11, - ||z|| and ||( ||, = ||z||}.

To characterize the noninclusion 0 0 d°f { x) , we need this expression and the following lemma.

L e m m a 2.25 Let X be a normed linear space and C be a nonempty weak*

compact subset of X*. Then there exists ( E C such that ||(|j* = dc(0).

P ro o f. Let {(n} be a sequence in C such th at l|(n||* —> dc(0). Then by the weak* compactness of C there exists a subsequence {(«,,} weak* convergent to some ( E C. Since the norm of X* is lower semicontinuous for the weak* topology, the norm of ( satisfies

11(11* < LûmiiLf||(m„ II, = dc(0).

K-~^-f-OC

Proposition 2.26 % 6e a BonocA apoce and / : % -4 B 6e IvipacWz near

X. Then the following are equivalent:

(i) 0 ^

(a) There exists ^ in d° f { x) such that

0 < ||( ||. = min{||7/||, : 77 e /(a;)} = dg«/(z)(0).

{iii) There exists fj, > 0 such that ||^||* > for all Ç G d°f { x) .

If also X is reflexive, then (z) — {in) are equivalent to each of the following:

(iv) There exist h^ £ X and // > 0 such that ||A^|| = 1 and /°(x ; hx) < —

(v) There exist ^ E d° f { x) and Vx E X such that ||^||* — ||%|| and

-zzg) = ((, - % ) = - ||( ||* - ||i;z|| < 0.

(vi) There exist ^ G d° f { x) and Vx E X such that ||^||* = ||%|| ond, for each

a G (0,1), there exists 6 > 0 such that

/(!/ - W < / W - (all^ll* - ll%ll < / W Vp G a; + 6B & V( G (0, d).

P ro o f. We prove this proposition by the following route:

(i) (ii) =4> (m ) => (z); (zz) => (v) {iv) {in)-, {v) {vi).

(z) => (zz) : Let 0 0 d° f { x) . Then

Since, by Proposition 2.4, d° f { x) is nonempty, convex and weak* compact, it follows from Lemma 2.25 th at there exists a ^ G d° f { x) such th a t

0 < 11(11, = min{||T7||. : E = dy/(z)(0).

{a) => (iii) follows immediately by taking p = ||( ||“ ^ and (in) => (i) is obvious.

(n) =>■ (r;) ; If statem ent (ii) is true, then ( is a minimiser of the convex function

/!(;?) = ^ lh ||2 subject to 7? E /( z ) .

Applying Proposition 2.13 to C = we have

O E a ° /t(() + Arc(().

This inclusion implies th a t there exists % E d°h(^) such th a t — % E 7Vc(()- Since C = d° f { x) is convex and closed, by Proposition 2.12,

; ? - ( > < 0, %.e., (77, -7 ;z ) < ( ( , -Uz> V77 E / ( z ) .

According to Proposition 2.4,

= max{(77, - n , ) : 77 E = ( ( , - % ) = - | | ( | | , - ||7;z|| < 0,

where the last equality is due to the fact th a t in a reflexive Banach space X

there holds

a ° /i ( ( ) = 9 A (() = {uz E A : ( ( , % ) = 11(11* ' ||% || a n d | | ( | | , =

ll^^zH}-{v) => {iv) : If there exist ( E d ° f { x ) and E A such th a t ||( ||, = ||% || and

then, taking n = ||C||r^ and hx = we have

Thus (n) => {iv) holds.

{iv) => {iii) is direct from Proposition 2.22.

(n) 4^ (n%) : Let ^ 6 9 ° /(z ) and n, E % be such th a t ||^||, = ||nz|| and

/"(a:; - t; i ) = ((, -% > = - Ikzll < 0.

Then, for a E (0,1) and for e = (1 —a)||^||*-||% ||, by the definition of f°{x; —%), there exists 6 > 0 such th a t for each y £ x + 6B and t E (0, Ô) we have

= —II6II* ■ ll%ll + (1 — «)||CI|* ■\\vx\\

= < 0,

from which it follows th at

/( l/ - tui) < / W - (allait, - ll^zll < / W E z + & Vt E (0,

Conversely, if ^ E d ° f{ x ) and Vx £ X are such th a t ||^||* = ||%|| and for each a £ (0,1) there exists <5 > 0 satisfying

/(%/ - < / W - ^«11(11* - 1|%|| < /(%/) Vy E a; + & Vt E (0,6),

then

B y th e d e h n itio n s o f / ° ( z ; —% ) a n d 9 ° / ( z ) , w e have

— ■ ll%ll < (Ç) ~ ^ x ) < f ° { x ] —Vx) < —o;||Ç|i* • ||% || < 0 V a e (0 , 1).

L e t ti n g a f 1 gives

-%;i) = ((, - t ; ,) = - | | ( | | , - ||t;z|| < 0.

R e m a r k 2.27 From the proof of Proposition 2.26 we see th a t if 0 0 d ° f{x )

and ^ is the element of least norm in d ° f { x ) then there exists Vx E X satisfying

(vi) in Proposition 2.26. It follows th at for each a G (0,1) there exists a 5 > 0

such th at for each y E x + SB and each t E (0,5) the point z — y — tvx satisfies

0 < a||2/ - z|| < - /(z )],

where

|1?||F^-2.2

Various D erivatives

We recall some concepts of classical derivatives and study their relations to the Clarke subdifferential.

D e fin itio n 2.28 Let F map a normed linear space X to a Banach space Y and

£ { X , Y ) be the space of continuous linear operators from X to Y .

• X is strictly (Hadamard) differentiable a t z E X if there is an operator

D sF[x) E £ { X , F ) such th a t for every v E X

Ihn t->-o+

and the convergence is uniform for u in any compact sets. The corre-sponding operator D sF{x) is said to be the strict derivative of F at x.

• F is sa id t o be Gâteaux {Fréchet) differentiable a t a: E X if th e r e is a n o p e r a to r D F {x ) E £ ( X , Y ) su ch t h a t for an y n in X

lim + _ o F { x ) { v )

t-»0+ t

and the convergence is uniform with respect to v in any finite (bounded) sets. The corresponding operator D F {x ) is called the Gâteaux {Fréchet)

derivative of F at x.

• F is continuously Gâteaux {Fréchet) differentiable at æ E X if there exists

f > 0 such th a t F is G âteaux (Fréchet) differentiable at each point y in

X + 5B and the mapping y -4 - D F {y ) is continuous at x. In particular, F

is Gâteaux {Fréchet) at z E X if F is continuously G âteaux (Fréchet)

differentiable at x and the mapping y - 4 D F {y ) is Lipschitz on æ 4 SB for some é > 0.

For a function there are close relations among the strict differentiability, the Clarke subdifferential and other differentiabilities of it.

P ro p o s itio n 2.29 ([16, Proposition 2.2.4]) A function f : X R is strictly

differentiable at x and D J { x ) = Ç iff f is Lipschitz near x and d° f { x ) ~

P ro p o s itio n 2.30 ([16, Proposition 2.3.6]) Let f be Lipschitz near x.

(b) If f admits a Gâteaux derivative D f { x ) and is regular at x, then d° f { x ) —

{ D /W } .

P r o p o s itio n 2.31 ([83, Proposition 2.17]) A function f : X R is strictly differentiable at x if and only if f is Lipschitz near x, Gâteaux differentiable and regular at x.

W ith the above propositions we derive the following simple but useful result:

P r o p o s itio n 2.32 ([87, Proposition 3.1]) Let f be Lipschitz near x € X . Then f is strictly differentiable at x if and only if both f and —f are regular at x.

P ro o f. If / is strictly differentiable at x, then —/ is also strictly differentiable at X, and hence by Proposition 2.30 they are both regular at x .

Conversely if f and —/ are both regular at x, then by Proposition 2.18 we have

a 7 W + ô ° ( - / ) ( j ) = {o}

which means th at d° f { x) is a singleton since both d ° f ( x) and d° { —f ) { x ) are

nonempty. Therefore / is strictly differentiable at x. ■

The condition for a mapping F : X Y t o he Fréchet differentiable at z E

X is equivalent to the assertion th a t there exists an operator DF { x ) E £ { X , Y)

such th at for any e > 0 there exists 5 > 0 such th a t

\\F{x + h) — F{x) — DF{x){h)\\ < e|]h|] whenever h E X and ||h|| < S

(cf. [70, Definition 1.12]). Hence if F is Fréchet differentiable at x then F is continuous at x. However F is not necessarily Lipschitz continuous at x.

E xam ple 2.33 The function / : A —y A given by

_

f

( 0 i f z = 0

is Fréchet differentiable at z — 0 with the Fréchet derivative /'(O) = 0. Since for re ^ 0 the derivative f' {x) = 2x sin ^ | cos f' {x) is unbounded near z = 0. This implies th a t / is not Lipschitz continuous near x — 0.

It is easy to see th at pointwise Fréchet (strict) differentiability implies G âteaux differentiability. The following example shows th a t its converse may not be true.

Exam ple 2.34 ([18, Exercise 11.20 (c), p.66]) Let / : ^ A be dehned by

It is easy to check th a t / is Gâteaux differentiable at (0,0) but th a t / is not continuous there. And hence / is neither strictly differentiable nor Fréchet differentiable at (0,0).

Although pointwise Gâteaux differentiability does not imply Fréchet differ­ entiability, a continuously Gâteaux differentiable function is always continuously Fréchet differentiable.

P r o p o s itio n 2.35 (e.g., see [25]) A function f : X R is continuously Gâteaux differentiable at x Ç. X if and only if f is continuously Fréchet dif­ ferentiable at this point. Moreover, these derivatives are identical.

Based on Proposition 2.35, / is simply said to be at x if it is continuously Gâteaux differentiable at x. Similarly f is at x provided / is G âteaux at a;.

2.3

T he M ichel-Penot SubdiSerential

The concept of the Michel-Penot snbdiSerential is very similar to th a t of the Clarke subdifferential. There are also analogous properties for these two subd­ ifferentials.

D e û n itio n 2.36 ([64, Dehnition 1.1]) Let f/ be an open subset of a locally convex topological vector space % and / : f/ — A be a function. The

Mtchel-Penot derivative of / at a; Ç 17 in the direction v ^ X is

r ( x - v )

:= su p lim su p i< i± M ± ^

l l i ( £ ± M .

y t-^o+ i

The Michel-Penot subdifferential of / at æ is the set

a ^ /(z ) := 6 %* : ((, u) < Vu E %}.

P r o p o s itio n 2.37 ([64, Proposition 1.2]) The mapping ff{x ; • ) : « - > f'^{x; v) is sublinear.

It is easy to see th a t if / is Lipschitz of rank L near x then for every v £ X

we have

/""(r;u) < /°(3:;i;) < (2.1)

This inequality with Proposition 2.37 implies th at

/""(a;; ï;i) < /^(a;; U2) + L||ui - ugH Vui, U2 E X.

Since the above inequality holds with v\ and ug switched, ff {x ] •) is Lipschitz of rank L on X . Besides it follows from inequality (2.1) th a t

However this inclusion may be strict as shown by the following example.

Exam ple 2.38 ([16, Example 2.2.3], [12, Example 6]) The function

( 0 ifT = 0

is Lipschitz near 0 and Gâteaux differentiable with

ao/(o) = { / ( o ) } = {0} c [-1 ,1 ] = a v ( o )

where the first equality is from the fact th a t d^f{x) reduces to the Gâteaux derivative when f is Gâteaux differentiable at x.

P r o p o s itio n 2.39 ([64, Proposition 1.5]) Let U be an open subset of a normed linear space X . Suppose f : U —> R attains its local minimum at x E U. Then

0 e a^/(a:).

P r o p o s itio n 2.40 ([64, Proposition 1.6]) Let U be an open subset of a normed

linear apoce %. Ginen / : [ / — one hoa, /o r onp % € [/, u E %,

( / + g)"'(a:;i;)< /''(a;;t;) + a'''(:r;n).

If f^{x] •) is finite and continuous at some point where g^{x; •) is finite, then

^ ( / + g ) ( 4 ^ + ^^(a;).

It is known th a t if / is Lipschitz near x then f^{x; •) is Lipschitz continuous. In this case the sum rule in Proposition 2.40 holds no m atter whether g’^{x] •) is finite at some point since if g'^{x; v) = +oo for each v E X then for each ^ E d^( f + g)(x) and rj € d^f (x) we always have

which implies ^ — 77 G d^g{x) and hence

P rop osition 2.41 ([64, Proposition 1.7]) (7 6e on open an^eet 0 / 0 normed

Zineor apoce %. For eoch (a;, n) E (7 x % one hoa

r ( a ; ; - n ) = ( - / ) ^ ( z ; n ), ^ ( - / ) ( z ) = - ^ / ( a ; ) .

For more properties of the Michel-Penot subdifferential we refer to [6, 12, 22, 37, 63, 64].

2.4

T he Proxim al Subdifferential

D e fin itio n 2.42 Let / : X —> ( —00,4-00] be lower semicontinuous and x E

dom f . A vector ^ E X* is said to be a proximal subgradient of / a t z if for

some M > 0 there exists 5 > 0 such th at

/ ( ! / ) - / ( 4 + M ||p - a;||^ > (^, P - a;) Vp E a; 4- 6 B .

Another way to say this is th at

y-^x

||p - rr|p

The proximal subdifferential of / a t x, denoted by d pf { x) , is the set of proximal

subgradients of / at a;.

R e m a rk 2.43 The concept of the proximal subdifferential was first introduced by Rockafellar [74] for lower semicontinuous functions in 72". The reason th a t

“proximal” is used is th a t in i?” (even in a Hilbert space) the proximal sub­ differential can be defined through the closest point to a set. For a general Banach space, d p f { x ) as defined is called the l-Holder-subdifferential (e.g., see [7]). For simplicity, in spite of slight abuse of terminology, we still call dp f { x )

the proximal subdifferential of / on a normed linear space.

By definition the proximal subdifferential has the following properties.

P r o p o s itio n 2.44 Let f : X (—oo, -t-oo] attain its local minimum at x. Then

0 e

L e m m a 2.45 ([47, Corollary 4A.5]) Let / : X —> (—oo, -t-oo] be lower semi­ continuous and Gâteaux differentiable at x. Then

Ç { D /(z )} ,

where D f { x ) is the Gâteaux derivative of f at x.

The following example shows th at a Gâteaux differentiable function may have no proximal subgradients.

E x a m p le 2.46 Consider the function f { x ) = —|ar|2, x E R . It is easy to see th a t Df { 0 ) = 0. But dpf { 0) = 0. Otherwise, suppose th a t dpf { 0) were nonempty. Then, by Lemma 2.45, d p f { 0 ) = {0}, th a t is, there exist M > 0

and d > 0 such th a t

This is a contradiction since the inequality fails to hold for any x e (—tW, 0) U_M2: M2,'

(0, J z ) '

Note th a t in this example / ( z ) is at 0. This shows th a t 9 f / ( z ) may be empty even for functions. However if a function / is at x, then d p f { x )

is nonem pty and coincides with { D /(z )} .

P r o p o s itio n 2.47 ([83, Proposition 2.29]) Tef / : {oo} a:.

TTien

^ / ( a ;) .

This condition is only sufficient but not necessary. Even for a function f

w ith both d p f { x ) and d p { —f ) { x) being nonempty, the function / may still not Lipschitz near x.

E x a m p le 2.48 It is easy to see th at for the function

\ _ f a;^8i n ^ i f z f O

{ 0 ifa; = 0

both d pf { 0) and d p{ —f ) {0) are equal to {0}. However as we have seen in Ex­ ample 2.33, the function is not Lipschitz continuous near a; = 0. So / is not at a; = 0.

Unlike the Clarke subdifferential and the Michel-Penot subdifferential, the proximal subdifferential usually does not satisfy the sum rule but has the fol­ lowing properties.

P rop osition 2.49 JLef ^ % -4^ 72 U { +00} 6e iotuer gemtcon^inuotw and a; w m (dom /) n (domg) aucA (/ia( J f /( z ) ond 9fg(a;) are 6otA nonemp%. TTien

^ f/(a ;) + 9fg(a;) Ç ( / + g)(a;).

P r o p o s itio n 2.50 ([83, Proposition 2.43]) Let f , g : X R U {00} be lower

semicontinuous. Suppose that d p f { x ) is nonempty and g is at x. Then

( / ± p ) (z ) = / ( z ) 4 : g(T).

T h e o re m 2.51 ([18, Theorem 8.3, p.56]) Let X be a Hilbert space and f , g : X - 4 ( —0 0 ,+ 0 0 ] be lower semicontinuous. If g is Lipschitz near x G dom f and

^ € 9 p ( / + g){x), then, for any e > 0, there exist Xi , X2 E x + eB such that

|/ ( a : i ) - /(a ;)] < e, |^(a;2) - g(a;)| < e

and

f € d p f ( x i ) + dpg{ x2) + eB.

2.5

T he Fréchet Subdifferential

D e fin itio n 2.52 Let C be a nonempty and closed subset of a Banach space X

and x & C. For any e > 0, the set of Fréchet e-normals to C at x, denoted by

Np{x; C), is defined to be the nonempty set

A ^ (T ; C ) : = e X ' : lim s u p ^ < e} y { € C ) - ^ x \\y - x\\

for z with C n (x + SB) ^ {z} for any <5 > 0 and Np( x] C) := X* otherwise. In particular, when e = 0, the corresponding set Np{x] C), which is a cone, is called the Fréchet normal cone to C at z and is denoted by Np{x-, C).

D e fin itio n 2.53 Let f : X ( —0 0 ,0 0 ] be a lower semicontinuous function and let x G dom / . For any e > 0, the convex set

: = e X * : ( ( , - 1 ) € ]V^((a;, /(a ;) ); e p z / ) }

is called the Fréchet e—subdifferential of f at x. In particular, for e = 0, the cor­ responding d p f { x ) is called the Fréchet subdifferential of / at a: and is denoted

by The function / is said to be at a; provided

th a t ^f'/(a;) is nonempty.

For a Lipschitz continuous function the Fréchet subdifferential is bounded.

P rop osition 2.54 Tef / : % — (—0 0,0 0] 6e TipacAitz 0/ L near z. Then,

/o r onp e > 0,

a ^ /(z )C (L -k e (l-h L ))5 ^ .

In particular, d p f i x ) Ç LB*.

P ro o f. Let ^ be in dp f { x ) . Then, for any Ci > 0, there exists 6 > 0 such th at,

for (y, u) e epi f with \\y — z|| < 6 and | u — f { x ) (< L8,

*/ - a;) - (n - /(a ;)) < (e + 6i)(||2/ - a;||+ | n - / ( z ) |)

from which it follows th at

<(, 3/ - 4 < ( / W - / ( 4 ) + (^ + - 3:||+ I / W - / ( 4 I).

Using the Lipschitz condition we obtain

Hence for any y ^ x the following inequality holds

which implies th at ||^||* < L + (e + ei)(l + L). Since ci > 0 is arbitrary, ||( ||* < j; + e(l + Z,).

This proves th at the inclusion stated holds. ■

Let / : X —> (—00,00] be a lower semicontinuous function and let x G dom f.

For any e > 0, we denote

4 /(x ) := {Ç . X - : > -e } .

Particularly, for e = 0, the corresponding dof{x) is denoted by df { x) .

Ioffe [36] proved th a t the Fréchet subdifferential d p f i x ) is the same as df { x ) .

P r o p o s itio n 2.55 ([36, Proposition 1]) Suppose that f is lower semicontinuous

at X and e > 0. If ^ E d p f i x ) , then ^ G dsf { x) with Ô= (e /(l — e))(l + ||^ ||* ).

C onneraefy, G / o r a o m e > 0, th en ^ G ^ / ( a ; ) . TTma

According to Proposition 2.55 we can further derive properties of the Fréchet subdifferential.

P r o p o s itio n 2.56 Suppose that f is lower semicontinuous at x. Then^ G d p f i x )

if and only if for each e > 0 there exists d > 0 such that

Hence 0 E d p f i x ) if and only if for every e > 0 the function /(•) 4- e|| • —x||

attains its local minimum at x. In particular if f attains its local minimum at

T (Aen 0 E / (a;).

P ro o f. By Proposition 2.55, f G d p f i x ) if and only if

11^ - :r||

This is equivalent to saying th a t for every e > 0 there holds

||i/ - a;||

th a t is, for every e > 0 there exists 5 > 0 such th at

/(%/) > / W + (^, 3/ - z) - e||3/ - a;|| Vy G a: + JB .

P r o p o s itio n 2.5 7 Suppose that f , g : X (—oo, +oo] are lower semicontinu­

ous at X e {dom f ) n ( domg) . Then

G ( / + p)(z).

P ro o f. The inclusion follows directly from Proposition 2.55 and the following inequality

l i ^ i ^ ( / + W - ( / + 3/ - :r) ^

\\y - x\\

lim inf + lim inf <l(v) ~ s{^) ~ { y ~ ^)

i | y - a : | | ||2/ - ^11

Prop osition 2.58 6'uppoae f/tof / : % (—0 0, +00] w (oiuer aemiconfinuottg

o( z E % . T /ien / w (fi^ eren fio b ie T ^ o n d on/;/ i / 6of/i o n d

dF( —f ) { x) are nonempty.

P ro o f. It is known th at / is Fréchet differentiable at x if and only if for some ^ e X* and any e > 0 there exists é > 0 such th a t

l / W - / W - ( ( , 2/ - z ) | < c||2/ - a;|| Vi/ E z +

This with Proposition 2.56 implies th at Ç E d p f i x ) and —Ç € dp^—f )( x) . Thus the necessity follows.

To prove the sufficiency, we suppose th a t both d p f i x ) and 9jp(—/)( x ) are nonempty. Then by Proposition 2.57 we have

+ a f ( - / ) W Ç {0}.

Thus d p f i x ) = {$} and d F{ —f ) { x ) — { —^} for some Ç E X * . It follows th a t

i t a i „ f M . r4 X X j ^ > o

II;/ - a;|| and

\\y — a;||

From these two inequalities we obtain

| | y - 4

th a t is, / is Fréchet differentiable at x. ■

C o ro lla ry 2.59 Suppose that f , g : X -> ( —00, + 0 0 ] are lower semicontinuous

at X E ( d o mf ) fl {dom g) and that f is Fréchet differentiable at x. Then

P ro o f. Suppose th a t f is Fréchet differentiable at x. Then, by Propositions 2.57 and 2.58, = {0}. Thus + G {o} + a f ( / + g)(T) = ( / + ^)(z) G + ajpp(z).

This completes the proof. m

It is interesting th a t some special Banach spaces can be characterized with the fuzzy sum rule of the Fréchet subdifferential.

D e fin itio n 2.60 A Banach space X is said to be an Asplund space provided every continuous convex function defined on a nonempty open convex subset U

of X is Fréchet differentiable at each point of some dense Gs subset of U.

If the dual space X* of the Banach space X is separable, then X is an Asplund space ([70, p.22]). Every reflexive Banach space is an Asplund space ([70, p.24]). But not all Banach spaces are Asplund spaces. For example neither P nor l°° is an Asplund space ([70, p .13]).

Fabian has characterized an Asplund space interestingly in terms of the Fréchet subdifferential as follows.

T h e o re m 2.61 ([28, Theorem 3]) X is an Asplund space {if and ) only if it

functions / i , • • •, /„ : X —> (—00, +00], n > 2, and for any z E X such that f i

w Zotuer and /z, " , A ofiG m a neig/i6orAood 0/ z (Ae

following inclusion holds

d e i f l + h f n ) { z ) Ç u{9i?/i(zi) H---h d p f n i ^ n ) '■

Zj E z + SB, \fj{zj) — fj {z)\ < 5, j = 1, ■ ■ ■ , n } + {e + j)B * .

2.6

T he D ini Subdifferentials

D e fin itio n 2.62 Let f : X (—00, +00] be lower semicontinuous a t a; €

dom f. The upper Dini derivative of / at x in the direction v E X is

/ + ( . ; » ) : = ü m s u p h £ ± t ï W M .

u—^v t

The upper Dini subdifferential of / at x is the set

a + / ( z ) : = { ( E X * : <(, n) < /+ ( a ;; u) Vu E X } .

Similarly the lower Dini derivative of / at x in the direction v E X is

/ - f e . ) : = = l i m m t h £ ± M z i M .

^ U-l-V t

t-^0+

The lower Dini subdifferential of / at a: is the set

: = (^, u) < / - ( z ; u) Vu E % } .

It is immediate from definition th a t the lower Dini subdifferential has the following properties.

P r o p o s itio n 2.63 Suppose that f is lower semicontinuous at x. If f attains

P r o p o sitio n 2.64 f/iat / , p : % (—00, +00] ore fower gemicontino-00g o( T € (dom /) n (domg). TAen

9 '/ ( z ) + 9 " ^ (z ) Ç ^ ( / + g)(z).

It is easy to see th a t if / is Lipschitz near x then

/+ ( a ;; n ) = / ^ ( a ; ; n ) : = lim su p

t->o+ t

/ - ( z ; n) = //" (a;; n) := Urn inf

9- / (3;) = a^ /(a;) : = { ( € % * : ( ( , u ) < / ^ ( a ; ; n ) Vu 6 %},

9 + / ( z ) = 9^ / (3;) : = 6 X * : (& u) < / ^ ( 3 ; ; u ) Vu e % } .

By the simpler form of the definitions the following proposition is immediate for a Lipschitz continuous function.

P r o p o s i t i o n 2 .6 5 Let / : X —>■ ( —00, 0 0] be Lipschitz of rank L near x, then

9/ (3;) Ç 9 f/(a ;) Ç 9 f/(a ;) Ç 9- / (3;) C 9+/(3;) C LB*.

P r o p o s i t i o n 2 .6 6 Let / : X —> ( —00, + 0 0 ] be lower semicontinuous at x E

d o m / . Then / is Gâteaux differentiable at x if and only if both d f f { x ) and d f { —f ) { x) are nonempty.

P r o o f . The necessity is obvious. We only need to prove the sufficiency. Sup­ pose th at both d f f { x ) and d f { —f ) { x ) are nonempty. Then similar to Proposi­ tion 2.64 we have

T h u s = {6} a n d ( —/ ) ( z ) = fo r so m e ^ E % *. I t follow s t h a t i i ^ i „ f / ( ^ + M .= i W > f c „ ) v . e x (-4.0+ t - \S, / a n d |i m s u p l ( l ± 5 ^ h L Z M < ( ç , „ ) v „ e x t-+o+ i

From these two inequalities we obtain

f-+o+ ÿ ' ^

th at is, / is G âteaux differentiable at x. ■

Note th a t for a concave continuous function f the lower Dini sub differen­ tial d l ( —f ) ( x) is always nonempty since d l ( —f ) { x ) coincides with d° { —f ) { x)

(which is nonem pty). Hence by Proposition 2.66 we have the following corollary.

C o r o l l a r y 2.67 Let f : X R be a concave continuous function. Then f is Gâteaux differentiable at x if and only if d f f { x ) is nonempty.

Similar to the Fréchet subdifferential, the lower Dini subdifferential has the following fuzzy sum rule:

P r o p o s i t i o n 2.68 ([35, Theorem 2]) Let / i , • • •, /n be lower semicontinuous at

X E i2” . Then

d (/i H h f n ) ir ) Ç n

U

s>o xj e u{ f j , x, s )

where [ / ( / , (^) = {^ E E " : ||t/ - z || < f, / ( y ) - / ( z ) < J }.

It is worth noting th a t the above property does not hold in a general Banach space. To explain this we consider the following example due to Ioffe [35].