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Metric regularity for strongly compactly Lipschitzian mappings
Pergamon
NonlinearAnalysis, Theory,Methods&Applications,Vol. 24, No. 2, pp. 229-240, 1995 Copyright © 1995ElsevierScienceLtd Printed in GreatBritain.All rightsreserved 0362-546X/95$9.50+ .00 0362-546X(94)E0061-K
METRIC REGULARITY FOR STRONGLY COMPACTLY LIPSCHITZIAN MAPPINGS A. JOURANH and L. THIBAULT~ tUniversit6 de Bourgogne, Laboratoire d'Analyse Appliqu6e et Optimisation, B.P. 138, 21004 Dijon C6dex, France; and ~Universit6 Montpellier II, Departement des Sciences Math6matiques, 34095 Montpellier Cedex 5, France (Received 25 May 1993; received in revised form 3 December 1993; received f o r publication 30 March 1994) Key words and phrases: Metric regularity, approximate subdifferential, strongly compactly Lipschitzian mappings, compactly epi-Lipschitzian sets, Fritz-John multipliers.
1. I N T R O D U C T I O N
Feasible sets of optimization constraint problems generally appear in the form C n g-l(D), where C and D are two subsets of two Banach spaces X and Y and g is a mapping from X into Y. It is now known (see [1]) that a general constraint qualification allowing us to get the existence of multipliers for optimization constraint problems with constraints in the form C n g-l(D) is the so-called metric regularity (of g with respect to C and D) whose importance is essentially due to Ljusternik and Robinson [2, 3]. Recall that g is metrically regular at Xo ~ C n g - l ( D ) with respect to C and D if there exist some real number a >_ 0 and two neighbourhoods X 0 and Y0 of Xo and zero in X and Y, respectively, such that d(x, C n g-l(D - y)) <_ ad(g(x) + y,D)
(1.1)
for all x ~ C n x0 and y e Yo. In this paper we are essentially concerned with easily verifiable conditions in terms of subdifferentials implying the metric regularity as initiated by Ioffe [4] who introduced the use of Ekeland variational principle in this study. This idea has been followed by Auslender [5] and largely extended by Borwein [6] to derive, for Y = ~P, conditions with Clarke generalized gradients ensuring relation (1.1). Mordukhovich [7] (see also [8, 9]) has shown that, when X and Y are finite dimensional, similar conditions, but with his notion of approximate subdifferentials instead of Clarke generalized gradients, are both necessary and sufficient for the metric regularity (1.1). So this paper is devoted to showing how the use of the extension by Ioffe [10-12] of the approximate subdifferentials to Banach spaces will allow us to generalize the above conditions when X and Y are infinite dimensional and D is compactly epi-Lipschitzian in the sense introduced by Borwein and Strojwas [13]. Our results are along the lines of our preceding ones [1, 14] and depend heavily on our lemma 3.2 establishing a new and important behaviour for compactly epi-Lipschitzian subsets. We shall also show that these results are too strong to allow us to derive the existence of Fritz-John multipliers for general infinite dimensional optimization problems. 2. APPROXIMATE SUBDIFFERENTIALS AND PRELIMINARIES Throughout we shall assume that X and Y are Banach spaces X* and Y* are their topological duals and ( ' , • ) is the pairing between the spaces. We denote by B x , B?c . . . . the closed unit balls 229
230
A. JOURANI
a n d L. T H I B A U L T
o f X, AT*. . . . . By d ( ' , S) we denote the usual distance function to the subset S C X
d(x, S) = infllx - ul[. u~S
f
s
We write x ~ x0 and x ~ xo to express x ~ Xo with f(x) -~ f(Xo) and x ~ x0 with x e S, respectively. We denote by G r F the graph of a multivalued function F: X ~ Y, that is, Gr F = [(x, y) e X x Y: y e F(x)l. I f f is an extended-real-valued function on X, we write for any subset S of X
x)
if x e S
fs(x) = (. +oo
otherwise.
The function
d - f ( x , h) = lim inf t - l ( f ( x + tu) - f(x)) u~h tJ, O
is the lower Dini directional derivative of f a t x and the Dini e-subdifferential of f a t x is the set
O~f(x) = Ix* ~ X*: (x*, h) < d - f ( x ; h) + ellhll, v h ~ x } for x e D o m f and O-;f(x) = O if x ¢ Dora f , where D o m f denotes the effective domain o f f . For e = 0 we write O-f(x). By 5:(X) we denote the collection o f finite dimensional subspaces o f X. The approximate subdifferentials of f a t x 0 e D o m f is defined by the following expressions (see Ioffe [11, 12])
014f(Xo) =
0 le~(X)
lim sup O-f.+L (x) = X~Xof
[") Left(X)
lim sup 07fx+ L (x), t
X~Xo
where lim sup O-fx+L (x) = Ix* e X*: x* = w* - lira x*, x* e O-fxi+L(xi), xi ~ Xo], that is, the f X--b Xo
set of w*-limits of all such nets. It is easily seen that the multivalued function
X ~ OAf(x ) is upper-semicontinuous in the following sense
0.4 f(xo) = lim sup 0A f(x) f
X -+ X0
and in [12] Ioffe has shown that when S is a closed subset o f X and x0 e S
04d(x o, S) =
0 L e ~(X)
lim sup Oldx+L(x, S).
(2.1)
s
X-~ Xo e~O
The following sum rule has been established by Ioffe in [11] for a more general situation. For the purpose of our discussion, a semi-Lipschitz case suffices.
Metric regularity
231
THEOREM 2.1 [11]. Let f : X ~ R be a function which is lower-semicontinuous near x0 and g: X ~ R be a function which is Lipschitz around x0. Then
a A ( f + g)(xo) C OAf(Xo) + aAg(XO). In the sequel we shall need the following class of mappings between Banach spaces.
Definition 2.2 [1]. A mapping g: X ~ Y is said to be strongly compactly Lipschitzian (s.c.L.) at a point Xo if there exist a multivalued function R: X ::* Comp(Y), where Comp(Y) denotes the set of all norm compact subsets of Y, and a function r: X x X --, [R+ satisfying: (i) lim r(x, h) = 0; X~X
0
h~O
(ii) there exists o~ > 0 such that
t - l ( g ( x + th) - g(x)) ~ R(h) +
Ilhllr(x,
th)By
for all x • Xo + ocBx, h • aBx and t e ]0, or[; (iii) R(O) = 10] and R is upper-semicontinuous.
It can be shown [15] that every s.c.L, mapping is locally Lipschitz. In finite dimensions the concepts coincide. Recently, we have developed in [16] a chain rule for this class of mappings. Let us note that this chain rule has been obtained before by Ioffe in [12] for maps with compact prederivatives. THEOREM 2.3 [16]. Let g: X --' Ybe s.c.L, at x0 and let f : Y --, ~ be locally Lipschitz at g(xo). Then f . g is locally Lipschitz at Xo and
OA(f o g)(Xo) C
U
On(y* ° g)(xo).
Y* • aAf(g(xo))
To complete this section we note the following property of s.c.L, mappings which is a direct consequence of lemma 2.5 in [17]. See also proposition 2.3 in [16] and theorem 1.5 in [18]. L E I ~ 2.4. Let g: X ~ Y be s.c.L, at x 0 and let (Yi*) any bounded net of Y* which w*converges to zero in Y* and let (xi) be a net IIII-converging to x0 in x . If x* • Oa(Y* o g)(xi), then (xi*) w*-converges to zero in X*.
Remark. Assuming that Y is separable, Glover and Craven [18] have independently shown later that, when g is s.c.L., the multivalued function (x, y*) ---* aA(Y*, g)(x) is closed (with respect to the w*-convergence in y * e Y*). This is also a direct consequence of lemma 2.5 in [17] without the separability assumption on Y. Let S be a subset o f a normed vector space Z and Zo e S. We recall the following definition.
232
A. JOURANI and L. THIBAULT
Definition 2.5. (1) S is epi-Lipschitz at zo (see [19]) if there exist ot > 0 and h e Z such that S n (Zo + otBz) + ]0, ot](h + aBz) C S. (2) S is epi-Lipschitz-like at z0 (see [20]) if there exist a > 0, h e Z and a nonempty convex subset ~ o f Z containing zero with ~o weak-star locally compact such that
S n (Zo + aBz) + ]0, ot[(h + f2) c S, where flo = {z* e Z*: 0 and a compact subset H such that
S n (Zo + aBz) + taBz C S - tH,
for all t e ]0, a[.
To close this section, let us recall the following important theorem of Borwein [20].
THEOREM 2.6 [20]. Let Z be a normed space, S be a closed subset of Z and Zo e S. (a) If S is epi-Lipschitz at Zo then S is epi-Lipschitz-like at Zo. (b) If S is epi-Lipschitz-like at Zo then S is compactly epi-Lipschitz at zo. (c) If Z is a finite dimensional space then each subset of Z is epi-Lipschitz-like at all its points.
3. M E T R I C R E G U L A R I T Y
Definition 3.1. Let F: X ~ Y be a multivalued function and (Xo, Yo) e Gr F. One says that F is metrically regular at (Xo, Yo) if there exist two real numbers k _ 0 and e > 0 such that d(x, F - l ( y ) ) -< kd(y, F(x)) for all x ~ Xo + eBx and y ~ Yo d(x, Q) = +0o.
+
eBy with
d ( y , F(x)) <_ e. Here we adopt the convention
Two of the key results used in the p r o o f of theorem 3.4 are the two following lemmas. LEMMA 3.2 [21]. Let F : X ::t y be a multivalued function of closed graph and let (xo, Yo) ~ Gr F. If F is not metrically regular at (Xo, Yo), then there are s n ~ 0, xn ~ x o, z~ --' Yo and Yn --' Yo such that O) (x~, Zn) ~ Gr F, for all n e N*; (ii) y~ ¢ F(x~) for all n e ~q*; and (iii) for all (x, y) e Gr F [Iz~ - y~l[ - [ly - y~[I + s~(I[x - x~[I + [[y - z~[[).
Remark. This lemma is mainly based on the ideas o f Borwein [ 6 ] .
Metric regularity
233
LEMMA 3.3. Let Z be a normed vector space and let S be a subset of Z. Suppose that S is compactly epi-Lipschitz at zo • S. Then, for H given by definition 2.5 there exist y > 0 and a neighbourhood V of Zo such that for each e > 0 there are h~ . . . . . hm • H satisfying Ilz*ll <_ e + y
max
i= l , . . . , m
I(z*, hi)[,
Y z • V,
Y Z* • OAd(Z, S ).
Proof. Let H be a compact subset of Z and r • ]0, 1] such that S tq (Zo + 3rBz) + trBz C S - tH,
for all t • ]0, r].
Choose an open neighbourhood V of Zo with V C Zo + rBz and d(z, S) <_ r for all z • V. Let h I .... , hm • Hwith
H C C) (hi + erBz). i=1
For each z • V and each t • ]0, r] we may select some p(z, t) ~ S such that Ilz -
p(z, t)[I ~ d(z, S) + t 2.
(3.1)
By the choice of V we have for each z • V and each t • ]0, r]
Ilzo - p(z, t) < IIz - p ( z , t)ll + IIz - zoll < d(z, S) + t 2 + r < 3r and, hence, p(z, t) • Zo + 3rBz. Fix any z • V a n d any z* • OAd(z, S). Fix any b • Bz and any L • 5:(X) with [b, hi . . . . . hm] C L. We may write z* = x* - l i m z * with z 7 • O-dzj+L(zj, S) j~J
and zj -~ z. Choose Jo • J such that zj • V for any j • J, j _> Jo. Let (t~) C ]0, r] converge to zero. For each n • N and each j • J, j _ Jo choose hi(,j) • [ h i . . . . . hml and b,,j • Bz with
p(zj, t.) + t.(rb + hit.,j) + erbn,j) • S.
(3.2)
For each j • J, j > Jo we may suppose that hi~.,j) = hq(j), for all n • N. Then for each j • J, J >- Jo (recalling that dzj+L(z, S) = d(z, S) i f z • zj + L and dzj+L(z, S) = +co i f z ¢ zj + L) we have
tn'tdzj+L(Z i + t.(rb + hqtj)), S) - dz,+L(ZJ, S)I = tyl[d(zy + t,(rb + hq
erllb.jll
<- tn + e r
t.)ll - d(zj, s)]
(by (3.1))
+ t£ld(p(zj, t.) + t.(rb + hqtj) + erb.,j), S)
(by (3.2)).
So for each j ___Jo we have
d-dzj+L(', S)(zj; rb + hqcj)) < er
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A. JOURANI and L. THIBAULT
which implies (zT, rb + hq(j)) <_ er and, hence, r(zT, b) < er +
max
I
i = 1, . . . , m
Therefore, we get for each b ~ Bz ( z * , b ) - < e + r -1
max
[(z*, hi)[
i = 1. . . . , m
which ensures that
IIz*ll
+ r-1
max
I
i= l , . . . , m
and the lemma is proved by setting 7 = r -1.
•
We can now establish the following regularity result for strongly compactly Lipschitzian mappings. TH~OgEU 3.4. Let C C X and D C Y be two closed subsets and g: X --* Y be s.c.L, at Xo ~ C O g-l(D). Suppose that D is compactly epi-Lipschitz at g(Xo). Suppose also that the following regularity condition holds at Xo [y* e #Ad(g(Xo),D) and 0 e OA(y** g + d(., C))(Xo)] = y* = 0.
(3.3)
Then the multivalued function F: X :~ Y defined by F(x)
f -g(x) + D Q
if x e C otherwise
is metrically regular at (x0,0) e Gr F, that is for some real numbers a _ 0 and r > 0 d(x, C A g-I(D - y)) < ad(g(x) + y , D ) for all x e C O (xo + rBx) and y ~ rBr. Proof. Suppose F is not metrically regular at (x0,0). Then by lemma 3.2, there exist sequences Sn ~ 0, (s~ < 1) xn ~ x0, Z~ ---' 0 and Yn --' 0 such that for all n ~ N* (x~, Zn) ~ G r F , Yn ~i F(x~) and for all (x, y) e Gr F ]lz~ -Ynl[-< ][Y- Y~[[ ÷ s ~ ( l l x - x~[I + [ ] y - zA). So (x~, ZD is a local minimum of the function (x,y) ~ [[y -Ynl[ + 2d(x,y, G r F ) + sn(llx - Xn[I + [lY - z~l[) [22, proposition 2.4.3]. As d(x,y, G r F ) _< d(g(x) + y , D ) + J(C,x), where J(C,x) is the indicator function of C, it follows that the function (x,y) ~ [[y -y~[[ ÷ 2d(g(x) + y , D ) ÷ sn([[x- xA ÷ [[y -z~[[)
Metric regularity
235
attains a local minimum at (x~, z.) on C x Y. Let kg be the Lipschitz constant of g at xo. Then the function (x,y) --* IlY - Y.II + 2d(g(x) + y , D ) + 2(kg + 1)d(x, C) + s.(llx - xnll ÷ Ily - z.[I) attains a local minimum at (x,, z,). So, by theorem 2.1, 0 ~ OAf(Xn, Z.) + [01 × S(xn, Zn) + sn(B~ × By*), where f ( x , y ) = 2d(g(x) + y , D ) + 2(kg + 1)d(x, C) and S(x,,, Zn) = {Y* E Y*: Ily*l[ = 1,
U
[OA(z* * g + 2(kg + 1)d(., C))(xn) × [z*}]
Z* E 2c~A d ( g ( x n ) +Yn, 1))
+ 101 × S(x., z.) + s. (a~ × By*). So there exist z* ~ 2OAd(g(x.) + Yn, D), x.• ~ OA(Z* ° g + 2(kg + l)d(., C))(x.) and y* S(Xn, z,,) such that
IIx~*ll ~ s.
(3.4)
Ily.* + z.*l[ -< s..
(3.5)
and
As the sequences (Yn*) and (zn*) are bounded, the sequence (-y*, z~*)n has some weak-star limit point (y*,z*). The relations (3.4) and (3.5) imply x~* ~ 0 and z * = y*. By the upper semicontinuity property of the approximate subdifferential and by lemma 3.4, we get y* E 2aAd(g(xo), D) and 0 e OA(y* * g + 2(ks + 1)d(', C))(Xo). Set v* = y*/(2(kg + 1)). Then 0 e Oa(V* *g + d(', C))(xo) and v* ~ OAd(g(xo),D) (since it is easily seen that [0, llOAd(g(xo), 19) C OAd(g(xo), D) because g(xo) e D). To get a contradiction with (3.3) it remains to show that v* # 0. Suppose that v* = 0 and fix some e e ]0, ½]. As D is compactly epi-Lipschitz at g(Xo), lemma 3.3 yields a real number y > 0, hi . . . . . hm e Y and a neighbourhood V of g(Xo) such that I[u*[[ < e +
y
max
[(u*,hi)[
forallu~
V
and
u*eOAd(u,D).
(3.6)
i= l,...,ra
As v* = 0, there exists a subnet (z*tj))~Es of (z~ w*-converging to zero. If we set 1 * uj• = ~z,~tj), uj = g(x,~tj ) + Y,~tJ)), then u* e OAd(ui, D) and, hence, there exists some Jo such that y. max [(u~, hi)[ -< e, s~,tjo) < e and t= l , . . . , m
Ilu~ll <- ~ + ~
max
i= l , . . . , r a
I(u~, hi)l <- 2~.
236
A. JOURANI and L. THIBAULT
However, (3.5) implies that
2llu~ll ->
1
-
Sjo >
1 -
e
and, hence, it follows from (3.6) that I < 5e which contradicts that e < ½. Thus v* # 0 and the proof is complete. • Using theorem 2.6, we obtain the following corollary of theorem 3.4. COROLLARY3.5. Let C, D, and g be as in theorem 3.4. Suppose that either D is epi-Lipschitz-like or epi-Lipschitz at g(Xo). Then under (3.3), the result of theorem 3.4 is valid. In order to state the regularity condition introduced by Borwein [6] let us recall that a map g: X --, Y is strictly differentiable at x0 if the Fr6chet derivative Vg(Xo) exists and satisfies lim
IIg(x)
- g(x') - Vg(xo)(X - x')[I
x-.x0
IIx - x'll
=0.
x' ~X 0
Using the constraint qualification Vg(xo)(T(C, Xo)) - T(D, g(Xo) ) = Y, where T(C, Xo) is the Clarke tangent cone to C at x0, Borwein [5] shows that the multivalued function
F(x)
f -g(x) + D O
if x e C otherwise
is metrically regular at (Xo, 0) with respect to C but with both sets C and D epi-Lipschitz at x0 c and g(Xo), respectively. Recall that h ~ T(C, Xo) iff for any (tn) ~ 0 and any xn ~ x0 there exists h, ---, h such that x n + t, h n ~ C, for all n e N*. We obtain Borwein's result as a consequence of the following corollary of theorem 3.4, where C is any closed subset of X and D is compactly epi-Lipschitz subset o f Y. COROLLARY 3.6. Let C and D be two closed subset of X and Y, respectively. If D is compactly epi-Lipschitz at g(Xo) and if Vg(xo)(T(C, Xo)) - T(D, g(xo)) = Y, then the conclusion of theorem 3.4 holds. Proof. We shall show that (3.3) is satisfied and theorem 3.4 will complete the proof. So let y * e Oad(g(Xo),D) C N(D,g(Xo)) and l e t x * e OAd(Xo, C) C N(C, xo) (here N(S, u) denotes the Clarke normal cone to S at u ~ S (see [22])) such that y * * Vg(xo) + x* = O.
Metric regularity
237
Let y e Y. Then, from the assumption, we have the existence of x ~ T(C, Xo) and
z ~ T(D, g(Xo)) such that y = Vg(xo)X - z and, hence, ( - y * , y ) = ( - y * o Vg(xo),X) + (y*,z) = (x*,x) + (y*,z) <- 0 which implies that y* -- 0 and the proof is complete.
•
The next two corollaries prepare the necessary optimality conditions in Section 4. COROI.LARV 3.7. Let C be a closed subset of X, D 1 and D 2 be two closed subsets of two Banach spaces YI and Y2, respectively. Let gi: X - ~ Y/, i = 1, 2, be s.c.L, at Xo. Suppose that D 1 and D E a r e compactly epi-Lipschitz at gl(X0) and gE(X0), respectively. Suppose also that
[y* ~ OAd(gi(Xo),Di), i = 1,2 and 0 e OA(y ~ *gl + Y~ °g2 + d ( ' , C))(Xo) ] y * = 0,
i = 1,2.
(3.7)
Then the multivalued function F: X ~ Y defined by
F(x)
f - - ( g l ( X ) , gE(X)) + D1 x D E
if x e C
(Q
otherwise
is metrically regular at (Xo, 0, 0) e Gr F.
Proof. Set g(x) = (gl(x), gZ(X)) and D = D 1 X 0 2 . It is easily seen that D is compactly epi-Lipschitz at g(Xo) and theorem 3.4 completes the proof. • COROLLARY 3.8. Let the assumptions of corollary 3.7 be satisfied with Y2 = Rv and D 2 = 101. Then the multivalued function F: X ~ Y defined by
F(x)= I;(gl(x)'g2(x))+Dl×[O}
ifxeC otherwise
is metrically regular at (xo, 0, 0) e Gr F.
Proof. It is enough to see that D 2 is compactly epi-Lipschitz at gz(xo) = 0 in [Rp and to apply corollary 3.7. • 4. APPLICATION TO OPTIMIZATION PROBLEMS In this section we shall discuss the abstract mathematical programming problem minimize f(x)
)
subject to x ~ C,
[
(4.1)
/
g(x) e D J where f : X ~ IR is locally Lipschitz at Xo, g" X ~ Yis s.c.L, at x o and C and D are two closed subsets of X and Y, respectively.
238
A. JOURANI and L. THIBAULT
In this abstract form (4.1) provides a generalization for both equality and inequality constrained optimization problems. The aim is to derive Lagrange multipliers of Fritz-John type for problem (4.1). TrmoREl~ 4.1. Let Xo be a local minimum for (4.1). Suppose that D is compactly epi-Lipschitz at g(Xo). Then there exist y > O, y* ~ aAd(g(xo), D) and 2 _ 0, with (A, y*) ~ (0, 0), such that 0 ~ cgA(2f+ y* . g + yd(', C))(x0). Proof. If the multivalued function F: X ~ Y defined by
F(x) =
I
-Qg(x) + D
ifxeC otherwise
is metrically regular at ( x 0 , 0 ) e G r F , the result follows from theorem 4.1 in [1]. If F is not metrically regular at ( X o , 0 ) e G r F , then by theorem 3.4, there exist y > 0 and y* ~ aAd(g(Xo), D) such that y* ~ 0 and 0 ~ OA(y* * g + yd(-, C))(Xo). So for 2 = 0 we have (2, y*) ;~ (0, 0) and
o e o A ( 2 f + y * o g + yd(.,C))(Xo).
•
If D is a closed convex cone with nonempty interior then theorem 4.1 yields the Fritz-John result due to Glover and Craven [18]. Observe that the result in [18] can also be derived from El Abdouni and Thibault [17]. It is possible to include equality constraints in theorem 4.1. COROLLARY4.2. Let g~ : X ~ Y be s.c.L, at x0, g2 : X --, [Rp be locally Lipschitz at Xo and D~ be a nonempty closed subset of Y which is compactly epi-Lipschitz at gl(xo). Suppose that x o E C tq g~l(Dl) tq g21(O) is a local minimum for (4. l) with D 1 × 10} in place of D and (gl, g2) in place of g. Then there exist 7 > O, y~ e OAd(gl(Xo),D1), Y~ ~ ~P and 2 _ 0, with (A, y*, Y2*) ;~ (0, 0, 0), such that 0 ~ 0,4 (2f + y~ o gl + Y~' * g2 + yd(-, C))(x0). Proof. This is a direct consequence of theorem 4.1.
•
The first dimensional version of this corollary was obtained in the paper of Kruger and Mordukhovich [23], for the case of finite numbers of equality and inequality constraints given by Lipschitz continuous mappings on a Banach space admitting an equivalent Fr6chet differentiable norm away from zero. Corollary 4.2 generalizes the result of Kruger [24], where D1 is assumed to be epi-Lipschitz at gl(xo).
Metric regularity
239
COROLL~Y 4.3. Let gl : X --~ Ybe s.c.L, at Xo, g2: X -* [Rp be locally Lipschitz at Xo and D 1 be a nonempty closed convex cone in Y. Assume that the polar cone D o is weak-star locally compact and that Xo ~ C N g~l(Dl) N g~l(0) is a local minimum for (4.1) with D 1 × [0} in place of D and (gl, g2) in place of g. Then there exist y > 0, y~ e D1°, y~ e [Rp and A __ 0, with (Y~, gl(xo)) = 0, such that (2, y~, Y2*) ~ (0, 0, 0) and
0 e 0A(;tf + y?o gl + Y~* g2 + yd(',
C))(xo).
Proof. Note that D1 is epi-Lipschitz-like (hence compactly epi-Lipschitzian by theorem 2.6) at Yo = g~(Xo) since D o is weak-star locally compact and D 1 n ( y o + By) + ]0, lID 1 c D 1 because Dl is a convex cone. We may apply corollary 4.2. As y* ~ aad(gl(Xo),DO and aAd(., D~) coincides with the subdifferential in the sense of convex analysis, it can be easily verified that ( y * , y ) <_ 0 for all y e D1 and (y~', gl(xo)) = 0. So the proof is complete. • When X and Y are finite dimensional we obtain the Fritz-John results of Mordukhovich [7] and Ioffe [10]. Acknowledgement--The authors gratefully acknowledge the comments made by the referee.
REFERENCES 1. JOURANI A. & THIBAULT L., Approximations and metric regularity in mathematical programming in Banach space, Math. Oper. Res. 18, 390-401 (1993). 2. ROBINSON S. M., Stability theorems for systems of inequality, Part If: differential nonlinear systems, S I A M J. numer. Analysis 13, 497-513 (1976). 3. ROBINSON S. M., Regularity and stability for convex multivalued functions, Math. Oper. Res. 1, 130-143 (1976). 4. IOFFE A. D., Regular points of Lipschitz mappings, Trans. Am. math. Soc. 251, 61-69 (1979). 5. AUSLENDER A., Stability in mathematical programming with nondifferentiable data, S I A M J. control Optim. 239-254 (1984). 6. BORWEIN J. M., Stability and regular point of inequality systems, J. Optim. theor. Appl. 48, 9-52 (1986) 7. MORDUKHOVICH B. S., Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk. SSSR 28, 976-979 (1984). 8. MORDUKHOVICH B. S., Approximation Methods in Problems o f Optimization and Control, Nauka, Moscow (1988); English translation (to appear). 9. MORDUKHOVICH B. S., Complete characterization of openness, metric regularity, and Lipschitzian property of multifunctions, Trans. Am. math. Soc. 340, 1-36 (1993). 10. IOFFE A. D., Approximate subdifferentials and applications 1: the finite dimensional theory, Trans. Am. math. Soc. 281,289-316 (1984). 11. IOFFE A. D., Approximate subdifferentials and applications 2: functions on locally convex spaces, Mathematika 33, 111-128 (1986). 12. IOFFE A. D., Approximate subdifferentials and applications 3: metric theory, Mathematika 36, 1-38 (1989). 13. BORWEIN J. M. & STROJWAS H. M., Tangential approximations, Nonlinear Analysis 9, 1347-1366 (1985) 14. JOURANI A. & THIBAULT L., Approximate subdifferential and metric regularity: finite dimensional case, math. Prog. 47, 203-218 (1990). 15. THIBAULT L., Subdifferential of compactly Lipschitzian vector-valued functions, Travaux du seminaire d'analyse convexe 8, (1978). 16. JOURANI A. & THIBAULT L., Approximate subdifferential of composite functions, Bull. A ustr. math. Soc. 47, 443-455 (1993). 17. EL ABDOUNI B. & THIBAULT L., Lagrange multipliers for Pareto nonsmooth programming problems in Banach spaces, Optimization 29, 277-285 (1992) 18. GLOVER B. M. & CRAVEN B. D., A Fritz John optimality condition using the approximate subdifferential, J. Optim. theory Appl. 82, 253-265 (1994).
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19. ROCKAFELLAR R. T., Directionally Lipschitz functions and subdifferential calculus, Proc. London math. Soc. 39, 331-355 (1978). 20. BORWEIN J. M., Epi-Lipschitz-like sets in Banach space: theorems and examples, Nonlinear Analysis 11, 1207-1217 (1987) 21. JOURANI A., Regularity and strong sufficient optimality conditions in differentiable optimization problems, Numer. funct. Anal. Optim. 14, 69-87 (1993). 22. CLARKE F. H., Optimization and Nonsmooth Analysis. Wiley, New York (1983). 23. KRUGER A. YA. & MORDUKHOVICH B. S., Extremai points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk. SSSR 24, 684-687 (1980). 24. KRUGER A. YA., Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Sib. math. J. 26, 370-379 (1985).
NonlinearAnalysis, Theory,Methods&Applications,Vol. 24, No. 2, pp. 229-240, 1995 Copyright © 1995ElsevierScienceLtd Printed in GreatBritain.All rightsreserved 0362-546X/95$9.50+ .00 0362-546X(94)E0061-K
METRIC REGULARITY FOR STRONGLY COMPACTLY LIPSCHITZIAN MAPPINGS A. JOURANH and L. THIBAULT~ tUniversit6 de Bourgogne, Laboratoire d'Analyse Appliqu6e et Optimisation, B.P. 138, 21004 Dijon C6dex, France; and ~Universit6 Montpellier II, Departement des Sciences Math6matiques, 34095 Montpellier Cedex 5, France (Received 25 May 1993; received in revised form 3 December 1993; received f o r publication 30 March 1994) Key words and phrases: Metric regularity, approximate subdifferential, strongly compactly Lipschitzian mappings, compactly epi-Lipschitzian sets, Fritz-John multipliers.
1. I N T R O D U C T I O N
Feasible sets of optimization constraint problems generally appear in the form C n g-l(D), where C and D are two subsets of two Banach spaces X and Y and g is a mapping from X into Y. It is now known (see [1]) that a general constraint qualification allowing us to get the existence of multipliers for optimization constraint problems with constraints in the form C n g-l(D) is the so-called metric regularity (of g with respect to C and D) whose importance is essentially due to Ljusternik and Robinson [2, 3]. Recall that g is metrically regular at Xo ~ C n g - l ( D ) with respect to C and D if there exist some real number a >_ 0 and two neighbourhoods X 0 and Y0 of Xo and zero in X and Y, respectively, such that d(x, C n g-l(D - y)) <_ ad(g(x) + y,D)
(1.1)
for all x ~ C n x0 and y e Yo. In this paper we are essentially concerned with easily verifiable conditions in terms of subdifferentials implying the metric regularity as initiated by Ioffe [4] who introduced the use of Ekeland variational principle in this study. This idea has been followed by Auslender [5] and largely extended by Borwein [6] to derive, for Y = ~P, conditions with Clarke generalized gradients ensuring relation (1.1). Mordukhovich [7] (see also [8, 9]) has shown that, when X and Y are finite dimensional, similar conditions, but with his notion of approximate subdifferentials instead of Clarke generalized gradients, are both necessary and sufficient for the metric regularity (1.1). So this paper is devoted to showing how the use of the extension by Ioffe [10-12] of the approximate subdifferentials to Banach spaces will allow us to generalize the above conditions when X and Y are infinite dimensional and D is compactly epi-Lipschitzian in the sense introduced by Borwein and Strojwas [13]. Our results are along the lines of our preceding ones [1, 14] and depend heavily on our lemma 3.2 establishing a new and important behaviour for compactly epi-Lipschitzian subsets. We shall also show that these results are too strong to allow us to derive the existence of Fritz-John multipliers for general infinite dimensional optimization problems. 2. APPROXIMATE SUBDIFFERENTIALS AND PRELIMINARIES Throughout we shall assume that X and Y are Banach spaces X* and Y* are their topological duals and ( ' , • ) is the pairing between the spaces. We denote by B x , B?c . . . . the closed unit balls 229
230
A. JOURANI
a n d L. T H I B A U L T
o f X, AT*. . . . . By d ( ' , S) we denote the usual distance function to the subset S C X
d(x, S) = infllx - ul[. u~S
f
s
We write x ~ x0 and x ~ xo to express x ~ Xo with f(x) -~ f(Xo) and x ~ x0 with x e S, respectively. We denote by G r F the graph of a multivalued function F: X ~ Y, that is, Gr F = [(x, y) e X x Y: y e F(x)l. I f f is an extended-real-valued function on X, we write for any subset S of X
x)
if x e S
fs(x) = (. +oo
otherwise.
The function
d - f ( x , h) = lim inf t - l ( f ( x + tu) - f(x)) u~h tJ, O
is the lower Dini directional derivative of f a t x and the Dini e-subdifferential of f a t x is the set
O~f(x) = Ix* ~ X*: (x*, h) < d - f ( x ; h) + ellhll, v h ~ x } for x e D o m f and O-;f(x) = O if x ¢ Dora f , where D o m f denotes the effective domain o f f . For e = 0 we write O-f(x). By 5:(X) we denote the collection o f finite dimensional subspaces o f X. The approximate subdifferentials of f a t x 0 e D o m f is defined by the following expressions (see Ioffe [11, 12])
014f(Xo) =
0 le~(X)
lim sup O-f.+L (x) = X~Xof
[") Left(X)
lim sup 07fx+ L (x), t
X~Xo
where lim sup O-fx+L (x) = Ix* e X*: x* = w* - lira x*, x* e O-fxi+L(xi), xi ~ Xo], that is, the f X--b Xo
set of w*-limits of all such nets. It is easily seen that the multivalued function
X ~ OAf(x ) is upper-semicontinuous in the following sense
0.4 f(xo) = lim sup 0A f(x) f
X -+ X0
and in [12] Ioffe has shown that when S is a closed subset o f X and x0 e S
04d(x o, S) =
0 L e ~(X)
lim sup Oldx+L(x, S).
(2.1)
s
X-~ Xo e~O
The following sum rule has been established by Ioffe in [11] for a more general situation. For the purpose of our discussion, a semi-Lipschitz case suffices.
Metric regularity
231
THEOREM 2.1 [11]. Let f : X ~ R be a function which is lower-semicontinuous near x0 and g: X ~ R be a function which is Lipschitz around x0. Then
a A ( f + g)(xo) C OAf(Xo) + aAg(XO). In the sequel we shall need the following class of mappings between Banach spaces.
Definition 2.2 [1]. A mapping g: X ~ Y is said to be strongly compactly Lipschitzian (s.c.L.) at a point Xo if there exist a multivalued function R: X ::* Comp(Y), where Comp(Y) denotes the set of all norm compact subsets of Y, and a function r: X x X --, [R+ satisfying: (i) lim r(x, h) = 0; X~X
0
h~O
(ii) there exists o~ > 0 such that
t - l ( g ( x + th) - g(x)) ~ R(h) +
Ilhllr(x,
th)By
for all x • Xo + ocBx, h • aBx and t e ]0, or[; (iii) R(O) = 10] and R is upper-semicontinuous.
It can be shown [15] that every s.c.L, mapping is locally Lipschitz. In finite dimensions the concepts coincide. Recently, we have developed in [16] a chain rule for this class of mappings. Let us note that this chain rule has been obtained before by Ioffe in [12] for maps with compact prederivatives. THEOREM 2.3 [16]. Let g: X --' Ybe s.c.L, at x0 and let f : Y --, ~ be locally Lipschitz at g(xo). Then f . g is locally Lipschitz at Xo and
OA(f o g)(Xo) C
U
On(y* ° g)(xo).
Y* • aAf(g(xo))
To complete this section we note the following property of s.c.L, mappings which is a direct consequence of lemma 2.5 in [17]. See also proposition 2.3 in [16] and theorem 1.5 in [18]. L E I ~ 2.4. Let g: X ~ Y be s.c.L, at x 0 and let (Yi*) any bounded net of Y* which w*converges to zero in Y* and let (xi) be a net IIII-converging to x0 in x . If x* • Oa(Y* o g)(xi), then (xi*) w*-converges to zero in X*.
Remark. Assuming that Y is separable, Glover and Craven [18] have independently shown later that, when g is s.c.L., the multivalued function (x, y*) ---* aA(Y*, g)(x) is closed (with respect to the w*-convergence in y * e Y*). This is also a direct consequence of lemma 2.5 in [17] without the separability assumption on Y. Let S be a subset o f a normed vector space Z and Zo e S. We recall the following definition.
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A. JOURANI and L. THIBAULT
Definition 2.5. (1) S is epi-Lipschitz at zo (see [19]) if there exist ot > 0 and h e Z such that S n (Zo + otBz) + ]0, ot](h + aBz) C S. (2) S is epi-Lipschitz-like at z0 (see [20]) if there exist a > 0, h e Z and a nonempty convex subset ~ o f Z containing zero with ~o weak-star locally compact such that
S n (Zo + aBz) + ]0, ot[(h + f2) c S, where flo = {z* e Z*:
S n (Zo + aBz) + taBz C S - tH,
for all t e ]0, a[.
To close this section, let us recall the following important theorem of Borwein [20].
THEOREM 2.6 [20]. Let Z be a normed space, S be a closed subset of Z and Zo e S. (a) If S is epi-Lipschitz at Zo then S is epi-Lipschitz-like at Zo. (b) If S is epi-Lipschitz-like at Zo then S is compactly epi-Lipschitz at zo. (c) If Z is a finite dimensional space then each subset of Z is epi-Lipschitz-like at all its points.
3. M E T R I C R E G U L A R I T Y
Definition 3.1. Let F: X ~ Y be a multivalued function and (Xo, Yo) e Gr F. One says that F is metrically regular at (Xo, Yo) if there exist two real numbers k _ 0 and e > 0 such that d(x, F - l ( y ) ) -< kd(y, F(x)) for all x ~ Xo + eBx and y ~ Yo d(x, Q) = +0o.
+
eBy with
d ( y , F(x)) <_ e. Here we adopt the convention
Two of the key results used in the p r o o f of theorem 3.4 are the two following lemmas. LEMMA 3.2 [21]. Let F : X ::t y be a multivalued function of closed graph and let (xo, Yo) ~ Gr F. If F is not metrically regular at (Xo, Yo), then there are s n ~ 0, xn ~ x o, z~ --' Yo and Yn --' Yo such that O) (x~, Zn) ~ Gr F, for all n e N*; (ii) y~ ¢ F(x~) for all n e ~q*; and (iii) for all (x, y) e Gr F [Iz~ - y~l[ - [ly - y~[I + s~(I[x - x~[I + [[y - z~[[).
Remark. This lemma is mainly based on the ideas o f Borwein [ 6 ] .
Metric regularity
233
LEMMA 3.3. Let Z be a normed vector space and let S be a subset of Z. Suppose that S is compactly epi-Lipschitz at zo • S. Then, for H given by definition 2.5 there exist y > 0 and a neighbourhood V of Zo such that for each e > 0 there are h~ . . . . . hm • H satisfying Ilz*ll <_ e + y
max
i= l , . . . , m
I(z*, hi)[,
Y z • V,
Y Z* • OAd(Z, S ).
Proof. Let H be a compact subset of Z and r • ]0, 1] such that S tq (Zo + 3rBz) + trBz C S - tH,
for all t • ]0, r].
Choose an open neighbourhood V of Zo with V C Zo + rBz and d(z, S) <_ r for all z • V. Let h I .... , hm • Hwith
H C C) (hi + erBz). i=1
For each z • V and each t • ]0, r] we may select some p(z, t) ~ S such that Ilz -
p(z, t)[I ~ d(z, S) + t 2.
(3.1)
By the choice of V we have for each z • V and each t • ]0, r]
Ilzo - p(z, t) < IIz - p ( z , t)ll + IIz - zoll < d(z, S) + t 2 + r < 3r and, hence, p(z, t) • Zo + 3rBz. Fix any z • V a n d any z* • OAd(z, S). Fix any b • Bz and any L • 5:(X) with [b, hi . . . . . hm] C L. We may write z* = x* - l i m z * with z 7 • O-dzj+L(zj, S) j~J
and zj -~ z. Choose Jo • J such that zj • V for any j • J, j _> Jo. Let (t~) C ]0, r] converge to zero. For each n • N and each j • J, j _ Jo choose hi(,j) • [ h i . . . . . hml and b,,j • Bz with
p(zj, t.) + t.(rb + hit.,j) + erbn,j) • S.
(3.2)
For each j • J, j > Jo we may suppose that hi~.,j) = hq(j), for all n • N. Then for each j • J, J >- Jo (recalling that dzj+L(z, S) = d(z, S) i f z • zj + L and dzj+L(z, S) = +co i f z ¢ zj + L) we have
tn'tdzj+L(Z i + t.(rb + hqtj)), S) - dz,+L(ZJ, S)I = tyl[d(zy + t,(rb + hq
erllb.jll
<- tn + e r
t.)ll - d(zj, s)]
(by (3.1))
+ t£ld(p(zj, t.) + t.(rb + hqtj) + erb.,j), S)
(by (3.2)).
So for each j ___Jo we have
d-dzj+L(', S)(zj; rb + hqcj)) < er
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A. JOURANI and L. THIBAULT
which implies (zT, rb + hq(j)) <_ er and, hence, r(zT, b) < er +
max
I
i = 1, . . . , m
Therefore, we get for each b ~ Bz ( z * , b ) - < e + r -1
max
[(z*, hi)[
i = 1. . . . , m
which ensures that
IIz*ll
+ r-1
max
I
i= l , . . . , m
and the lemma is proved by setting 7 = r -1.
•
We can now establish the following regularity result for strongly compactly Lipschitzian mappings. TH~OgEU 3.4. Let C C X and D C Y be two closed subsets and g: X --* Y be s.c.L, at Xo ~ C O g-l(D). Suppose that D is compactly epi-Lipschitz at g(Xo). Suppose also that the following regularity condition holds at Xo [y* e #Ad(g(Xo),D) and 0 e OA(y** g + d(., C))(Xo)] = y* = 0.
(3.3)
Then the multivalued function F: X :~ Y defined by F(x)
f -g(x) + D Q
if x e C otherwise
is metrically regular at (x0,0) e Gr F, that is for some real numbers a _ 0 and r > 0 d(x, C A g-I(D - y)) < ad(g(x) + y , D ) for all x e C O (xo + rBx) and y ~ rBr. Proof. Suppose F is not metrically regular at (x0,0). Then by lemma 3.2, there exist sequences Sn ~ 0, (s~ < 1) xn ~ x0, Z~ ---' 0 and Yn --' 0 such that for all n ~ N* (x~, Zn) ~ G r F , Yn ~i F(x~) and for all (x, y) e Gr F ]lz~ -Ynl[-< ][Y- Y~[[ ÷ s ~ ( l l x - x~[I + [ ] y - zA). So (x~, ZD is a local minimum of the function (x,y) ~ [[y -Ynl[ + 2d(x,y, G r F ) + sn(llx - Xn[I + [lY - z~l[) [22, proposition 2.4.3]. As d(x,y, G r F ) _< d(g(x) + y , D ) + J(C,x), where J(C,x) is the indicator function of C, it follows that the function (x,y) ~ [[y -y~[[ ÷ 2d(g(x) + y , D ) ÷ sn([[x- xA ÷ [[y -z~[[)
Metric regularity
235
attains a local minimum at (x~, z.) on C x Y. Let kg be the Lipschitz constant of g at xo. Then the function (x,y) --* IlY - Y.II + 2d(g(x) + y , D ) + 2(kg + 1)d(x, C) + s.(llx - xnll ÷ Ily - z.[I) attains a local minimum at (x,, z,). So, by theorem 2.1, 0 ~ OAf(Xn, Z.) + [01 × S(xn, Zn) + sn(B~ × By*), where f ( x , y ) = 2d(g(x) + y , D ) + 2(kg + 1)d(x, C) and S(x,,, Zn) = {Y* E Y*: Ily*l[ = 1,
U
[OA(z* * g + 2(kg + 1)d(., C))(xn) × [z*}]
Z* E 2c~A d ( g ( x n ) +Yn, 1))
+ 101 × S(x., z.) + s. (a~ × By*). So there exist z* ~ 2OAd(g(x.) + Yn, D), x.• ~ OA(Z* ° g + 2(kg + l)d(., C))(x.) and y* S(Xn, z,,) such that
IIx~*ll ~ s.
(3.4)
Ily.* + z.*l[ -< s..
(3.5)
and
As the sequences (Yn*) and (zn*) are bounded, the sequence (-y*, z~*)n has some weak-star limit point (y*,z*). The relations (3.4) and (3.5) imply x~* ~ 0 and z * = y*. By the upper semicontinuity property of the approximate subdifferential and by lemma 3.4, we get y* E 2aAd(g(xo), D) and 0 e OA(y* * g + 2(ks + 1)d(', C))(Xo). Set v* = y*/(2(kg + 1)). Then 0 e Oa(V* *g + d(', C))(xo) and v* ~ OAd(g(xo),D) (since it is easily seen that [0, llOAd(g(xo), 19) C OAd(g(xo), D) because g(xo) e D). To get a contradiction with (3.3) it remains to show that v* # 0. Suppose that v* = 0 and fix some e e ]0, ½]. As D is compactly epi-Lipschitz at g(Xo), lemma 3.3 yields a real number y > 0, hi . . . . . hm e Y and a neighbourhood V of g(Xo) such that I[u*[[ < e +
y
max
[(u*,hi)[
forallu~
V
and
u*eOAd(u,D).
(3.6)
i= l,...,ra
As v* = 0, there exists a subnet (z*tj))~Es of (z~ w*-converging to zero. If we set 1 * uj• = ~z,~tj), uj = g(x,~tj ) + Y,~tJ)), then u* e OAd(ui, D) and, hence, there exists some Jo such that y. max [(u~, hi)[ -< e, s~,tjo) < e and t= l , . . . , m
Ilu~ll <- ~ + ~
max
i= l , . . . , r a
I(u~, hi)l <- 2~.
236
A. JOURANI and L. THIBAULT
However, (3.5) implies that
2llu~ll ->
1
-
Sjo >
1 -
e
and, hence, it follows from (3.6) that I < 5e which contradicts that e < ½. Thus v* # 0 and the proof is complete. • Using theorem 2.6, we obtain the following corollary of theorem 3.4. COROLLARY3.5. Let C, D, and g be as in theorem 3.4. Suppose that either D is epi-Lipschitz-like or epi-Lipschitz at g(Xo). Then under (3.3), the result of theorem 3.4 is valid. In order to state the regularity condition introduced by Borwein [6] let us recall that a map g: X --, Y is strictly differentiable at x0 if the Fr6chet derivative Vg(Xo) exists and satisfies lim
IIg(x)
- g(x') - Vg(xo)(X - x')[I
x-.x0
IIx - x'll
=0.
x' ~X 0
Using the constraint qualification Vg(xo)(T(C, Xo)) - T(D, g(Xo) ) = Y, where T(C, Xo) is the Clarke tangent cone to C at x0, Borwein [5] shows that the multivalued function
F(x)
f -g(x) + D O
if x e C otherwise
is metrically regular at (Xo, 0) with respect to C but with both sets C and D epi-Lipschitz at x0 c and g(Xo), respectively. Recall that h ~ T(C, Xo) iff for any (tn) ~ 0 and any xn ~ x0 there exists h, ---, h such that x n + t, h n ~ C, for all n e N*. We obtain Borwein's result as a consequence of the following corollary of theorem 3.4, where C is any closed subset of X and D is compactly epi-Lipschitz subset o f Y. COROLLARY 3.6. Let C and D be two closed subset of X and Y, respectively. If D is compactly epi-Lipschitz at g(Xo) and if Vg(xo)(T(C, Xo)) - T(D, g(xo)) = Y, then the conclusion of theorem 3.4 holds. Proof. We shall show that (3.3) is satisfied and theorem 3.4 will complete the proof. So let y * e Oad(g(Xo),D) C N(D,g(Xo)) and l e t x * e OAd(Xo, C) C N(C, xo) (here N(S, u) denotes the Clarke normal cone to S at u ~ S (see [22])) such that y * * Vg(xo) + x* = O.
Metric regularity
237
Let y e Y. Then, from the assumption, we have the existence of x ~ T(C, Xo) and
z ~ T(D, g(Xo)) such that y = Vg(xo)X - z and, hence, ( - y * , y ) = ( - y * o Vg(xo),X) + (y*,z) = (x*,x) + (y*,z) <- 0 which implies that y* -- 0 and the proof is complete.
•
The next two corollaries prepare the necessary optimality conditions in Section 4. COROI.LARV 3.7. Let C be a closed subset of X, D 1 and D 2 be two closed subsets of two Banach spaces YI and Y2, respectively. Let gi: X - ~ Y/, i = 1, 2, be s.c.L, at Xo. Suppose that D 1 and D E a r e compactly epi-Lipschitz at gl(X0) and gE(X0), respectively. Suppose also that
[y* ~ OAd(gi(Xo),Di), i = 1,2 and 0 e OA(y ~ *gl + Y~ °g2 + d ( ' , C))(Xo) ] y * = 0,
i = 1,2.
(3.7)
Then the multivalued function F: X ~ Y defined by
F(x)
f - - ( g l ( X ) , gE(X)) + D1 x D E
if x e C
(Q
otherwise
is metrically regular at (Xo, 0, 0) e Gr F.
Proof. Set g(x) = (gl(x), gZ(X)) and D = D 1 X 0 2 . It is easily seen that D is compactly epi-Lipschitz at g(Xo) and theorem 3.4 completes the proof. • COROLLARY 3.8. Let the assumptions of corollary 3.7 be satisfied with Y2 = Rv and D 2 = 101. Then the multivalued function F: X ~ Y defined by
F(x)= I;(gl(x)'g2(x))+Dl×[O}
ifxeC otherwise
is metrically regular at (xo, 0, 0) e Gr F.
Proof. It is enough to see that D 2 is compactly epi-Lipschitz at gz(xo) = 0 in [Rp and to apply corollary 3.7. • 4. APPLICATION TO OPTIMIZATION PROBLEMS In this section we shall discuss the abstract mathematical programming problem minimize f(x)
)
subject to x ~ C,
[
(4.1)
/
g(x) e D J where f : X ~ IR is locally Lipschitz at Xo, g" X ~ Yis s.c.L, at x o and C and D are two closed subsets of X and Y, respectively.
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In this abstract form (4.1) provides a generalization for both equality and inequality constrained optimization problems. The aim is to derive Lagrange multipliers of Fritz-John type for problem (4.1). TrmoREl~ 4.1. Let Xo be a local minimum for (4.1). Suppose that D is compactly epi-Lipschitz at g(Xo). Then there exist y > O, y* ~ aAd(g(xo), D) and 2 _ 0, with (A, y*) ~ (0, 0), such that 0 ~ cgA(2f+ y* . g + yd(', C))(x0). Proof. If the multivalued function F: X ~ Y defined by
F(x) =
I
-Qg(x) + D
ifxeC otherwise
is metrically regular at ( x 0 , 0 ) e G r F , the result follows from theorem 4.1 in [1]. If F is not metrically regular at ( X o , 0 ) e G r F , then by theorem 3.4, there exist y > 0 and y* ~ aAd(g(Xo), D) such that y* ~ 0 and 0 ~ OA(y* * g + yd(-, C))(Xo). So for 2 = 0 we have (2, y*) ;~ (0, 0) and
o e o A ( 2 f + y * o g + yd(.,C))(Xo).
•
If D is a closed convex cone with nonempty interior then theorem 4.1 yields the Fritz-John result due to Glover and Craven [18]. Observe that the result in [18] can also be derived from El Abdouni and Thibault [17]. It is possible to include equality constraints in theorem 4.1. COROLLARY4.2. Let g~ : X ~ Y be s.c.L, at x0, g2 : X --, [Rp be locally Lipschitz at Xo and D~ be a nonempty closed subset of Y which is compactly epi-Lipschitz at gl(xo). Suppose that x o E C tq g~l(Dl) tq g21(O) is a local minimum for (4. l) with D 1 × 10} in place of D and (gl, g2) in place of g. Then there exist 7 > O, y~ e OAd(gl(Xo),D1), Y~ ~ ~P and 2 _ 0, with (A, y*, Y2*) ;~ (0, 0, 0), such that 0 ~ 0,4 (2f + y~ o gl + Y~' * g2 + yd(-, C))(x0). Proof. This is a direct consequence of theorem 4.1.
•
The first dimensional version of this corollary was obtained in the paper of Kruger and Mordukhovich [23], for the case of finite numbers of equality and inequality constraints given by Lipschitz continuous mappings on a Banach space admitting an equivalent Fr6chet differentiable norm away from zero. Corollary 4.2 generalizes the result of Kruger [24], where D1 is assumed to be epi-Lipschitz at gl(xo).
Metric regularity
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COROLL~Y 4.3. Let gl : X --~ Ybe s.c.L, at Xo, g2: X -* [Rp be locally Lipschitz at Xo and D 1 be a nonempty closed convex cone in Y. Assume that the polar cone D o is weak-star locally compact and that Xo ~ C N g~l(Dl) N g~l(0) is a local minimum for (4.1) with D 1 × [0} in place of D and (gl, g2) in place of g. Then there exist y > 0, y~ e D1°, y~ e [Rp and A __ 0, with (Y~, gl(xo)) = 0, such that (2, y~, Y2*) ~ (0, 0, 0) and
0 e 0A(;tf + y?o gl + Y~* g2 + yd(',
C))(xo).
Proof. Note that D1 is epi-Lipschitz-like (hence compactly epi-Lipschitzian by theorem 2.6) at Yo = g~(Xo) since D o is weak-star locally compact and D 1 n ( y o + By) + ]0, lID 1 c D 1 because Dl is a convex cone. We may apply corollary 4.2. As y* ~ aad(gl(Xo),DO and aAd(., D~) coincides with the subdifferential in the sense of convex analysis, it can be easily verified that ( y * , y ) <_ 0 for all y e D1 and (y~', gl(xo)) = 0. So the proof is complete. • When X and Y are finite dimensional we obtain the Fritz-John results of Mordukhovich [7] and Ioffe [10]. Acknowledgement--The authors gratefully acknowledge the comments made by the referee.
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