Characterization of stability for cone increasing constraint mappings

Characterization of stability for cone increasing constraint mappings

Ren e Henrion

Abstract

We investigate stability (in terms of metric regularity) for the specic class of cone increasing constraint mappings. This class is of interest in problems with additional knowledge on some nondecreasing behavior of the constraints (e.g. in chance constraints, where the distribution function of some measure is automatically nondecreasing). It is demonstrated, how this extra information may lead to sharper characterizations. In the rst part, rather general cone increasing constraint mappings are studied by exploiting criteria for metric regularity, as recently developed by Mordukhovich. The second part focusses on genericity investigations for global metric regularity (i.e. metric regularity at all feasible points) of nondecreasing constraints in nite dimensions. Applications to chance constraints are given.

Keywords:

cone increasing constraints, nonsmooth analysis, metric regularity, chance constraints, genericity

AMS subject classications:

90C15, 90C30, 49J52

The author thanks Prof. R.T. Rockafellar (University of Washington, Seattle) and Prof. J.M. Borwein (Simon Fraser University, Burnaby) for helpful discussion. Financial support by a grant of the 'Deutsche Forschungsgemeinschaft' is gratefully acknowledged.

1

1 Introduction

The concept of metric regularity as introduced by Robinson 26] is fundamental for deriving stability results in parametric programming. It is closely related to several other well-known conditions in stability analysis. Recall, for instance, the equivalence between metric regularity and pseudo-lipschitzness (see 1] and 28]) of multifunctions which was established by Borwein and Zhuang 4] and Penot 25]. For many dierent areas of optimization theory (smooth, convex, nonsmooth, nite-, innite-dimensional, semi-innite etc.) characterizations of metric regularity in terms of constraint qualications have been found (e.g. 2], 3], 9], 10], 15], 24], 26], 27], 28], 31]). Signicant progress in the nonsmooth setting was made by Mordukhovich who found an injectivity condition for his coderivative of multifunctions which is an equivalent criterion of metric regularity in nite dimensions 21] and, under additional hypotheses, is at least sucient in innite dimensions 23]. For closely related investigations involving Ioe's ap- proximate coderivative 13], which is the topological counterpart of Mordukhovich's sequentially dened coderivative, we refer to Jourani and Thibault 16], 17].

The purpose of this paper is to demonstrate how the characterization of metric regularity of constraint systems may be improved in case that the constraint mapping has the additional property of being cone increasing. By this, we mean a mapping f : X ^! Y together with cones Kx X and Ky Y such that x^{1 ;}x^{2 2} Kx implies f(x¹)^;f(x²) ² Ky. The motivation for this investigation came from stability analysis of chance constraints 11]. To give a simplied idea, assume that h:IR^{n !}IR^m is a mapping which indicates the production hj(x) (j = 1 ::: m) of a certain good (e.g. energy) as a function of n decision variables xi at m dierent times. Of course, decisions have to be taken in such a way that the production meets the demand j for this good at all times, so h(x) is a natural requirement. Unfortunately, in general the demand is a random variable which can be observed only after decisions have been taken. Therefore, it is not reasonable to model the constraint in the deterministic way above but rather to replace it by a stochastic formulation like (h(x) )p⁰ where is a probability measure for the m- dimensional random variable and p⁰ is some xed probability level. So, the constraint has to be fullled with a certain probability at least, i.e. it is a chance constraint. In addition, some non-stochastic constraints (e.g. capacity constraints for the decision variables) may enter the model in the form x ² C where C IRⁿ is some closed subset. It is convenient to reformulate this chance constraint by introducing the distribution function F corresponding to which is dened for y²IR^m as F (y) = ( y):

(F h)(x)p⁰ x²C (1)

Since the true underlying measure of is not known in general, one usually replaces it by empirical measures which are based on observations of and which may be understood as perturbations of . Then, the question of (Lipschitzian) stability of optimal values and local minimizers with respect to such pertubations arises in a problem with a corresponding cost function. As a key result in this direction, Romisch and Schultz 30] showed that the question of stability of the chance constraint w.r.t. perturbations of may be reduced to metric regularity of the constraint mapping (F h)(x) w.r.t. perturbations of the right-hand side probability level (in 30] an equivalent formulation in terms of Pseudo-Lipschitzness was used).

2

The study of metric regularity of (1) in a nonsmooth context (note that F is only upper semicontinuous in general and also h might be nonsmooth) has several specic features. First, the constraint mapping has the structure of a composite function, hence nonsmooth chain rules are of interest. Second, as a distribution function, F is automatically nondecreasing, i.e. in the above terminology, it is (IR^m+ IR⁺)- cone increasing. In 11], these particular properties were combined with Mordukhovich's injectivity condition to arrive at veriable criteria of metric regularity, namely conditions for the density of and constraint qualications for h.

In the rst part of this paper, certain ideas of 11] are generalized to a partially innite dimensional setting, this means to a nite number of inequality constraints in an innite dimen- sional space. In particular, the information on nondecreasing behaviour is used to get a more precise constraint qualication ensuring metric regularity for composite mappings or to char- acterize metric regularity w.r.t. some unperturbed, xed set. It is also shown that, for certain cone increasing constraint mappings, the verication of metric regularity via Mordukhovich's coderivative is equivalent to the corresponding injectivity condition using Clarke's coderivative, which might be easier to handle. Of course, both criteria dier signicantly in general. For other papers, also considering nondecreasing mappings in the context of subdierentiation, we refer to 7], 8], 20].

The second part of the paper re-addresses the nite dimensional situation, but from a dierent viewpoint: Usually, metric regularity of a feasible set mapping is required to hold at the local minimzers of some optimization problem. Since this condition is hard to verify, one could substitute it by a global version, namely metric regularity at all feasible points.

This requirement seems extremely strong. On the other hand, it is known, that such global metric regularity is a generic property of smooth constraint functions, i.e. in some sense it is typically fullled. This follows from the well-known equivalence of metric regularity with the Mangasarian-Fromovitz Constraint Qualication and the fact, that even the stronger Linear Independence Constraint Qualication holds globally for a generic set of smooth constraint functions (see 14]). A similar result does not hold in the locally Lipschitzian case (see Example 3.8 below). On the other hand, for the particular class of nondecreasing, locally Lipschitzian constraint mappings, genericity properties may be derived again. As an interesting aspect, it turns out that the results are sensitive to the structure of some possible additional xed constraint set (not subject to perturbations), usually reecting simple capacity limitations.

Referring back to the application in chance constraints of the type (1), special attention is devoted to the subclass of distribution functions.

2 Preliminaries

In this section, some basic concepts from multivalued analysis shall be recalled. Let X Y be arbitrary sets. For a multifunction :X^!!Y put

Ker = ^fx²X ^j0²(x)^g

Im = ^fy ²Y ^jy²(x) x²X^g Gph = ^f(x y)²XY ^jy²(x)^g

3

^;1(y) = ^fx²X ^jy²(x)^g

Now let X Y be two normed spaces. A multifunction : X^!!Y is called metrically regular at some point (x⁰ y⁰)²Gph if there are constants a >0 and " >0 such that

dist(x ^;1(y))adist(y (x)) ⁸(x y)²B"(x⁰)B"(y⁰):

The abstract form of constraint sets writes as C^\F^;1(K), where C X and K Y are closed subsets of the respective spaces (K usually being a closed convex cone) and F :X ^!Y is the constraint function. Then, F is said to be metrically regular with respect to C at some feasible point x^{0 2}C^\F^;1(K), if the associated multifunction

(x) =

(

;F(x) +K for x²C

else

is metrically regular at (x⁰ 0). It is easily seen that this is equivalent to the conventional denition of metric regularity for constrained systems:

9" >0⁹a >0⁸(x y)²(C^\B"(x⁰))B"(0) : dist(x C^\F^;1(K^;y))adist(F(x) K^;y) F is simply called metrically regular in the case C =X .

Given two cones Kx X and Ky Y, a mapping f :X ^!Y shall be called (Kx Ky)- increasing at some point x²X if there exists some " >0 such that

x^1;x^{2 2}Kx =⁾f(x¹)^;f(x²)²Ky ⁸x¹ x^{2 2}B(x ") For a Banach space X with dual X and a multifunction :X^!!X denote by

limsup_x

!x =^fx ²X ^j9xn^!x ⁹xn * x xn²(xn)^g

the sequential Kuratowski-Painleve upper limit with respect to the norm topology in X and the weak-star topology in X. To a cone K X its polar cone K⁰ X is assigned by K⁰ =^fx ²X ^jhx xⁱ0 ⁸x²K^g.

Next, we introduce Mordukhovich's normal cone which is based on the set of Frechet "- normals:

De nition 2.1

^Let CXbe a nonempty subset of a Banach space X and "0.

1. The set of Frechet "- normals (" 0) to C at some x²cl C is N^"(Cx) =^fx ²X ^jlimsup

u2C

u!x

hx u^;xⁱ

ku^;x^k "^g 2. The (Mordukhovich-) normal cone to C at some x²cl C is

N(Cx) = limsup

x!x

x2C

"#0

N^"(Cx)

4

In 22] it is shown that for Asplund spaces (i.e. those Banach spaces on which every continuous convex function is Frechet dierentiable at a dense set of points) one can let " = 0 in the denition of the normal cone. It is noted, that in innite dimensions, this normal cone lacks the property of being weak star closed unless a normal compactness assumption introduced by Loewen 19] is made for the set C:

De nition 2.2

A closed set C X is said to be normally compact around x ² C if there exist >0 and a compact set S X such that

^kx^kmax_s

2S ^hx s^{i 8}x ²N^{^0}(Cx) ⁸x²B(x )^\C

In 19] Loewen showed that the multifunction x^7!N(Cx) is closed near x²C in the norm x weak star topology of XX provided that C is normally compact around x and that X is a reexive Banach space. In particular, N(C x) is a weak star closed set then.

We also make use of Clarke's tangent cone (see 6]) to a set C at some point x²C: Tc(Cx) =^fh²X ^j8xn^!x (^fxn^gC) ⁸tn^#0 ⁹hn ^!h: xn+tnhn ²C^g

and of its polar, the Clarke's normal cone Nc(Cx) =T_c⁰(Cx). In any Banach space, one has N(Cx)Nc(C x), while in Asplund spaces, the two introduced normal cones are related by (see 22]) Nc(Cx) =coN(Cx), where co denotes the weak star closed, convex hull.

With a multifunction :X^!!Y one may associate a multifunction D(x y) :Y^!!X at some point (x y)²Gph which is called the coderivative of and is dened by

D(x y)(y) =^fx ²X ^j(x ^;y)²N(Gph(x y))^g

The just dened coderivative relates to Mordukhovich's normal cone. If, instead, it relates to Clarke's normal cone Nc, then we shall use the symbol D_c for distinction. From the inclusions for the corresponding normal cones it follows that ImD(x y) ImD_c(x y) and KerD(x y) KerD_c(x y). For special multifunctions (x) = f(x) +IR⁺ = epif, where f :X ^!IR and 'epi' denotes the epigraph, one gets the corresponding Mordukhovich's and Clarke's subdierentials D(x f(x))(1) =@f(x) and D_c(x f(x))(1) =@cf(x).

The following results by Mordukhovich are collected from 22] and 23]. The statement of the rst theorem is a nite dimensional reduction of the original result.

Theorem 2.3

^Let X be an Asplund space and : X^!!IR^m a multifunction with closed graph such that (x y)²Gph. Then, the injectivity condition

KerD(x y) = ^f0^g

is sucient to imply metric regularity of at (x y). If, moreover, X is nite dimensional, then it is both necessary and sucient for metric regularity.

Theorem 2.4

^Let C¹ C² be two closed subsets of an Asplund space X such that x²C^1\C². If one of these sets is normally compact in the sense of De nition 2.2 and if the condition

N(C¹ x)^\;N(C² x) =^f0^g holds, then one has N(C^1\C² x)N(C¹ x) +N(C² x).

5

Theorem 2.5

^Let F :X ^!Y be a continuous function between Asplund spaces and f :Y ^! IR a locally Lipschitzian function. Then, at any xed x²X, one has

@(f F)(x)

y²@f⁽F⁽x⁾⁾DF(x F(x))(y)

3 Results

3.1 Metric regularity for cone increasing constraint mappings

In this section, we deal with constraint mappings modelling a nite number of inequalities in an innite dimensional space with additional cone increasing behaviour. The following simple observation is basic for introducing this information into the characterization of metric regularity:

Proposition 3.1

^Let X Y be Banach spaces, Kx X a closed cone, Ky Y a closed, convex cone and f : X ^! Y a (Kx Ky)- increasing function around x ² X. Then, the associated multifunction :X^!!Y de ned by (x) :=^;f(x) +Ky satis es:

ImD(x y)ImD_c(x y)K_x^{0 8}y²(x)

Proof:

Only the second inclusion has to be shown. Assume that x ²ImD_c(x y), that is, there exists some y ²Y such that (x ^;y)²Nc(Gph(x y)). We show that (h 0)²Tc(Gph(x y)) for all h²Kx. For any (x y)²Gph in a small neighborhood of (x y) we have f(x+h)^; f(x)²Ky and f(x)+y²Ky, hence, by convexity of Ky it holds f(x+h)+y²Ky. Therefore, (x+h y) ² Gph. Now consider arbitrary sequences (xn yn) ^! (x y) ((xn yn) ² Gph) and tn ^#0. Then (xn yn)+tn(h 0) = (xn+tnh yn)²Gph (since tnh²Kx for all n ²IN), so (h 0) ² Tc(Gph(x y)) and we conclude that ^hx hⁱ = ^h(x ^;y) (h 0)ⁱ 0 for all

h²Kx. Therefore x ²K_x⁰ as was to be proved. ²

Corollary 3.2

^Let X be a Banach space, Kx X a closed cone and f :X ^!IR a (Kx IR⁺)- increasing function around x² X. Then @f(x)@cf(x) ^;K_x⁰. In particular, for X =IRⁿ and Kx=IRⁿ⁺, one has @f(x)@cf(x)IRⁿ⁺.

Proof:

Since ^;f is (^;Kx IR⁺)- increasing around x, it follows from Proposition 3.1 that

@f(x) =D(epif)(x f(x))(1)D_c(epif)(x f(x))(1) =@cf(x)^;K_x⁰

2

The next lemma deals with a constraint mapping having the structure of a composite function with the outer function being cone increasing. This structure is motivated by the chance constraint (1) discussed in the introductory section (recall, that in (1) F as a distribution function is (IR^m+ IR⁺)- increasing).

6

Lemma 3.3

^Let F : X ^! Y be a continuous function between Asplund spaces, Ky Y a closed cone, and f : Y ^! IR a locally Lipschitzian function which is (Ky IR⁺)- increasing.

Then, the constraint (fF)(x)0 is metrically regular at some feasible point x if f(F(x))>0 or if, in the binding case, the following two conditions are satis ed:

1. 0²= @(^;f)(F(x))

2. 0²= DF(x F(x))(y) ⁸y ²K_y^0nf0^g

If, in addition, Y =IR^m Ky =IR^m+ X is reexive and F is locally Lipschitzian and (Kx IR^m+)- increasing, where Kx is a closed cone with the property

9x^ ²X : ^hx x^ⁱ>0⁸x ²K_x^0nf0^g (2) then condition 2. reduces to

0²= @Fi(x) i= 1 ::: m

Proof:

According to the denitions, we have to verify metric regularity of the multifunction (x) =

;(fF)(x)+IR⁺ at (x 0)²Gph. This is clear in the nonbinding case f(F(x))>0 where, due to continuity, both distances occuring in the denition of metric regularity equal zero locally.

For the binding case we apply Theorem 2.3. The sucient criterion KerD(x 0) =^f0^g for metric regularity is equivalent in the present context to 0 ²= @(^;(f F))(x). Now, condition 1. above along with Corollary 3.2 (applied to ^;f) give @(^;f)(F(x)) K_y^{0 nf}0^g. Therefore, 0²= ^fDF(x F(x))(y)^jy ²@(^;f)(F(x))^g due to condition 2. above, and Theorem 2.5 yields 0²= @(^;(fF))(x) as was to be shown.

Now, consider the additional assumptions of the lemma. It follows that DF(x F(x))(y) =@^hyFⁱ(x)^Xm

i⁼¹y_i@Fi(x)

The equation is the so-called scalarization formula proved in 22] while the inclusion comes from the sum rule. So we are done, if we can show that zero does not belong to the set on the right hand side whenever y ²K_y^0nf0^g=IR^{m; nf}0^g. Obviously, this amounts to the relation 0²= co^f@Fi(x)^ji= 1 ::: m^g where co refers to the convex hull.

From the reexivity of X it follows, that the subdierentials @Fi(x) are weak star closed (see Theorem 9.2 in 22]), so, due to boundedness (recall that F is locally Lipschitzian) they are weak star compact. Consequently, there exist ^x_i ²@Fi(x) such that

i = max^fhx x^^ijx ²@Fi(x)^g=^hx^_i x^ⁱ

where ^x refers to (2). On the other hand, each component Fi is (Kx IR⁺)- increasing, so Corollary 3.2 along with the assumption 0²= @Fi(x) gives ^;x^_i ²K_x^0nf0^g. Then, (2) provides i <0. Setting := maxi <0 we arrive at

@Fi(x)^fx ²X ^jhx x^^ig\;K_x⁰ =:H i = 1 ::: m 7

where H is a convex set not containing zero. Therefore, 0²= co^f@Fi(x)^ji= 1 ::: m^g as was

to be shown. ²

Note, that Lemma 3.3 provides separate constraint qualications for the two functions in the composition. While this could also be obtained without cone increasing behaviour, condition 2. is substantially improved by introducing additional information. In order to illustrate this fact, assume for a moment, that F :IR^{n !}IR^m is a continuously dierentiable, nondecreasing mapping. Without exploiting the nondecreasing behaviour, condition 2. would reduce to the linear independence of the gradients ^rFi(x). But in fact, it is sucient to restrict condition 2. to y ² K_y^{0 n f}0^g = IR^{m; n f}0^g, which only means negative (or, equivalently, positive) linear independence of these gradients. Although the concept of active indices does not make sense in this context, one may compare this dierence with the dierence between the Linear Independence and the Mangasarian Fromovitz Constraint Qualication, where the latter is substantially weaker. It is also noted, that the additional assumptions of Lemma 3.3 are met, for instance, in the chance constraint (1) in case that the production function h is nondecreasing too (then X =IRⁿ, and Kx =IR⁺ⁿ meets (2)).

The following proposition is technical and similar versions of it are proved in 12] or 15].

Proposition 3.4

^Let X be a Banach space. With some locally Lipschitzian mapping f :X ^! IR^m associate the multifunction : X^!!IR^m de ned by (x) :=^;f(x) +IR^+m. Then, at any x with f(x)²IR^m+ one has (x 0)²Gph and

1. y ²IR^m;

2. ^kx^{k k}y^{k 8}(x y)²N(Gph(x 0)) for some >0 3. Gph is normally compact around (x 0)

Proof:

Let L " > 0 be such that L is a Lipschitz modulus of f in B(x "). Consider any (x y) ² Gph^\(B(x "=2)B(0 "=2)). Choose an arbitrary h ²X^nf0^g. Then, for 0< t < "=(2^kh^k) one has

f(x+th) +yf(x+th)^;f(x)^;Lt^kh^k

1

⁽

1

^{= (1 ::: 1)2IRm):}

Consequently, (x+th y+Lt^kh^k

1

) ² Gph and for any (x y) ² N^(Gph(x y)) (with arbitrary 0) it follows:

hx hⁱ+L^kh^khy

1

ⁱ = ^k(h L^kh^k

1

)^k lim_t

#0

h(x y) (x+th y+Lt^kh^k

1

)^;(x y) t^k(h L^kh^k

1

^{)k i}

k(h L^kh^k

1

^{)k limsup}

(x 0

y 0

)!(xy)

(x 0

y 0

)2Gph

h(x y) (x⁰ y⁰)^;(x y)

k(x⁰ y⁰)^;(x y)^ki

^k(h L^kh^k

1

)^k (3)

8

Next, x any index i with 1im and observe that (x y+tei)²Gph, where ei denotes the i^th standard unit vector in IR^m (the remaining variables xed as above). It follows for any (x y)²N^(Gph(x y)) (where 0 is arbitrary and brackets refer to the components):

yi] = lim_t

#0

h(x y) t^;1((x y+tei)^;(x y))ⁱ (4) with the same argumentation as in (3). Now, corresponding to some (x y)²N(Gph(x 0)) there exist sequences (xn yn) ^! (x 0) (x_n y_n) * (x y) and n ^# 0 such that (xn yn) ² Gph and (x_n y_n) ² N^ⁿ(Gph(xn yn)). Consequently, for all h ² X (the excluded case h= 0 follows trivially) one gets by (3)

hx hⁱ= lim_n ^hx_n hⁱlim_n ^fn^k(h L^kh^k

1

)^k;L^kh^khy_n

1

^ig=^;L^kh^khy

1

ⁱ

and by (4): yi] = lim_n y_ni] lim_n n = 0, hence y ² IR^m;. Finally, interchanging h and ^;h provides

jhx h^ijL^kh^kky^k1 L^kh^kky^{k 8}h²X

where ^{k k1} refers to the sum norm and is some modulus of norm equivalence in IR^m. Putting := L, one arrives at ^kx^{k k}y^k. It remains to check the last assertion of the Proposition. If we reconsider (3) and (4) but with = 0, then the same reasoning as in the lines before gives that

y 0 and ^kx^kky^k

8(x y)²N^⁰(Gph(x 0)) ⁸(x y)²Gph^\(B(x "=2)B(0 "=2))

Therefore, ^k(x y)^k = ^kx^k+^ky^k (1 +)^ky^k (1 +)^h;

1

yⁱ, where is another modulus of norm equivalence in IR^m. Now, normal compactness of Gph around (x 0) follows according to Denition 2.2 with :=^;1(1 +)^;1 :="=2 S :=^f(0 ^;

1

)^g. ² The subsequent theorem relates the injectivity conditions for Mordukhovich's and Clarke's coderivative in the case of cone increasing constraint mappings. It is known that, in gen- eral, the injectivity condition KerD_c(x 0) = ^f0^g based on Clarke's coderivative, is too strong as a criterion for metric regularity. Take, for instance, the one-dimensional multifunc- tion (x) = ^;jx^j+IR⁺ which is metrically regular at (0 0) but where KerD_c(0 0) = IR⁺ (note, however, that KerD(0 0) = ^f0^g). On the other hand, the theorem shows that, for certain cone increasing constraints (modelling a nite number of inequalities in an innite di- mensional space), both injectivity conditions are equivalent in order to check metric regularity of the associated multifunction. In such constellations, there is no advantage of using the one or the other coderivative, it might actually be more convenient to work with Clarke's concepts of subdierentiation.

Theorem 3.5

^Let X be a reexive Banach space, Kx X a closed cone with the property (2) and f : X ^! IR^m a (Kx IR^m+)- increasing, locally Lipschitzian mapping. Then, the multifunction :X^!!IR^m de ned by (x) :=^;f(x) +IR^m+ satis es

KerDc(x 0) =^f0^g()KerD(x 0) =^f0^g 9

Proof:

Due to KerD(x 0) KerD_c(x 0) one has to show the direction ^{0 (0}, so assume that KerD(x 0) = ^f0^g. This is equivalent to 0 ²= D(x 0)S^m;1] where S^m;1 = ^fy ² IR^{m j}

ky^k1 = 1^g and ^kk1 refers to the sum norm in IR^m. First note, that D(x 0)S^m;1] is weak- compact. In fact, from Proposition 3.4 we derive that D(x 0)S^m;1]B(0 ) for some >

0, so it is bounded. It remains to show weak- closedness. Let x * x (x ²D(x 0)S^m;1]) be a convergent net. By denition, there exists a net y ² S^m;1 such that (x ^;y) ² N(Gph(x 0)). By compactness of S^m;1 there is a convergent subnet y^{0 !} y ² S^m;1, so (x^{0 ;}y⁰)* (x ^;y) with (x^{0 ;}y⁰)²N(Gph(x 0)). According to Proposition 3.4, Gph is normally compact around (x 0) (compare Denition 2.2), hence N(Gph(x 0)) is weak star closed. It follows that (x ^;y)²N(Gph(x 0)), so x ²D(x 0)S^m;1], as was to be shown. As a consequence of weak- compactness, there is some ^x ² D(x 0)S^m;1] with ^hx^ x^ⁱ = min^fhx x^^{i j} x ² D(x 0)S^m;1]^g, where ^x refers to (2). Proposition 3.1 provides D(x 0)S^m;1]K_x⁰, so, by assumption, ^x ²K_x^0nf0^g. Now (2) yields ^hx^ x^ⁱ>0.

Summarizing, it follows D(x 0)S^m;1] H = ^fx ² X ^{j h}x x^^{i h}x^ x^^ig and 0 ²= H. We are done if we can show that

D_c(x 0)S^m;1]coD(x 0)S^m;1] (5) since then, due to convexity and weak- closedness of H one gets D_c(x 0)S^m;1] H, in particular 0²= D_c(x 0)S^m;1] from where the desired relation KerD_c(x 0) =^f0^g follows.

Now, rst consider any (x ^;y)² coN(Gph(x 0)) with y ²S^m;1. This means exis- tence of some i 0 (i= 1 ::: k) and of (x_i ^;y_i)²N(Gph(x 0)) such that ^P_ki⁼¹i = 1 and (x ^;y) = ^P_ki⁼¹i(x_i ^;y_i). We may assume that y_i ⁶= 0, since otherwise the second assertion of Proposition 3.4 implies x_i = 0 and the term (x_i ^;y_i) may then be removed from the sum. Also from Proposition 3.4, we know that ^;y_i ²IR^m; (i= 1 ::: k), so y =^P_ki⁼¹iy_i implies ^ky^k1 =^P_ki⁼¹i^ky_i^k1. By the cone property of N one has

(^ky_i^k1]^;1x_i ^;ky_i^k1]^;1y_i)²N(Gph(x 0)): Therefore, w_i := ^ky_i^k1]^;1x_i ²D(x 0)S^m;1]. It results

x =^Xk

i⁼¹ix_i =^Xk

i⁼¹i^ky_i^k1ky_i^k1]^;1x_i =^Xk

i⁼¹iw_i

where i 0 (i= 1 ::: k) and ^P_ki⁼¹i = 1. Consequently, x ²coD(x 0)S^m;1].

In order to verify (5), let x ²D_c(x 0)(y) with y ²S^m;1. Then, (x ^;y)²N^c(Gph(x 0)) =co N(Gph(x 0))

so there is a net (x ^;y) * (x ^;y) with (x ^;y)² co N(Gph(x 0)). Then, we also have

(x ^;y)( (^ky^k1]^;1ky^k1x ^;ky^k1]^;1ky^k1y) =: (v ^;r)²co N(Gph(x 0)) but r ²S^m;1. As it was proved above, it follows that v ²co D(x 0)S^m;1]. Consequently,

x ²co D(x 0)S^m;1] which terminates the proof. ²

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Corollary 3.6

^Let f : IR^{n !} IR^m be a (IRⁿ⁺ IR^+m)- increasing, locally Lipschitzian mapping de ning the constraint f(x)0. Then f is metrically regular at some feasible point x²IRⁿ, if and only if KerD_c(x 0) =^f0^g for (x) =^;f(x) +IR^m+.

Proof:

By Theorem 2.3 f is metrically regular at x, if and only if KerD(x 0) = ^f0^g. Apply

Theorem 3.5. ²

As an application, we consider the chance constraint (1) with locally Lipschitzian distribution function and continuous production function. The corollary provides, that checking the rst condition in Lemma 3.3, which in the context of (1) reads as 0²= @(^;F )(h(x)), is equivalent to verifying the condition 0²= @cF (h(x)) for the nondecreasing distribution function F .

It is clear, that in Theorem 3.5 some cone property has to be required for Kx. Otherwise, one could take the example f(x) =^jx^j discussed before the statement of the theorem. Here, f is trivially (0 IR⁺)- increasing, but the equivalence between the two injectivity conditions does not hold for the associated multifunction . Of course, Kx =^f0^g violates (2). On the other hand, the required cone property is not too restrictive. It holds, in particular for the usual positivity cones IR⁺ⁿ or l^p+ L^p+ with p²(1 ¹), so it is not necessary - although sucient - to have nonempty interior. Note, however, that cones of the type R^n+f0^gm do not meet (2), so the consideration of equality constraints in the setting of Theorem 3.5 is excluded.

The next lemma shows, that metric regularity of constraint systems itself may be character- ized by cone increasing behaviour. To this aim, we call a closed subset CX to be generating at some x²C, if Clarke's tangent cone is a generating cone there, i.e. Tc(C x)^;Tc(C x) =X. This may be understood as a kind of constraint qualication for the set C.

Lemma 3.7

^Let X be an Asplund space, f : X ^! IR^m a locally Lipschitzian mapping and CX a closed subset which is generating at some point x²C also ful lling f(x)0. Then, the constraint f(x)0 is metrically regular at x with respect to C if

1. KerD(x 0) =^f0^g, where (x) :=^;f(x) +IR^m+ 2. f is (Tc(C x) IR^m+)- increasing around x.

Proof:

We have to show metric regularity of the multifunction ¹(x) :=

(

;f(x) +IR^m+ if x² C

else

at the point (x 0) ² Gph¹. By Theorem 2.3 it remains to check that KerD¹(x 0) =

f0^g. To see this, choose any y ² IR^m with (0 ^;y) ² N(Gph¹(x 0)). Obviously, we can write Gph¹ = Gph ^\ (C IR^m) with as introduced in the statement of the lemma. Now, for arbitrary (x z) ² N(Gph(x 0))^\;N(C IR^m(x 0)) one has z = 0 due to N(C IR^m(x 0)) = N(C x) ^f0^g. But then, Proposition 3.4 provides x = 0, hence N(Gph(x 0))^\;N(CIR^m(x 0)) =^f0^g. But we also know from Propo- sition 3.4 that Gph is normally compact around (x 0). Therefore, Theorem 2.4 provides

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(0 ^;y) ² N(Gph(x 0)) + N(C x)^f0^g]. This means existence of some x ^{2 ;}N(C x) such that (x ^;y) ² N(Gph(x 0)), i.e. x = D(x 0)(y). By assumption 2. of this lemma and by Proposition 3.1 we know that x ²(Tc(C x))⁰ =Nc(C x). On the other hand, x ^2;Nc(C x) (since always N Nc). But, since Tc(C x)) is a generating cone due to the assumption of C being a generating set at x, its polar Nc(C x)) must be a pointed cone, therefore x = 0 and y ² KerD(x 0). Now, assumption 1. of this lemma gives y = 0

which completes the proof. ²

Lemma 3.7 provides an alternative criterion for metric regularity as compared to the usual constraint qualications, directly relating subdierentials of the components fi to the normal cone of C.

3.2 Global metric regularity of nite dimensional, nondecreasing constraint mappings

In this section, we study global metric regularity of nite dimensional, nondecreasing (i.e.

(IRⁿ⁺ IR^m+)- increasing) constraint mappings. More precisely, we mean metric regularity w.r.t.

C at all feasible points of the constraint

M =^fx²IR^njf(x)0 and x²C^g (6) where C IRⁿ is closed, f ^{2 C01}(IRⁿ IR^m) and f satises x y ⁾ f(x) f(y) with the partial orders of IRⁿ IR^m, respectively. By ^C01(IRⁿ IR^m) we denote the space of locally Lipschitzian mappings f :IR^n!IR^m. In the case m= 1, the symbol ^C01(IRⁿ) will be used.

As mentioned in the introductory section, global metric regularity is a typical or generic property of continuously dierentiable constraint mappings. Here, 'generic' refers to the fact, that it is fullled for a dense G- set (a countable intersection of open sets) in the space of continuously dierentiable mappings from IRⁿ to IR^m endowed with a suitable topology.

As we shall see from an example below, a similar statement does not hold true for locally Lipschitzian constraint mappings. First, we endow ^C01(IRⁿ) with a metric. For f ^2C01(IRⁿ) dene the function f(x) = max^fky^{k j} y ² @cf(x)^g. Obviously, f is nonnegative and it is uppersemicontinuous due to the uppersemicontinuity of the set-valued mapping @cf().

Furthermore, it has the folllowing properties (for arbitrary f g ^{2 C01}(IRⁿ) and arbitrary x²IRⁿ):

f⁺g(x) f(x) +g(x) (7) d^H(@cf(x) @cg(x)) f^;g(x) (8) Here d^H refers to the Hausdor distance of closed subsets of IRⁿ. Relation (7) is based on the sum rule for Clarke's subdierential. To see (8), recall the representation of the Haus- dor distance between compact, convex sets by means of their support functionals, which for Clarke's subdierential is the generalized directional derivative d⁰. Since d⁰ fullls a triangular inequality w.r.t. f g for xed point and direction, we have

d^H(@cf(x) @cg(x))

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