on the qualitative approximation of lipschitzian functions

Nonlinear Analysis 56 (2004) 289–307www.elsevier.com/locate/na

On the qualitative approximation ofLipschitzian functions

Mar)*a Alonso∗, Luis Rodr)*guez Mar)*nDpto. Matematica Aplicada I. E.T.S.I. Industriales, Universidad Nacional

de Educacion a Distancia, Despacho No. 2.49 Aptdo. 60149, 28080 Madrid, Spain

Received 10 July 2003; accepted 14 August 2003

Abstract

Problem of topological equivalence between a function and certain local approximations isstudied. The study is carried out in a neighbourhood of a critical point with the concept ofcritical point of Clarke’s theory. The function belongs to a particular class of non B-di:erentiablefunctions. The local approximations are positively homogeneous maps. Using the concept oftopological equivalence we establish the existence of a local coordinate transformation betweenthe original function and the positively homogeneous function. As a consequence we obtainsu<cient conditions for the existence of local extremes for the initial map.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Nonsmooth function; Topological equivalence; Critical point; Continuous selection

1. Introduction

Problems about qualitative behaviour of local approximations of nonsmooth func-tions, have motivated di:erent authors as Kuntz and Scholtes [9,10], Jongen [7], Chaney[3] or Bougeard [2]. For a function f :U → R deDned on an open set of Rn, and apoint x0 ∈U , this behaviour is analyzed through the concept of local topological equiv-alence, which consists in the existence of a local coordinate transformation such thatlocally around the point x0 of investigation, the approximation as a function of the newcoordinates coincides with the original function in the original coordinates.These studies are specially complex if the point under investigation is a critical

point. The concept of critical point for nondi:erentiable functions is based on Clarke’stheory of locally Lipschitzian functions. Kuntz and Scholtes deal with this problem

∗ Corresponding author. Tel.: +91-3987990; fax: +91-3986012.E-mail address: [email protected] (M. Alonso).

0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2003.09.017

mailto:[email protected]

290 M. Alonso, L.R. Mar+n /Nonlinear Analysis 56 (2004) 289–307

for real-valued piecewise di:erentiable functions. In particular, these functions areB-di:erentiable at x0, i.e., they are Lipschitz-continuous on a neighbourhood of x0,and the B-derivative or Bouligand derivative, deDned by

f′(x0; v) =f(x0 + v)− f(x0)

texists for all v∈Rn. Therefore the local approximation for these functions is determinedby the B-derivative.The aim of this article is to investigate the qualitative behaviour of certain approxi-

mation functions when f is not B-di:erentiable. To accomplish this we make use ofthe notion of topological equivalence deDned by Jongen (cf. [7]):

De�nition 1. Two continuous functions f; g :U → R deDned on an open set U ⊂ Rn

are called topologically equivalent at x0 ∈U , if there exist open neighbourhoods of x0,V and W , and a homeomorphism � :V → W such that �(x0) = x0 and

f = g ◦ � on V:

If � and �−1 are Lipschitzian functions, then f and g are called Lipschitz-equivalent.

We will mainly study this problem for a class of non-B-di:erentiable functionsaround a critical point.On the other hand, we will obtain su<cient conditions for the existence of local

extremes for these functions. The function class which we deDne and study (PL func-tions), is comprised by continuous selections of Lipschitzian functions. A PL functionis Lipschitzian but generally not B-di:erentiable. The concept of local approximationwhich we have deDned for this class is based on the one for nondi:erentiable functionswhich we have established in [1]. In the case of Lipschitzian functions we can describethis concept as follows: we consider Clarke’s derivative deDned by

f0(x0; v) = lim supt→0+y→x0

f(y + tv)− f(y)t

;

furthermore, by computing the lower limit we obtain another derivative

Mf(x0; v) = lim inft→0+y→x0

f(y + tv)− f(y)t

:

We will refer to f0 and Mf as Clarke’s upper and lower derivatives, respectively. Bothderivatives satisfy:

(i) they are Dnite and Mf(x0; v)6f0(x0; v) for all v∈Rn,(ii) f0(x0;−v) = (−f)(x0; v) for all v∈Rn.

Then we will call the pair (f0(x0; ·); Mf(x0; ·)) a coherent pair of derivatives of f atx0. In this way, we give the deDnition of local approximation for a Lipschitzian func-tion around x0 (i.e., for a function which is Lipschitz-continuous on a neighbourhoodof x0):

M. Alonso, L.R. Mar+n /Nonlinear Analysis 56 (2004) 289–307 291

De�nition 2. Let f be Lipschitzian around x0. A positively homogeneous functiong :Rn → R will be called a local approximation of f at x0 if

Mf(x0; v)6 g(v)6f0(x0; v) for all v∈Rn

According with this deDnition, there is a family of functions which are localapproximations for the function f, in particular Clarke’s derivatives. We will investi-gate qualitative behaviour of this family. In this analysis we will focus on the conceptof critical point, determined by the derivative f0(x0; ·). This function is sublinear,hence it is the support function of a convex compact set @f(x0) deDned by

@f(x0) = {�∈Rn | � · v6f0(x0; v) ∀v∈Rn}:This set gives rise to the following deDnition:

De�nition 3. A point x0 is called a critical point of a Lipschitzian function around x0,f, if 0∈ @f(x0).

To carry out our research in the case of a critical point, we need to use a sub-division of the domain into constraint sets. In Section 2 we study these partitions byanalyzing certain characters of the constraint sets. Section 3 contains some propertiesof Lipschitzian functions deDned on these subdivisions. In Section 4, we state su<cientconditions for a PL function to be topologically equivalent to its local approximations(Theorem 27). We will use the results about partitions and functions obtained in Sec-tion 3. Finally, the statements of Section 4 are applied to establish su<cient conditionsfor a critical point to be a local extremum of f (Theorem 30).Throughout this paper, f :U → R will be deDned on an open set U ⊂ Rn and x0

will be a point of U .

2. Constraint sets and tangent cones

For a noncritical point x0, we have obtained in [1] a result about topological equiv-alence:

Proposition 4. Let f :U → R be Lipschitzian around x0 ∈U and x0 a noncriticalpoint of f. Then f is Lipschitz-equivalent to any function hx0 :Rn → R de7ned byhx0 (x) = f(x0) + h(x − x0), where h :Rn → R is a continuous local approximation off at x0, such that h0(0; v)6f0(x0; v) for all v∈Rn.

Remark 5. Note that from the inequality h0(0; v)6f0(x0; v), we can deduce that−f0(x0; v)6−h0(0; v). But by Clarke’s derivatives deDnitions −f0(x0; v)= Mf(x0;−v),so we obtain Mf(x0;−v)6 Mh(0;−v) and then Mf(x0; v)6 Mh(0; v)6 h0(0; v)6f0(x0; v)for all v∈Rn.

In the case that the point under investigation is a critical point, we need moreinformation about the function f and the domain U . We will restrict ourselves to a


class of Lipschitzian functions. In order to deDne this class we require the existenceof a certain partition of the neighbourhood U . Following Kuntz and Scholtes [9] wewill consider partitions deDned by constraint sets. A constraint set

M = {x∈U | hi(x)6 0; j = 1; : : : ; m}is called Cr-regular if h1; : : : ; hm :U → R are Cr-functions and, for each x∈M , the set{∇hi(x) | i∈ I(x)} is a linearly independent vectors set, where I(x)={ j | hj(x)=0} (see[6]). For a Cr-regular constraint set M , the tangent cone of M at x0 ∈M is deDned by

C(x0) = {y∈Rn | ∇hj(x0)y6 0; j∈ I(x0)}It is a convex polyhedral cone. Recall that the linear space of a polyhedral cone C isthe largest linear subspace contained in C and it coincides with the linear space of eachof its faces. We will denote this space by L. If L= {0} the cone C is called pointed.

De�nition 6 (cf. Kuntz and Scholtes [9]). Let U be an open set of Rn and let x0 ∈U .A collection M = {M1; : : : ; Mk} of subsets of U is called a Cr-subdivision of U atx0 if there exists a natural number m6 n and a collection of Cr-functions {hij :U →R | i = 1; : : : ; k; j = 1; : : : ; m}, such that

(1) each Mi is representable as Mi={x∈U | hij(x)6 0; j=1; : : : ; m} and it is a regularconstraint set,

(2) x0 ∈⋂k

i=1 Mi,(3) M forms a partition of U , i.e.

k⋃i=1

Mi = U;

int(Mi) ∩ int(Mj) = ∅ if and only if i �= j;

(4) the collection {C1; : : : ; Ck} of tangent cones to each Mi forms a partition of Rn,such that two distinct cones Ci and Ci′ intersect in a common face,

(5) IfCi∩Ci′={y∈Rn | ∇hij(x0)y=0; j∈ I;∇hij(x0)y6 0; j �∈ I}, where I ⊂ {1; : : : ; m},then Mi ∩Mi′ = {x∈U | hij(x) = 0; j∈ I; hij(x)6 0; j �∈ I}.The subdivision is called pointed if m= n.

Note that in virtue of this deDnition, the linear space L of all tangents cones Ci

of a Cr-subdivision coincide. We will call L the linear space of the subdivision. Itturns out that for a Cr-subdivision, every tangent cone Ci is representable as Ci=Ci + L where Ci ⊂ L⊥ is a pointed cone and the space L and the cone Ci are unique(cf. [12], Chapter 8).We will denote by {zij} j=1; :::;m the normalized extremal rays of the cone Ci and by

Zi the matrix with columns zij.Next, we deDne a particular property of certain subdivisions. We will designate with

〈ei; ei+1; : : : ; en〉 the linear subspace generated by the vectors of the canonical base ofRn {ei; ei+1; : : : ; en}.


De�nition 7. Let M be a Cr-subdivision of U at x0. M is said provided with theupper (lower) triangular property, if for every i = 1; : : : ; k, j = 1; : : : ; m the functionshij satisfy ∇hij(x0)∈ 〈en−j+1; : : : ; en〉 (∇hij(x0)∈ 〈ej; : : : ; en〉). We will say that M isprovided by the triangular property if it is provided by the upper triangular propertyor by the lower one.

Remark 8. Note that for a pointed subdivision (m= n), then L= {0} and Ci = Ci forall i=1; : : : ; n. It is easy to prove that for a pointed subdivision with the upper (lower)triangular property, the matrixes Zi are upper (lower) triangular matrixes.

Lemma 9. Let M be a Cr-subdivision of U at x0 with the triangular property. Thenthere exists a pointed Cr-subdivision M of U at x0, associated to M, with thetriangular property.

Proof. We will suppose that M has the upper triangular property (the lower triangularcase is similar), and that M={M1; : : : ; Mp} where Mi={x∈U | hij6 0; j=1; : : : ; m}.IfM is not pointed, thenm¡n and for every i=1; : : : ; p the vectors ∇hi1 (0); : : : ;∇him(0)form a basis for the subspace L⊥. Let {am+1; : : : ; an} be a basis for L. Since M hasthe upper triangular property then ∇hij(x0)∈ 〈en−j+1; : : : ; en〉 for every j=1; : : : ; m andal ∈ 〈e1; : : : ; en−m〉 for every l = m + 1; : : : ; n. We can choose am+1 = en−m, am+2 =en−m−1; : : : ; an= e1. Following Kuntz and Scholtes [9, Theorem 3.1] we deDne the sets

Mi;" = {x∈U | hij(x)6 0; "(l)al · x6 0; j = 1; : : : ; m; l= m+ 1 : : : ; n};

where " is a map from the set {m + 1; : : : ; n} to the set {−1; 1}. It is easy to provethat the collection M of these sets Mi;" is a pointed Cr-subdivision of U at x0 withthe upper triangular property.

Remark 10. Note that for a Cr-subdivision of U at x0, M= {M1; : : : ; Mp} and for anorthogonal map #, the collection #(M) = {#(M1); : : : ; #(Mp)} is a Cr-subdivision of#(U ) at #(x0). We will say that M is a triangular subdivision of U at x0, if M hasthe triangular property or if there exists an orthogonal map # such that #(M) has thetriangular property.

3. Lipschitzian functions on polyhedral cones

In order to provide conditions for topological equivalence between a Lipschitzianfunction f and its local approximations, we will Drst study some properties aboutthese functions. We will begin with the following proposition which is a generalizationof a lemma due to Kuntz and Scholtes [9, Lemma 3.1]. For an index set I ⊂ {1; : : : ; m}we will denote by LI the linear space deDned by LI = {y∈Rn |yi = 0; i �∈ I} and by$LI the projection matrix onto LI . We recall that the set $p@f(x0) is deDned by

$p@f(x0) = {�p | �= (�1; : : : ; �p; : : : �n)∈ @f(x0)}:


Proposition 11. Let f :U → R be Lipschitzian around x0 such that there existsi0 ∈ Im={1; : : : ; m} (m6 n) with 0 �∈ $i0@f(x0). Let H be a continuous local approx-imation of f at x0 such that H 0(0; v)6f0(x0; v) for all v∈Rn. Suppose that forevery I ⊂ Im with |I |6m− 1 there is a Lipschitzian around x0 function �I :U → Rn

satisfying

(A1) �I (x0) = x0.(A2) �J (y) = �I ($LJ (y)) for every J ⊆ I and every y∈U .(A3) f(�I (x)) = f(x0) + H ($LI (x)− x0) for every su:ciently small x∈Rn.

Then there exists a neighbourhood U ⊂ U of x0 and a Lipschitzian around x0function �m : U → Rn satisfying properties (A1)–(A3) with respect to the index setI = Im.

Before the proof we will run through the following lemma.

Lemma 12. Let f :U → R be Lipschitzian around x0 ∈U . Let H be a local approx-imation of f at x0 such that H is continuous and H 0(0; v)6f0(x0; v) for all v∈Rn.Then H is Lipschitzian around 0.

Proof. Since f is Lipschitzian around x0, f0(x0; v) and Mf(x0; v) are Dnite for all v∈Rn.From H 0(0; v)6f0(x0; v) we deduce that Mf(x0; v)6 MH (0; v) (Remark 5), so H 0(0; v),MH (0; v) are Dnite for all v∈Rn. From this fact and from that H is continuous, weconclude that H is Lipschitzian around 0 (see [11, Proposition 2]).

Proof of Proposition 11. We assume without loss of generality that x0 = f(x0)= 0. Following Kuntz and Scholtes [9, Lemma 3.1], we use the function ' deDnedon U by

'(y) =m−1∑l=1

(−1)m−l−1∑J⊂Im|J |=l

�J ($LJ (y))

and we deDne on an appropriate neighbourhood of 0∈Rm+1 the function g by

g(y) = f('(y1; : : : ; ym; 0; : : : ; 0) + ym+1ei0 )− H

(m∑i=1

yiei

):

By Lemma 12, H is Lipschitzian around 0 and @g(0) is a nonempty set. Furthermore,by the calculus of the subdi:erential of a sum and the generalized Jacobian chain rule(see [4, Proposition 2.3.3 and Theoorem 2.6.6]) we obtain

$m+1@g(0) ⊂ $i0@f(0)

As 0 �∈ $i0@f(0) and g(0) = 0, we can apply the Clarke’s implicit function theo-rem to the equation g(y1; : : : ; ym; ym+1) = 0, to obtain the existence and uniquenessof a Lipschitzian function ’ deDned on a neighbourhood of 0, U ∩ Rm, such that


U ⊂ U and

g(y1; : : : ; ym; ’(y1; : : : ; ym)) = 0; ’(0) = 0:

We deDne �Im : U → Rn by

�Im(y) ='(y) + ’(y1; : : : ; ym)ei0 :

It remains to verify properties (A1)–(A3) for the function �Im . The details of thisproof are analogous to Lemma 3.1 in [9].

Remark 13. We note that, in the upper proof, a direct calculation shows that '($LI (y))=�I ($LI (y)) for every I ⊂ Im with |I |6 (m−1). Taking this into account and property(A3) we obtain that, for x su<ciently small and xi = 0 for some index i∈{1; : : : ; m},then g(x1; : : : ; xm; 0)=0. Thus by the local uniqueness of the function ’ it follows that’(x1; : : : ; xm) = 0 for x su<ciently small with xi = 0 for some i∈{1; : : : ; m}.

In the next lemma we show a “chain rule” for a particular composition of functions.We denote by JG(x) the jacobian matrix of a function G :Rm → Rn and by d−f(·; ·),d+f(·; ·) the lower and upper Dini derivatives, respectively of a function f :Rn → R.These derivatives are deDned by

d−f(x; v) = lim inft→0+

f(x + tv)− f(x)t

; d+f(x; v) = lim supt→0+

f(x + tv)− f(x)t

:

Lemma 14. Let G :Rm → Rn, (n; m¿ 1) be di;erentiable in the sense of Frechet atx∈Rm. Let f :Rn → R be Lipschitzian around G(x). Then, for all v∈Rn:

(a) d−(f ◦G)(x; v)=d−f(G(x); JG(x)v), (b) d+(f ◦G)(x; v)=d+f(G(x); JG(x)v).

Proof. (a) For v∈Rn we have

f(G(x + +v))− f(G(x))+

=f(G(x) + +u(+; v))− f(G(x))

+;

where u(+; v) = JG(x)v+ w(+; v), w(+; v) = 1=+(G(x + +v)− G(x)− +JG(x)v).Since G is Fr)echet-di:erentiable at x, then w(+; v) → 0 and u(+; v) → JG(x)v when

+ → 0+. And as f is Lipschitzian around G(x) we obtain

lim inf+→0+

f(G(x) + +u(+; v))− f(G(x))+

= lim inf+→0+

f(G(x) + +JG(x)v)− f(G(x))+

:

Therefore d−(f ◦ G)(x; v) = d−f(G(x); JG(x)v).(b) We apply (a) to the function (−f), taking into account that d+f(G(x); ·) =

−d−(−f)(G(x); ·).

Next, we will study some properties about Lipschitzian functions deDned on poly-hedral cones. We will suppose that C is a convex polyhedral pointed cone of Rn

(L = {0}), with n normalized extremal rays z1; : : : ; zn, so C is the convex hull of its


extremal rays (see [14])

C =

y∈Rn |y =

n∑j=1

/jzj; /j¿ 0

:

In the propositions below we obtain Lipschitzian homeomorphisms. To clariDcate thenomenclature we give the following deDnition.

De�nition 15. Let Mx∈Rn. A map � is a Lipschitzian homeomorphism at Mx, if thereexists a neighbourhood V of Mx such that:

(i) � :V → �(V ) is an homeomorphism,(ii) � is Lipschitzian on V and �−1 is Lipschitzian on �(V ).

Proposition 16. Let U ⊂ Rn be an open neighbourhood of 0. Let h :U → R beLipschitzian around 0 and let l be a continuous local approximation of h at 0. If forall j = 1; : : : ; n

(a) l0(0; zj)6 h0(0; zj),(b) h0(0; zj)h0(0;−zj)¡ 0,

then for every j = 1; : : : ; n there exists 0¿ 0, and a Lipschitzian homeomorphismat 0, 2j : (−0; 0) → 2j(−0; 0) ⊂ R such that

(i) 2j(0) = 0,(ii) h(2j(t)zj) = h(0) + l(tzj) for all t ∈ (−0; 0),(iii) for every sequence {tk} → 0 with tk �= 0 for all k ∈N, such that there exist

2′j(tk) and limk→∞ 2′j(tk), it satis7es limk→∞ 2′j(tk)¿ 0,(iv) 2j(t)¿ 0 for all t ∈ (0; 0).

Proof. We will suppose without loss of generality that h(0)=0. For every j=1; : : : ; nwe deDne from a neighbourhood of 0∈R2 to R the function gj by

gj(t; u) = h(uzj)− l(tzj):

We assume that h0(0; zj)¿ 0, h0(0;−zj)¡ 0 (the other case is similar). From Lemma 12we know that l is Lipschitzian around 0, then @gj(0) is not empty and it satisDes (see[4, Proposition 2.3.3 and Theoorem 2.6.6]) $u@gj(0) ⊂ co{� · zj | �∈ @h(0)}. Fromh0(0;−zj)¡ 0, we deduce that � · zj ¿ 0 for all �∈ @h(0), and then that 0 �∈ $u@gj(0).We apply the Clarke’s implicit function theorem to obtain the existence and uniquenessof a Lipschitzian around 0 function 2j deDned from an interval (−0; 0) to R satisfying(i) and (ii).(iii) By Rademacher’s theorem locally Lipschitz functions on Rn are almost every-

where di:erentiable, then we can choose tk ∈ (−0; 0), tk �= 0, such that there exists


2′j(tk). As l is positively homogeneous, from (ii) it results

h(2j(tk)zj) =

{tk l(zj) if tk ¿ 0;

−tk l(−zj) if tk ¡ 0

and by calculating the lower Dini derivative

d−(h ◦ 2jzj)(tk ; 1) ={

l(zj) if tk ¿ 0;

−l(−zj) if tk ¡ 0:

In virtue of Lemma 14

d−h(2j(tk)zj; 2′j(tk)zj) =

{l(zj) if tk ¿ 0;

−l(−zj) if tk ¡ 0:(3.1)

We consider a sequence {tk} → 0 with tk �= 0 for all k ∈N, such that there exists2′j(tk) for all k and there exists limk→∞ 2′j(tk).

Since l is a local approximation of h, we have Mh(0; zj)6 l(zj)6 h0(0; zj). Fur-thermore Mh(0; zj) = −h0(0;−zj). By considering as well (3.1) and hypothesis (b) weconclude that 2′j(tk) �= 0. We will prove that 2′j(tk)¿ 0 for k su<ciently large. In othercase there exists a subsequence {tkN } with 2′j(tkN )¡ 0 and such that

−2′j(tkN ) · d−h(2j(tkN )zj;−zj) =

{l(zj) if tkN ¿ 0;

−l(−zj) if tkN ¡ 0:

By hypothesis h0(0;−zj)¡ 0. Then Mh(0; zj) = −h0(0;−zj)¿ 0 and since l is a localapproximation of h

Mh(0; zj)6 l(zj)6 h0(0; zj);

− h0(0;−zj)6− l(−zj)6− Mh(0;−zj); (3.2)

therefore l(zj)¿ 0, −l(−zj)¿ 0 and in consequence

0¡d−h(2j(tkN )zj;−zj)¡h0(2j(tkN )zj;−zj):

By taking upper limits in this inequality and by using upper semicontinuity of h0(·;−zj)we obtain

06 lim supkN→∞

d−h(2j(tkN )zj;−zj)6 lim supkN→∞

h0(2j(tkN )zj;−zj)6 h0(0;−zj)

in contradiction with the hypothesis h0(0;−zj)¡ 0. Consequently, 2′j(tk)¿ 0 and from(3.1) we can deduce

2′j(tk)d−h(2j(tk)zj; zj) =

{l(zj) if tk ¿ 0;

−l(−zj) if tk ¡ 0;

then from (3.2)

2′j(tk) · d−h(2j(tk)zj; zj)¿ Mh(0; zj):


By considering upper limits we obtain

lim supk→∞

(2′j(tk) · d−h(2j(tk)zj; zj))¿ Mh(0; zj):

therefore (see [13, Lemma 5.2])

limk→∞

2′j(tk) lim supk→∞

d−h(2j(tk)zj; zj)¿ Mh(0; zj)¿ 0: (3.3)

On the other hand, by the lower semicontinuity of Mh(·; zj) we get

lim supk→∞

d−h(2j(tk)zj; zj)¿ lim infk→∞

Mh(2j(tk)zj; zj)¿ Mh(0; zj)¿ 0: (3.4)

From (3.3) and (3.4) we conclude:

limk→∞

2′j(tk)¿ 0:

Let us show that 2j is a Lipschitzian homeomorphism at 0. We check the Clarke’ssubdi:erential, @2j(0), deDned by @2j(0)=co{lim k→∞

tk �∈S∪72j

2′j(tk)}, with 72j the set where

2j is not di:erentiable and S a set of cero measure. Since limk→∞2′j(tk)¿ 0 then �¿ 0for every �∈ @2j(0) and 0 �∈ @2j(0). In virtue of Clarke’s inverse theorem we get that2j is a homeomorphism between two neighbourhoods of 0 with 2−1

j Lipschitzian.(iv) As 2j is Lipschitzian around 0, we can assume that it is Lipschitzian on (−0; 0)

and then absolutely continuous on [0; 0). Furthermore, in a similar way that in (iii) wededuce that 2′j(s)¿ 0 for all s in some neighbourhood of 0, and we can suppose that2′j(s)¿ 0 for all s∈ (0; 0). Therefore for every t ∈ (0; 0) we have

2j(t) = 2j(t)− 2j(0) =∫ t

02′j(s) ds¿ 0:

In the next proposition, we consider the matrix Z which columns are the vectors zj,j = 1; : : : ; n.

Proposition 17. Under the hypothesis of Proposition 16, there exists a Lipschitzianfunction around 0, 9, associated to h, de7ned between two neighbourhoods U , V of0∈Rn such that

(i) 9(0) = 0.(ii) (h ◦ Z ◦ 9)(x) = h(0) + (l ◦ Z)(x) for all x∈U .(iii) The kth coordinate (9(x))k of 9(x) only depends on the coordinates (xk ; xk+1; : : : ;

xn) of x.

Proof. We suppose without loss of generality that h(0)= 0. For every j=1; : : : ; n andfor x su<ciently small we deDne �{j}(x)=2j(xj)ej where 2j is the map of Proposition16. Starting with these maps �{j} we apply inductively Proposition 11 to the functionh ◦ Z , and we consider at each step m the subindex i0 as the Drst subindex of Im. Forinstance, if Im = {i1; : : : ; im} with i1 ¡ · · ·¡im, then i0 = i1.Let us check the hypothesis of Proposition 11. As Z is an isomorphism, we apply the

chain rule for generalized directional derivative (see [4, Theorem 2.3.10]) to obtain that


l◦Z is a continuous local approximation of h◦Z with (l◦Z)0(0; v)6 (h◦Z)0(0; v) forall v∈Rn and 0 �∈ $i0@(h◦Z)(0) for all i0 =1; : : : ; n. Furthermore, it is easy to see thatthe maps �{j} verify hypothesis (A1)–(A3). Then we obtain a Lipschitzian function�In deDned between two neighbourhoods of 0. Let 9=�In . 9 satisDes properties (A1)–(A3) of Proposition 11, so it satisDes (i), (ii). By construction of �In the kth coordinateof �In(x) only depends of (xk ; : : : ; xn), so we have (iii).

Remark 18. Note that, in the upper proof, for an index set Im = {i1; : : : ; im} withi1 ¡ · · ·¡im we can choose i0 = im. In this case the coordinate (9(x))k only dependson (x1; : : : ; xk).

Proposition 19. Let 9 be the map of Proposition 17. Let x∈U such that there existsthe matrix J9(x). Let alj(x) be the element of the row l, column j of the matrixJ9(x). Then:

(i) J9(x) is an upper triangular matrix.(ii) If {yk} is a sequence of Rn, {yk} → 0, such that for every yk there exists the

matrix J9(yk) and such that there exists limk→∞ J9(yk), then limk→∞ alj(yk)¿ 0 if l= j.

(iii) 9 is a Lipschitzian homeomorphism at 0.

Proof. We suppose without loss of generality that h(0)=0. (i) Since 9 is Lipschitzianaround 0, by Rademacher’s theorem, 9 is almost everywhere di:erentiable (in the senseof Lebesgue measure) on a neighbourhood of 0, and from property (iii) of Proposition17, we conclude that J9(x) is an upper triangular matrix.(ii) We note that by deDnition of 9,

9(x) = 21(x1)e1 + · · ·+ 2n(xn)en +∑

J⊂{1;:::; n}J={i0 ;:::; ik};|J |¿2

’J (x)ei0 ;

where 21; : : : ; 2n are the maps of Proposition 16 and ’J is the implicit function whichwe obtain in the application of Proposition 11 to the index set J . We consider a genericelement all(x) of the main diagonal of J9(x). On the other hand, in the process ofdeDnition of 9, we Dx the step corresponding to the set I = {l; l + 1; : : : ; n} and themap �I . Then

�I (x) = 2l(xl)el + 2l+1(xl+1)el+1 + · · ·+ 2n(xn)en +∑

J⊂{l;:::; n}J={i0 ;:::; ik};|J |¿2

’J (x)ei0

and it is easy to check that

J�I (x)el = all(x)el: (3.5)

From property (A3) of Proposition 11 we have

(h ◦ Z)(�I (x)) = (l ◦ Z)($LI (x)) for all x∈U:


Let yk be such that �I is di:erentiable in yk . On the upper equality we take the lowerDini derivative in the direction of el and, via Lemma 14, it results

d−h(Z(�I (yk));ZJ�I (yk)el) = d−(l ◦ Z)($LI (yk); el):

Then by (3.5)

d−h(Z(�I (yk)); all(yk)zl) = d−(l ◦ Z)($LI (yk); el):

Similar to the proof of Proposition 16 and using chain rules for Clarke’s derivatives,we conclude that all(yk)¿ 0 for k su<ciently large and that

limk→∞

all(yk)¿ 0

when this limit exists.(iii) From (i) and (ii) we deduce that, when limk→∞ J9(yk) exists, it is an upper

triangular matrix with positive numbers on the main diagonal. As @9(0) is deDned by

@9(0) = co

lim

yk→0yk �∈79

J9(yk)

with 79 the set where 9 fails to be di:erentiable, then if A∈ @9(0) , A is regular. ByClarke’s inverse function theorem, we conclude that 9 is a homeomorphism betweentwo neighbourhoods of 0 with 9−1 Lipschitzian around 0.

Remark 20. If 9 veriDes the properties of Remark 18 then it veriDes Proposition 19with J9(x) a lower triangular matrix.

Proposition 21. Under the hypothesis of Proposition 17, there exists a Lipschitzianhomeomorphism 9 at 0 associated to h and de7ned between two neighbourhoods U ; Vof 0∈Rn such that

(i) 9(0) = 0.(ii) (h ◦ 9)(x) = h(0) + l(x) for all x∈ U .(iii) 9(U ∩ C) ⊂ V ∩ C.

Proof. We suppose without loss of generality that h(0) = 0. Let 9 be the homeomor-phism of Proposition 19. From Z(U ) to Rn we deDne the function 9 = Z ◦ 9 ◦ Z−1,then 9 is a Lipschitzian homeomorphism at 0 with 9(0) = 0. We will assume thatU = Z(U ).

(ii) By Proposition 17(ii) we have

(h ◦ 9)(x) = h(0) + (l ◦ Z)(Z−1(x)) = h(0) + l(x) for all x∈ U :

(iii) We note that C={y∈Rn |y=∑nj=1 /jzj; /j¿ 0} and 9(x)=(Z ◦9◦Z−1)(x)=

(Z ◦ 9)(Z−1(x)).If x∈ C, then Z−1(x) has nonnegative coordinates. We will check that for x =

(x1; : : : ; xn) su<ciently small with xi¿ 0 for all i = 1; : : : ; n, 9(x) has nonnegativecoordinates. Let us suppose that it is false. So, let 91(x) be the Drst coordinate of 9(x)


such that 91(x)¡ 0 (other cases are similar). We consider the function F : [0; 1] → RdeDned by

F(t) = 91(x1; (1− t)x2; : : : ; (1− t)xn)):

F is a continuous function such that F(0)¡ 0. On the other hand, by deDnition of 9and Remark 13, F(1) = 91(x1; 0; : : : ; 0) = 21(x1). But x1 ¿ 0, because 91(0; x2; : : : ; xn)= 21(0) = 0 in contradiction with 91(x)¡ 0. Hence by Proposition 16(iv) we obtain21(x1)¿ 0 and therefore F(1)¿ 0. Consequently, there exists t0 ∈ (0; 1) such that

F(t0) = 91(x1; (1− t0)x2; : : : ; (1− t0)xn)) = 0: (3.6)

Furthermore, by deDnition of 9,

91(0; (1− t0)x2; : : : ; (1− t0)xn) = 0: (3.7)

From (3.6), (3.7) and Proposition 17(iii), we deduce that

9(x1; (1− t0)x2; : : : ; (1− t0)xn)) = 9(0; (1− t0)x2; : : : ; (1− t0)xn):

Then since 9 is a homeomorphism we deduce that x1=0, in contradiction with x1 ¿ 0.Therefore if x∈ C, (9 ◦ Z−1)(x) has nonnegative coordinates and 9(x)∈ C.

In order to deDne an homeomorphism which provides the topological equivalencebetween the function f and its local approximations, we will use the partition of U andwe will obtain an homeomorphism 9i associated with each Mi. Under these conditions,the next proposition shows how to deDne an homeomorphism associated to the wholeset U . This construction is based on the concept of continuous selection which we canDnd for example in [7].

De�nition 22. A function f :U → Rn is called a continuous selection of the functionsf1; : : : ; fp, if f and f1; : : : ; fp are continuous and the active index set I(x)={i |f(x)=fi(x)} is nonempty for every x∈U .

Proposition 23. Let M be a Cr-subdivision of U at x0 = 0. We assume that M ispointed and has the triangular property, with Mi = Ci for all i = 1; : : : ; k. Let ussuppose that for each i there exists a Lipschitzian homeomorphism at 0, 9i, suchthat, for all sequence {xk} → 0 for which there exists limk→∞ J9i(xk), this matrixis triangular with positive numbers on the main diagonal. If there is a continuousfunction X such that

X(x) = (Zi9iZ−1i )(x) if x∈Ci;

then X is a Lipschitzian homeomorphism at 0.

Proof. We suppose that M has the upper triangular property and limk→∞ J9i(xk)is an upper triangular matrix. (The other case is similar.) Let 9i = Zi9iZ−1

i . ThenX is a continuous selection of the functions {9i}i=1; :::; k . Since continuous selectionsof a Dnite number of locally Lipschitzian functions are locally Lipschitzian (see


[5, Theorem 2.2]), we deduce that X is locally Lipschitzian around 0. To prove thatX is a homeomorphism, we consider the generalized Jacobian deDned by

@X(0) = co

lim

xj→0xj �∈7X

JX(xj)

with 7X the set where X fails to be di:erentiable. We assume that the sequence{xj} ⊂ Ci for some cone Ci and hence

@X(0) = co

Zi lim

xj→0xj �∈7X

J9i(Z−1i (xj))Z−1

i

;

where Z−1i (xj) → 0. By hypothesis, lim xj→0

xj �∈7X

J9i(Z−1i (xj)) is an upper triangular matrix

with positive numbers on the main diagonal and as Zi y Z−1i are upper triangular,

then the matrixes of @X(0) are upper triangular with positive numbers on the maindiagonal. So @X(0) is of maximal rank and by Clarke’s inverse function theorem, Xis a Lipschitzian homeomorphism at 0.

4. The PL functions

From now on M= {Mi}i=1; :::; k will be a Cr-subdivision of U at x0 and L its linearspace. To deDne a new class of functions we will use the concept of continuousselection (DeDnition 22).

De�nition 24. Let f :U → R be a continuous selection of Lipschitzian around x0functions f1; : : : ; fp. f is called a PL function associated to M if it veriDes

(i) |I(x)|= 1 for all x∈⋃ki=1 int(Mi).

(ii) f(x) = f(x0) for all x∈ (x0 + L) ∩ U .(iii) 0 �∈ @fi(x0) for all i = 1; : : : ; p.

Remark 25. (a) Note that as we have mentioned before, a continuous selection of lo-cally Lipschitzian functions is in turn a locally Lipschitzian function [5, Theorem 2.2]so a PL function at x0 is Lipschitzian around x0.

(b) According to DeDnition 24(i) we can suppose that p= k and that f(x) =fi(x)for all x∈Mi.

Proposition 26. Let f be a PL function associated to M. Let # be an orthogonalmap, # :U → Rn. Then f ◦ #−1 is a PL function associated to #(M).

Proof. By Remark 10, #(M) is a Cr subdivision of #(U ) at #(x0). Since f ◦ #−1

is continuous and # is an isomorphism, it is easy to see that f ◦#−1 is a continuousselection of functions f1 ◦#−1; : : : ; fp ◦#−1 which are Lipschitzian around #(x0) andthat they satisfy conditions (i)–(iii) of DeDnition 24.


In the next theorem we consider a local approximation H of the function f at x0,which is a PL function. Assume that H is a continuous selection of the functionsH1; : : : ; Hp.

Theorem 27. Let M be a Cr-triangular subdivision of U at x0. Let f and H be PLfunctions associated to M such that H is a local approximation of f at x0 and forall i = 1; : : : ; p, Hi is a local approximation of fi at x0. If for each i = 1; : : : ; p andfor each j = 1; : : : ; n conditions (A)–(D) below holds:

(A) H (x) = Hi(x) for all x∈Ci,(B) Hi(x − x0) = 0 for all x∈ (x0 + L) ∩ U ,(C) (Hi)0(0; zij)6f0

i (x0; zij),(D) f0

i (x0; zij)f0i (x0;−zij)¡ 0,

then there exists a Lipschitzian homeomorphism at x0 de7ned from a neighbourhoodV ⊂ U to another neighbourhood W of x0 such that:

(1) X(x0) = x0.(2) f(X(x)) = f(x0) + H (x − x0) for all x∈V .

Proof. We assume that M is an upper triangular subdivision. (The lower triangularcase is similar.) We suppose without loss of generality that x0 = f(x0) = 0 and weconsider two cases: (I) M with the upper triangular property and (II) there exists anorthogonal map in the sense of Remark 10.(I) Let M be with the upper triangular property. The proof is carried out in three

steps. In the Drst step we prove the theorem under the assumptions that L = {0} andMi =Ci ∩U . In the second one we suppose that Mi �= Ci ∩U . Step 3 yields the resultfor L �= {0}.Step 1: We suppose that L = {0} and that Mi = Ci ∩ U for all i = 1; : : : ; p. As we

noted in Remark 25(b), we can assume that f coincides with fi on the set Mi. Onthe other hand, since L= {0}, the cones Ci are equal to Ci and we have:

Ci =

y∈Rn |y =

n∑j=1

/jzij; /j¿ 0

:

Furthermore, for each i = 1; : : : ; p by hypotheses (C) and (D), fi and Hi satisfy theconditions of Proposition 16 and consequently of Proposition 21. Then for each i =1; : : : ; p, there exists a Lipschitzian homeomorphism at 0, 9i : Ui → Vi.On the set V =

⋂pi=1 U i we deDne the function X by

X(x) = 9i(x) if x∈Ci

X is well deDned because the construction of 9i ensures that its values on somelower-dimensional face of Ci depends only on the parametrization of the extremal raysof this face by means of the maps 2ij (Proposition 16) as well as the values of f onthis face. By condition (4) of DeDnition 6, it follows that for all x su<ciently small,


x∈Ci ∩ Cj, then 9i(x) = 9j(x). Let us prove that (1) and (2) hold:

(1) X(0) = 0 because 9i(0) = 0 for all i = 1; : : : ; p.(2) If x∈V , then x∈Ci ∩ V for some i and hence f(X(x)) = f(9i(x)). By Proposi-

tion 21(iii), 9i(x)∈Ci and, since f coincides with fi on Ci, we have f(9i(x))=fi(9i(x)). We apply Proposition 21(ii) to obtain: fi(9i(x)) = Hi(x) and hypoth-esis (A) to get, Hi(x) = H (x) for all x∈Ci. Therefore f(X(x)) = H (x) for allx∈V .

Let us show that X is a Lipschitzian homeomorphism at 0. For each i=1; : : : ; p, byProposition 21, we have 9i=Zi◦9i◦Z−1

i , where 9i is a Lipschitzian homeomorphism at0. Furthermore, for all sequence {xk} → 0 for which there exists limk→∞ J9i(xk), it isan upper triangular matrix with positive elements on the main diagonal (Proposition 19).By Proposition 23 we obtain that X is a Lipschitzian homeomorphism at 0.Step 2: If Mi �= Ci ∩ U , we apply a similar method as in the Proposition 11

and a theorem due to Kuntz and Scholtes [8, Theorem 3.1] to obtain a Lipschitzianhomeomorphism at 0, < such that:

(b1) <(0) = 0 and <(Ci) =Mi for all i = 1; : : : ; p.(b2) < is strictly di:erentiable at 0 in the Bourbaki sense (see [4]) with strict derivative

DS<(0)x = x for all x∈Rn.

By (b2) and the chain rule for generalized directional derivatives [4, Theorem2.3.10], we have that (fi ◦ <)0(0; x) = f0

i (0;DS<(0)(x)) = f0i (0; x). Then f ◦ < is a

PL function associated to M, H is a local approximation of f at 0 and Hi is a localapproximation of fi ◦ < at 0, for all i = 1; : : : ; p. It is easy to see that f ◦ < satisDeshypothesis (C) and (D), so the homeomorphism X we set out to construct is locallyaround the origin deDned by X = < ◦X1 where X1 is the homeomorphism of step 1.Step 3: We suppose that L �= {0}. From Lemma 9 we obtain the pointed subdivision

M with the upper triangular property and deDned by the sets

Mi;" = {x∈U | hij(x)6 0; "(l)alx6 0; j = 1; : : : ; m; l= m+ 1; : : : ; n}whose tangent cones are given by

Ci;"(0) = {y∈Rn | ∇hij(0)y6 0; "(l)aly6 0}:In order to apply step 1 to M, we consider the extremal rays of Ci;". If zij is anextremal ray of Ci;" then it is an extremal ray of Ci or it is an element of L. In theDrst case, we construct the map 2ij in the same way as in Proposition 16. If zij ∈L, wedeDne 2ij(t) = t. Let us check that this 2ij satisDes the properties of Proposition 16. Itis obvious that 2ij is a Lipschitzian homeomorphism at 0 with 2ij(0)=0 and 2ij(t)¿ 0for all t ∈ (0; 0).(ii) For all t ∈ (−0; 0), since f coincides with fi on Ci and L ⊂ Ci we have

fi(2ij(t)zij) = fi(tzij) = f(tzij);

but if tzij ∈L, as f is a PL function, then f(tzij) = f(0) = 0 and by hypothesis (B)Hi(tzij) = 0, so we conclude that fi(2ij(t)zij) = Hi(tzij).


(iii) Is obvious since limk→∞ 2′ij(tk)=1. By applying steps 1 and 2 to the subdivisionM, we obtain the homeomorphism X.(II) If there exists an orthogonal map # such that #(M) has the upper triangular

property, we suppose without loss of generality that #(0)=0. We consider the functionf◦#−1. Since # is an isomorphism, the maps f◦#−1, H ◦#−1 and, for all i=1; : : : ; p,Hi◦#−1, fi◦#−1 satisfy hypothesis (A)–(D). Then we apply steps 1–3 to f◦#−1.

We note that, in particular, the function H can be some of Dini or Clarke derivativesof f. In the next corollary we denote by fx0 (x0; ·) any of the derivatives: d−f(x0; ·),d+f(x0; ·), f0(x0; ·), Mf(x0; ·).

Corollary 28. Let M be a Cr-triangular subdivision of U at x0. Let f be a PLfunction associated to M such that for all i = 1; : : : ; p:

(a) fx0 (x0; x) = (fi)x0 (x0; x) for every x∈Ci,(b) (fi)x0 (x0; x − x0) = 0 for every x∈ (x0 + L) ∩ U ,(c) f0

i (x0; zij)f0i (x0;−zij)¡ 0 for every j = 1; : : : ; n,

then there exists a Lipschitzian homeomorphism at x0X :V → W de7ned on a neigh-bourhood V ⊂ U of x0, such that:

(1) X(x0) = x0,(2) f(X(x)) = f(x0) + fx0 (x0; x − x0) for every x∈V .

Proof. It only remains to verify hypothesis (C) of Theorem 27. For Clarke derivativesit is obvious that (f0

i (x0; ·))0(0; x)=f0i (x0; x) and analogously ( Mf)(x0; ·)(0; x)= Mf(x0; x)

for all x∈Rn. Furthermore, Dini derivatives hold (see [1, Proposition 3.3]):

Mfi(x0; x)6d−fi(x0; ·)(0; x)6 (d−fi(x0; ·))0(0; x)6 (fi)0(x0; x)

and analogously for d+fi(x0; ·).

Remark 29. If fx0 (x0; ·)=d−f(x0; ·) or fx0 (x0; ·)=d+f(x0; ·), hypothesis (b) is redun-dant because as f∈PL, then f(x)=f(x0) for all x∈ (x0+L)∩U and Dini derivativessatisfy (b).

Next we show that, in the case of a critical point, the last theorem provides su<cientconditions for the existence of a local extremum of f.

Theorem 30. Let f and H be functions satisfying the hypothesis of Theorem 27.Then

(i) if H (x)¿ 0 for all x in a neighbourhood of 0, f has a local minimum at x0,(ii) if H (x)6 0 for all x in a neighbourhood of 0, f has a local maximum at x0,(iii) if H (x) changes the sign in any neighbourhood of 0, f has no local extremum

at x0.


Proof. From Theorem 27 we obtain (f ◦X)(x)=f(x0)+H (x−x0) and, since X is anhomeomorphism on a neighbourhood of x0, it holds: f(x) = f(x0) +H (X−1(x)− x0),then the statements are obvious.

Next, we present an example of a function which in particular veriDes hypothesis ofCorollary 28.

Example 31. Let f : [− 1; 1]× R→ R be deDned by:

f(x; y) =

2=3x + 1=(3 · 23n) + xy if 1=22+3n6 x6 1=23n;

3x − 1=22+3n + xy if 1=23+3n6 x6 1=22+3n;

−2=3x + 1=(3 · 23n) + xy if − 1=23n6 x6− 1=22+3n;

−3x − 1=22+3n + xy if − 1=22+3n6 x6− 1=23+3n;

0 if x = 0;

n= 0; 1; 2; : : :

f is a continuous selection of the functions

f1(x; y) =

{f(x; y) if 06 x6 1

−f(x;−y) if − 16 x¡ 0and f2(x; y) = f1(−x;−y):

Both functions are Lipschitzian around (0; 0) and satisfy (0; 0) �∈ @f1(0; 0), and (0; 0) �∈@f2(0; 0). We consider the subdivision of U = (−1; 1)× R deDned by the sets M1 =[0; 1)×R and M2=(−1; 0]×R, whose tangent cones are, respectively, C1=[0;∞)×R,C2=(−∞; 0]×R. The collection M={M1; M2} is a C∞-subdivision of U at (0; 0) withthe lower triangular property and linear subspace L= {(x; y)∈R2 | x=0}. Then f is aPL function associated to M. It is easy to verify the remaining hypothesis of Theorem27 for f and H =f0((0; 0); ·), therefore f and f0((0; 0); ·) are Lipschitzian-equivalentat (0; 0). Furthermore, since f0((0; 0); v)¿ 0 for all v∈R2, f has a local minimumat (0; 0).

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on the qualitative approximation of lipschitzian functions

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