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Error Estimates and Lipschitz Constants for BestApproximation in Continuous Function SpacesM. Bartelt

W. LiOld Dominion University

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Repository CitationBartelt, M. and Li, W., "Error Estimates and Lipschitz Constants for Best Approximation in Continuous Function Spaces" (1995).Mathematics & Statistics Faculty Publications. 138.https://digitalcommons.odu.edu/mathstat_fac_pubs/138

Original Publication CitationBartelt, M., & Li, W. (1995). Error-estimates and Lipschitz-constants for best approximation in continuous function spaces. Computers& Mathematics with Applications, 30(3-6), 255-268. doi:10.1016/0898-1221(95)00104-2

Pergamon Computers Math. Applic. Vol. 30, No. 3-6, pp. 255-268, 1995

Copyright~)1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0898-1221(95)00104-2 0898-1221/95 $9.50 + 0.00

Error Estimates and Lipschitz Constants for Best Approximation

in Continuous Function Spaces

M. BARTELT Department of Mathematics, Christopher Newport University

Newport News, VA 23606, U.S.A. mbaxtelt@pcs, cnu. edu

W. LI Department of Mathematics and Statistics, Old Dominion University

Norfolk, VA 23529, U.S.A. wuli~math, odu. edu

A b s t r a c t - - W e use a structural characterization of the metric projection PG(f), from the con- tinuous function space to its one-dimensional subspace G, to derive a lower bound of the Hausdorff strong unicity constant (or weak sharp minimum constant) for PG and then show this lower bound can be attained. Then the exact value of Lipschitz constant for PG is computed. The process is a quantitative analysis based on the G~teaux derivative of PG, a representation of local Lipschitz constants, the equivalence of local and global Lipschitz constants for lower semicontinuous mappings, and construction of functions.

g e y w o r d s - - E r r o r bounds, Lipschitz constants, G£teaux derivatives, Metric projections, Strong uniqueness.

1. I N T R O D U C T I O N

Consider the following minimizat ion problem:

inf ~(g) , (1) gEG

where G, called the feasible set, is a subset of a normed linear space Y with norm [[. [[, and ~ , called

the objective function, is a real-valued function defined on Y. Assume tha t ~min := infgec O(g)

is finite and the opt imal solution set S :-- {g E G : ~(g) = ( I ) m i n } is not empty. Then there are two fundamenta l problems associated with (1)---error est imates and stabili ty analysis [1-3].

Error es t imates refer to est imates of the distance from an approximate solution to the opti-

mal solution set. Error est imates are extremely impor tan t in convergence analysis of iterative

a lgori thms for finding an opt imal solution of (1), as shown in recent l i terature [4-21]. Another impor tan t application of error est imates is to provide a priori information on how far an ap-

proximate solution is from the opt imal solution set [20,22-46]. Such a priori information can be

used as a reliable te rminat ion criterion of an iterative method for solving (1). Stabil i ty analysis

(or sensitivity analysis) refers to the s tudy of the behavior of the opt imal solution set under

pe r tu rba t ion of parameters (or data) involved in the definition of • a n d / o r G [1-3]. Here we are interested in the following type of error estimates:

dist(g, S) < ~/((I)(g) - ( ~ ) m i n ) , for g e G, (2)

Typeset by ~4A/~-TEX

255

256 M. BARTELT AND W. LI

where 7 is some positive number and dist(g, S) is the distance from g to the optimal solution set S defined as

dist(g, S) := ~ f ][g - sll.

If (2) holds, then one can say that (1) has a weak sharp minimum (cf. [5,47-50]). See [4,5,9,21,50] for applications of the weak sharp minimum property in convergence analysis of iterative methods for solving (1). The existence of 7 is sufficient for qualitative applications of weak sharp minimum properties, such as in the convergence analysis of algorithms. However, in order to obtain a priori error estimates, one must also have a quantitative analysis of 7. For this purpose, it is important to derive an explicit expression for the smallest -y which satisfies (2):

~min : = inf ~(g)--(I)min gea\s dist(g, S) (3)

In this paper, we give a quantitative analysis of ~'rnin for a special optimization problem--the best approximation problem in continuous function spaces. For this special problem, ")'min is closely related to the Lipschitz constant of S with respect to perturbations of the data function involved. Therefore, we also give a quantitative analysis of the related Lipschitz constant.

Let G be a finite-dimensional subspace of the Banach space Co(T) of all real-valued contin- uous functions on a locally compact Hausdorff space T which vanish at infinity (i.e., {x c T : If(x)[ _> e} is compact for f 6 Co(T) and e > 0). The supremum norm of Co(T) is defined as Hf[[ :-- supxeT [f(x)[ for f 6 Co(T) and the objective function for the best approximation problem is (I)(g) := I[f - g[I which depends on a (data) function f in Co(T). In this setting, the optimal solution set is actually a set-valued mapping PG(') from Co(T) to subsets of G, called the range of the metric projection and defined as

PG(f) := {g ~ a : I l f -- gll = d i s t ( f , G ) } .

See [51] for set-valued analysis. Hausdorff strong uniqueness [52], ness property of Haar subspaces Hausdorff strong unicity constant

Note that weak sharp minimum in this case was also called because it is a set-valued version of the classical strong unique- [53-55]. Here we want to find the exact values of the uniform F of Pc and the Lipschitz constant A of PG, respectively, where

F := inf { [[f -g[[ -d i s t ( f 'G) } dist (g, Pv(f)) : f C Co(T), g C G with g q[ Pc(f) , (4)

A := sup { H (PG(f)'PG(h)) } [ [ f_h [ I : f, h 6 Co(T) with f # h , (5)

where H(A, B) is the Hausdorff distance between two sets A and B defined as

H(A, B ) : = max I.aeA~SUp dist(a, B), besSUp dist(b, A)} .

A special case of the best approximation problem in Co(T) is data regression in ]~n with the supremum norm [56,57]. Note that Co(T) - (R n, I1" ]]oo), the n-dimensional vector space with the norm ]lYlloc := maxl<i<n lYd, if T consists of n isolated points. It is well known that the best approximation problem in (N n, ]]. IIoo) can be reformulated as a linear programming problem (cf. [56,58]). In [29,31], sharp Lipschitz constants for (basic) optimal solutions and (basic) feasible solutions of a linear program with right-hand side perturbations are given in terms of seminorms of pseudoinverses of certain submatrices. However, we do not know whether the analysis given in [29,31] can be modified to find the exact values of F and A if G is a closed polyhedral subset of (N n, II • [Ioo). By using Hoffman's error estimate, Li proved that F > 0 and A < c~ for any closed convex polyhedral subset G of (R n, II" Iloo) [49]. However, for a finite-dimensional subspace

Error Estimates 257

of Co(T), it is not necessary that F > 0 or A < oo. For any finite-dimensional subspace G of Co(T), Li proved that the following statements are equivalent [59]:

(a) r > 0 . (b) A < ~ . (c) supp(g) := {x : g(x) # 0} is compact for any g E G.

Therefore, we should only consider a finite-dimensional subspace G whose elements have compact supports. Due to difficulty of the problem, we will only treat the one-dimensional case in the present paper. Therefore, we make the following assumption throughout this paper, unless stated otherwise.

ASSUMPTION 1. Let G := span{gl} be a one-dimensional subspace of Co(T) such that {x : gl(x) -# O} is compact.

The paper is organized as follows. In Section 2, we use a structural characterization of P v ( f ) to derive a lower bound of F and then show this lower bound can be attained by constructing a function. Section 3 is devoted to finding the exact value of A. The process is a quantitative analysis based on the Ggteaux derivative of Pc, a representation of local Lipschitz constants, the equivalence of local and global Lipschitz constants for lower semicontinuous mappings, and con- struction of functions. In order to give a clean presentation, we put the complicated construction of functions with certain desirable properties in Section 4.

2. H A U S D O R F F S T R O N G U N I C I T Y

In this section, we first give a structural characterization of Pa( f ) . Using this characterization, we can derive a lower bound for F. Then, by constructing a function, we show that this lower bound can be attained. Thus, we obtain the exact value of F.

First, we establish a structural characterization of Pc.

LEMMA 2. Let 1 << u. Then PG(f) = {C91 : l < C < u } if and only i f there exist two points xz and xu such that 91(xl) # 0, 91(xu) # O, and

dist(f , G) = D[f - lg lN = sgn (g l (x l ) ) ( f (xt) - l g l (zt)), (6)

gist(f , G) = ]If - u9, [1 = - s g n (gl (Xu)) ( f (xu) - u91 (Xu)), (7)

where sgn(a) denote the sign of a number a.

PROOF. First assume Pa( f ) = {cgl : 1 < c < u}. Since supp(gl) is compact, sgn(gl(x)) is a continuous function on supp(gl). Therefore, there exists xt 6 supp(gl) such that

sgn (91 (xl)) ( f (xt) - Igl (xl)) = xEsumppaXgi)sgn (91(x)) ( f (x) - - lgl(x)).

We claim that

dist(f , G) = sgn (91 (xl) ) ( f (xl) - lgl (xl) ) . (8)

If (8) does not hold, then

6 := dis t ( f , G) - sgn (gl (Xl)) ( f (Xl) -- lgl (Xl)) > O.

Let 0 < l - 16 < 611gill. Then

sgn (gl(x) ) ( f (x) -16gl(x)) <_ sgn (g l (x ) ) ( f (x ) - l g l ( x ) ) + (1 -16)I9(z)l

_< sgn (91 (xl)) ( f (xt) - 191 (zl)) + 6 <_ dist(f , G)

and

- s g n (gx(x) ) i f (x) -- 16gl(x)) <_ - sgn (g l (x ) ) ( f (x ) - Igx(x)) -4- (l - 16)Ig(x)l

_< sgn (gl (x) ) ( f (x) - I g l ( x ) ) < IIf - lg l (x )H = dist(f , G). 30:M6-R

258 M. BARTELT AND W. LI

As a consequence, [If-l~gl(x)[[ _< dist(f, G) and l~gl E Pc( f ) , a contradiction to the assumption that Pa( f ) = {cgl : l < c < u}. Therefore, (8) holds.

Similarly, we can prove that there exists a point xu such that gl(x,,) ~ 0 and

dist(f, G) = -sgn (gl (zu) ) ( f (xu) - ugl (x,,) ) .

Obviously, we also have

IIf - zglll = IIf - uglll = d i s t ( f , G).

On the other hand, if (6) and (7) hold, then, by convexity of Pc( f ) , we get

PG(f) D {cgl : I < c < u}.

Since gl(xl) ~ 0 and gl(xu) ¢ 0, the two equations (6) and (7) imply that cgl • PG(f) i f c < l or c > u. Thus,

PG(f) = {cgl: l < c < u}. II

Now we can derive the exact value of F.

THEOREM 3.

F = i n f { 'g~(x)' } Ilgll-----~ : z E T with gl(x) ~ 0 . (9)

PROOF. First we show that, if gl(xt) ~ 0, then

F < Igl (Xl)l (10) - I l g l l l

In fact, if Igl(Xl)l = Ilglll, then (10) holds, since F always satisfies F _< 1 (cf. [53, page 83]). Otherwise, by Proposition (13), there exists a function f ( x ) in Co(T) such that P c ( f ) = {0} and

Ill - g l II -< dist(f, G) + Igl (Xl)l • (11)

Since IIf -g l [ ] > dist(f, G) + F . dist (gl, PG(f)) = dist(f, G) + F . Ilglll, (12)

inequality (10) follows from (11) and (12). Now let P c ( f ) = {cgl : 1 < c < u}. By Lemma (2), there exist two points xt and x~ such that

gl(xt) ¢ 0, gl(xu) ~ 0, and equations (6) and (7) hold. Let g = agl ¢ Pa( f ) . Assume a < l. Then

II f - gl l -> I f ( x t ) - g ( x l ) l

> sgn (gl (xl)) ( f (xt) - g (xl))

= sgn (gl (xl)) ( ( f (xt) - lgl (xt)) 4- (l - ~)gl (xt)) (13)

= d i s t ( f , G) + (1 - a ) I g l (x~)t

= d i s t ( f , G) + Iga (xt)._.____~l dist (g, Pc(f)) Itglll

where the second equality follows from (7). Similarly, when a > u, we can prove that

]If - gll ~ dist(f, G) q- - - ]gl (Xu)] dist (g, Pa(f)). (14) IIg~ll

Error Estimates 259

It follows from (13) and (14) that

r - > i n f { / g l ( z ) ' } ]Ig~N : x • T with gx(x) ¢ 0 . (15)

It is easy to see that (9) follows from (15) and (10) and the proof is complete. 1

If G = span{g1} is a Haar space and T is a compact Hausdorff space, then supp(gl) = T is compact. Therefore, the following result is a special case of Theorem 3.

COROLLARY 4. Let T be a compact Hausdorff space and G = span{g1} be a one-dimensional Haar space in C(T). Then

F = i n f { [gl(x)[ } Ilgll] : z • T .

In particular, r = 1 when G = span{l}.

The result in Theorem 3 holds for a line segment in Co(T) with a similar proof.

COROLLARY 5. Let G = {agl : A < a < B} be a line segment in Co(T) with {x : gl(x) ¢ 0} compact. Then

F = i n f { 'g'(x)[ } Ilglll : x • T w i t h gl(x) 7 ~ 0 .

PROOF. Let P c ( f ) = {agl : l <_ c~ <_ u}. Then it follows as in Lemma (2) that if A < l then (6) holds, and if u < B then (7) holds. Now F is invariant under translation, i.e., if Gfl = {agl : A - fl < c~ < B - fl}, then Fa = rao. Thus, we may assume that A < 0 < B so that 0 • G. Then the conclusion of Proposition (13) holds. Now the proof of Theorem 3 holds, where, to verify (13) and (14), it is only required that we consider g = agl f~ Pa( f ) when A <_ c~ < l and when u < a _< fl so that Lemma (2) in this case can be applied.

3. L I P S C H I T Z C O N S T A N T S

It is well known that the uniform Hausdorff strong unicity constant F provides an upper bound 2/F for the uniform Lipschitz constant A. That is,

2 A < ~. (16)

The above inequality was first established by Cheney [53] for a Haar space G and then extended by Park [60] to general cases. However, it was not clear whether the estimate (16) was sharp or not. Our first main result in this section is to show that the equality holds in (16) if G is not a Haar space (i.e., supp(gl) ~ T). In this case, we prove A = 2/F by constructing two functions f , h in Co(T) such that

2 H ( P c ( f ) , P c ( h ) ) >_ ~ Ill - hH > O.

However, if G is a Haar space and T is not a singleton, then we always have A < 1/F and it is not easy to find the exact value of A. Fortunately, the G£teaux derivative formula of Kolushov [61] provides some information on the exact value of A. The existence of the G£teaux derivative of Pc for a Haar space G was first discovered by Kro5 [62]. Later, Kolushov derived a formula for the G£teaux derivative of PG [61]:

lim P c ( f + re) - P c ( f ) = p(f , ¢), (17) t-.0+ t

where p(f , ¢) is the unique solution of the following minimax problem:

min max (¢(x) - g(x)) , sgn(f(x) - PG(f) (x) ) , (18) gEG xEE(f-Pc(f))

260 M. BARTZLT AND W . LX

where E ( f - Pc(f)) := {x • T : [(f - PG(f))(x)l = Ilf - PG(:)II}. Note that, by (17),

liP(f, ¢)1_______~ _ lim I lPc(f + t¢) - Pc(f)l[ < A. II¢II t-~o+ tll¢ll

Therefore, s u p { ][p(f' ¢)[[ } II¢I-------F : f ' ¢ • Co(T) (19)

provides a seemly tight lower bound for A. It turns out that the expression (19) is the so-called uniform local Lipschitz constant of P c [63]:

A t = sup { [[P(f'¢)[[[[¢[[ : f , ¢ • Co(T) with ¢ ~ 0} , (20)

where

:= sup inf sup ~ H(Pc(f)-2'PG(h)) } A t fECo(T) 5>0 [ [If -- hi[ : h • Co(T) with 0 < [If - hi[ _< 5 .

Even though for a specific function the local Lipschitz constant need not equal the (global) Lip- schitz constant, it is known that the uniform local Lipschitz constant of any Lipschitz continuous mapping is the same as the uniform Lipschitz constant of the mapping (cf. [31, Theorem 2.1] or Lemma (7). As a consequence, A t = A. Therefore, in order to get the exact value of A, we only need to compute the norm of p(f, ¢) and to do this, we will use the following explicit representation of p(f, ¢):

sgn (gl (Xl)) • ~)(Xl) q- sgn (gl (x2)) " ~b (X2)gl(x) ' p(f, ¢)(x) = [gx(xl)[ + [gl (x2)l

(21)

where xl and x2 are two distinct points in E ( f - Pa(f)). In short, when G is a one-dimensional Haar space, by using (20), (21), and A l = A, we are able

to prove that 2 [Iglll (22) A =

inf{Igl (Xl)] + Igl(x2)l: Xl,X2 • T with zl ~ z2}"

The first main result of this section shows that A -- 2/F if G is not a Haar space.

THEOREM 6. Suppose that G = span{g1}, supp(gl) is compact, and Z(gl) := {x : gl(x) = 0} is not empty. Then

2 Ilgl II 2 Ilglll A _ _ _ _ - -

r inf {[gl(x)[ : gl(x) ~ 0}"

PROOF. By inequality (16), we have 2

A < - . - F

Thus, by Theorem 3, it only remains to show that there exist f , h • Co(T) such that

H(Po(f) ,Po(h)) > rllY - hll > 0.

By the assumption, there exists a point xo such that gl(xo) = 0. Let xl • T such that

r = Igl (x~)l Ilgxll

Then, by Proposition (14), there exist f, h • Co(T) such that

H(Pc(f ) ,Pc(h)) > 21I f - hi[ > 0. | t

Error Estimates 261

From now on, we proceed to establish the identity (22). First we show that A l = A. For a finite-dimensional subspace G of Co(T), we say that P c is locally upper Lipschitz continuous with modulo A, denoted by PC E UL(A) (cf. [64]), if, for any f E Co(T), there exists a positive constant 6 > 0 such that

dist (Pc(h), Pc(f)) ~ Aiif - hi[, for h E Co(T) with [If - hi[ -< 6,

where dist( . , -) is defined as

dist (Pc(h) ,Pc(f)) := sup inf l iP - g[[. pEPG(h) gePc(f)

Note that , if P c is Lipschitz continuous, then P c E UL(A). However, the converse is also true if P c is also Hausdorff lower semicontinuous, i.e.,

lira dist (Pa(f), Pa(h)) : 0, for every f • Co(T). h---.f

Note [52] that P c is Hausdorff lower semicontinuous if and only if, for any nonzero function g E G,

card(bdZ(g)) < dim{p • G : intZ(g) c Z(p)} - 1, (23)

where Z(g) := {x • T : g(x) = 0}, bdZ(g) and intZ(g) are the boundary and the interior of Z(g), respectively, card(K) denotes the number of points in a set K. Therefore, the following result is a consequence of [31, Theorem 2.1].

LEMMA 7. Suppose that G is a finite-dimensional subspace of Co(T) such that (23) holds for every nonzero function g in G. Then PG E UL(A)/if and only if A <_ A.

Using Lemma 7, we can easily show that A = A l. In fact, we can prove the following more general result.

LEMMA 8. Suppose that G is a finite-dimensional subspace of Co(T) such that (23) holds for every nonzero function g in G. Then

A ~ = A z = A,

where A ~ := sup !n fsup{ dist(Pa(h) 'Pc(f)) }

SeCo(T) 0>0 h : 0 < IIf - hll < 6 (24)

PROOF. It is easy to see that A u < A t _< A. On the other hand, let e > 0. Then, by the definition of A u, Pc E UL(A ~ + e). By Lemma 7, A <_ A u + e. Since e > 0 is arbitrary, we have A _< A u. I

Next we give a representation of A by using the norms of the G£teaux derivatives of Pc-

THEOREM 9. Let T be a compact Hausdorff space and G -- span{g1} a one-dimensional Haar subspace of C(T). Then

A = sup{llp(f,¢)ll : f, cb • C(T) with I1¢11 ~ 1}. (25)

PROOF. proved that , for any f E C(T),

inf sup (~ H (PG(h), PG(f)) C(T) with h / l :he

= sup{liP(f, ¢)1[ : ~ E C(T) with II•l[ -< 1}.

Thus, equation (25) follows from (26) and Lemma 8.

In fact, for any finite-dimensional Haar subspace G of C(T), Bartelt and Swetits [63]

(26)

I

262 M. BARTELT AND W. LI

LEMMA 10. Suppose that T is a compact Hausdorff space and G = span{g1} is a one-dimensional Haax subspace of C(T). Let f in C(T) \ G and ¢ in C(T). Then there are two distinct points x l

and x2 in E ( f - Pa ( f ) ) such that

p(f, ¢)(x) = sgn (gl (Xl)) • • (Xl) ÷ sgn (gl (X2))" ¢ (X2) gl(X) Igl (xl)l + Im (x2)l

and ( f -- P G ( f ) ) (Xl) "gl (Xl)" ( f -- P G ( f ) ) (X2) "gl (X2) < 0.

PROOF. By Kolushov's representation of the Ghteaux derivative of PG (cf. (17) and (18) or [61]), one can easily verify that there exist two distinct points xl and x2 in E ( f - Pa ( f ) ) such that

- sgn (gl (Xl)) ((~ (Xl) - P ( f , ¢ ) (Xl)) ----- sgn (gl (x2)) (¢ (x2) - p(f , ¢) (x2)) (27)

and ( f - P c ( f ) ) (Xl) "g l (Xl)" ( f - P c ( f ) ) (x2) " gl (x2) < O.

Substituting p(f , ¢) := Agl into (27), we have a linear equation in A with the solution

A = sgn (gl (Xl)) " ¢ (Xl) ÷ sgn (gl (x2)) • ~b (x2) Igl (Xl)l ÷ Igl (x2)]

THEOREM 11. Suppose that T is a compact Hausdorff space and G = span{gl} is a one-

dimensional Haar subspace of C(T). I f G = C(T), then A = 1; otherwise,

2 Ilgl II 1 A -- inf {Igl (xl)[ + Igl (x2)l : x l , x2 E T with xl # x2} -< F"

(28)

PROOF. If G = C(T), then PG is the identity mapping and, obviously, A = 1. Otherwise, it

follows from Theorem 9, and Lemma 10 that

A = sup{llP(¢,f)ll : f , ¢ e Co(T) with I1¢11 < 1}

< sup sgn(gl (Xl)) . ¢(Xl) +sgn (g l (x2)) • ¢(x2) . Hgl[[

- jml<1,~1~ ]m (xl)l + ]m (x2)l { 2 Ilglll }

_< sup Igl (xl)l + ]gl (x2)l : x l , x2 e T, x 1 • x2 •

Now, let Xl, x2 E T with Xl # x2. By Proposition (15), there exist functions f, ¢ in C(T) such

that I1¢]1 = 1 and 2 IIglll (29)

lip(f, ¢)11 -> Im (x~)l + Im (x2)l"

By Theorem 9 and (29), we get

A>_sup Ig l (x l ) [+lg l (x2) l : x t , x 2 E T w i t h x l ¢ x 2 •

This completes the proof. |

COROLLARY 12. Suppose that T is a compact Hausdorff space with no isolated points and

G = span{g1 } is a one-dimensional Haar subspace of C(T). Then

1 Hmll r i n f { l g l ( z ) l : z e T}"

Error F~timates 263

4. CONSTRUCTION OF F U N C T I O N S

In this section, we construct several functions with certain desirable properties. Let

T1 : = { x • T : 9t (x) ¢ 0 } .

For convenience, we use the following notation:

0,

a, [fl(x)]ba : = b,

f l ( x ) ,

if x q~ T1,

if x • T1 and f l ( x ) < a,

if x • 7'1 and f l (x) > b,

i f x • T 1 a n d a < _ f l ( x ) < _ b

for scalars a,b with a < b and a function f l defined on T1. Note tha t [f l(x)] b is actually the t runca t ion of f l (X) on T1 by the lower bound a and the upper bound b which is na tura l ly extended to a funct ion on T with values 0 outside T1. If fx (x) is continuous on T1, then [fl (x)]b a is in Co (T) for any a < b, due to the fact tha t 7"1 is bo th open and compact .

PROPOSITION 13. For any xl E T with gl(Xl) ¢ 0, there exists a function f ( x ) in Co(T) such that Pc(. f ) = {0} and

Ill - roll < dist ( f , G) + Igx (xl)l • (30)

PROOF. If 91(X) ---- 0 for all x ~ Xl, then f = 0 is the required function; otherwise, let x2 • 7"1 \ {xl}. Wi thou t loss of generality, we may assume Ilgxll - 1 and g l (x l ) > 0.

Define a function ]1 on {Xl,X2} by ] l (Z l ) := - 1 and ]1(x2) := sgn(gx(x2)). Since Igl(x2)l <_ 1, it follows t ha t

] l ( X i ) - g l ( x i ) _< 1 + ]gl ( z l ) l , for i = 1,2.

By the Tie tze Extension Theorem, there exists a continuous extension ( f l - 9 1 ) of (]1 - 9 1 ) on TI such t ha t

i l l (X) -- gl(X)l ~ 1 -+-[gl (XI)[ , for x E T1. (31)

Let f ( x ) := [fa(x)]l_l. Then f ( x ) E Co(T) and [[fll <- 1. Since f l ( x l ) = ]1(xl ) = - 1 and

fl(X2) = ] l (x2 )=sgn(91(x2) ) , we have f ( x x ) = - 1 and f (x2)=sgn(gl (x2)) . Therefore, for a > 0,

I l l - ~glH ~ I ( f - ~ m ) (xl)[ = [1 + O~g I (Xl) [ > 1,

and, for a < 0,

[If - a911[ > I(f - c~91)(x2)l = [sgn (91 (x2)) - ag l (x2)l > 1.

As a consequence, Ilfll = 1 < IIf - aglll for a ~ 0 and PG( f ) = {0}. Now we claim tha t

I f (z ) - gx(x)l <_ 1 + Igl (Xl)], for x • T. (32)

In fact, (32) is trivially t rue if x • Ti. If x • Ti and Il l(x)] _< 1, then (32) follows from (31). For x • T1 with f l ( x ) > 1, by (31) and [91(x)[ <_ 1,

0 < 1 - 91(x) = f ( x ) - gl(x) <_ f l ( x ) - gl(x) < 1 + [gl(Xl)l •

For x • T1 with f l(X) < - 1 , by (31) and I91(x)l <_ 1,

- (1 + Igl ( X l ) [ ) <_ f l ( x ) - g l ( x ) <_ f ( x ) - g l ( x ) = - 1 - g l ( x ) <_ O.

Thus, (32) holds. The inequality (30) follows from (32) and dis t ( f , G) = 1. |

264 M. BARTELT AND W. LI

PROPOSITION 14. Let xo and x l be two distinct points in T such that gl ( xo ) = 0 and gl ( x l ) # O. Then there exist f , h E Co (T) such that

2 Ilglll H(PG(f),PG(h)) >_ Igl ( X l ) " " ' - ~ I l l - hll > o.

PROOF. Suppose first that gl(x) = 0 for x ~ 51. Let f l (x) be a nonzero function in C o ( T \ {Xl}) and define ( f l ( X ) , if x ~ Xl,

I(5) := / ~11~(5)1 , i f 5=51 .

Then it is easy to verify that P c ( f ) = {cgl(x) : 0 <_ c <_ 2f(xl)} . Let h(x) = 0. Then

Pc(h) = {0} and

2 [[glJ] Ilf - hll > 0. H(PG(f) ,PG(h)) = 2 If (xl)[ = 2 [If - hll = Igl (Zl)-----~

Now suppose that gl(x2) ¢ 0 for some x2 c T \ {x0,xl}. Without loss of generality, we may assume that Ilglll = 1 and gl(xl) > 0. Let ~ := [gl(xl)l and ~ := (1/7) + 1. Define

{ /3+1, x = x o ,

]l(X) = -,6', 5 = 51, (33)

/~. sgn (gl (52)), x = x2.

Then it is easy to verify that

fil (xi) gl ~xi) _< ~ + 1, for 0 < i < 2.

According to the Tietze Extension Theorem, there exists a continuous extension (f l - (gl/~))

o f ( ] 1 -- (if1/?'})) on 7"1 such that

(x) - g-~ (x) < 13 + 1, for x E T1. (34) f l

Let f ( x ) := [fl(Z)]Z_-~ 1. Then f e Co(T) and f ( z i ) = ]l(xi) for 0 < i < 2. Obviously, for any

scalar a, I l l - O~fflll --> [f (x0) - - agl (X0)[ : I]1 (X0) - - agl (50) : ~ + 1. (35)

We claim that

f ( x ) __gl 1 - -~ - (x ) ~ + 1 , f o r x 6 T 1 . (36)

In fact, (36) follows from (34) if -/3 ~ fl(X) ~ /~q- 1. For f l (x) > ~q-1, by (34) and Igl(x)l < 1, we have [ \

0 < ~ + 1 g1~5) _ f ( 5 ) - g1~5) <_ ] 1 ( 5 ) - gl~5---AJ < Z + 1.

If f l (x) < -/3, it follows from (34) and Igl(x)l < 1 that

--(f~ q- ]) _< f l (5) -- gl(X) ~ _ / 3 _ 91(5__.__~) ---- f (x ) -- gl(x-----!) ~ O.

Thus, (36) holds. Since f ( x ) : gl(x) = 0 for x ¢ T1, by (35) and (36), we get

f _ gl

Error Estimates 265

which, along with (35), implies

g~l E Pc( f ) . rl

This completes the analysis of f (x) . Next we construct a function h(x). Let 0 < e < 1/2 and h(x) := Il l(x) - ¢]~_-~1_-~. Since

(37)

h ( x ) = I l l ( x ) - ¢ 1 G ' - 7 = [ f , ( x ) l G x - ¢ = - ,

for all x E T1, we have

and

IlY - hll = ¢

h ( X l ) = f ( X l ) - - ¢ = - - f l - - ¢ .

Now we show that

Po(h) C {agl : a <_ l --~2¢ } .

By the definition of h(x), dist(h, G) < ]]hll < fl + 1 - ¢.

If a > (1 - 2¢)/~?, by (38), we get

(38)

(39)

(40)

h (Xl) - agl (Xl) = - - f l - - ¢ - - O~?~ < --fl - - ¢ - - (1 - 2¢) = - f l - 1 + ¢.

Thus, for a > (1 - 2¢)/~, I Ih - a g ~ l l > fl + 1 - ¢ and, by (40), agl • Pa(h). This proves (39). By (37) and (39), we see that

H (Pc( f ) ,Pc(h) ) >_ dist ( g-~,Po(h))

_>min{ ~ - a g l : a < l - 2 e } _

_ gl 1 - 2eg 1 2e

2 Ilalll = lal ( x l ) - - - - ~ IlY - hll > 0.

This completes the proof of Proposition 14. |

PROPOSITION 15. Let T be a compact Hausdorff space and G = span{g1} a one-dimensional Haar subspace of C(T). Then, for any two distinct points xl and x2 in T, there exist [unctions f , ¢ in C(T) such that I1¢11 = 1 and

lip(Y, ¢)11-> 2 IIg~ II

191 (xl)l + Ig~ (x~)l"

PROOF. Without loss of generality, we may assume that gl(xl) > 0. Let V1 and V2 be disjoint neighborhoods of xl and x2. By Urysohn's Lemma, there is a function hi such that hi(xi) = 1, O < h < l, and hi(x) = O for x ~ V~. Let

f i(x) := max {hi(x) - I g l (xi) - gl(x)l, 0}, for i = 1,2.

Then 0 < f i(x) <_ hi(x) _ 1. If fi(x) = 1, then

1 = fi(x) = hi(x) - Igl (xi) - gl(X)l <_ 1 - Igl (xi) - gl(x)l ,

266 M. BARTELT AND W. LI

which implies hi(x) = 1 and gl(x) = gl(xi ) . Thus,

{ z : / ~ ( x ) = 1} c {z • y~: gl(z) = gl (x~)}. (41)

Define

f ( x ) := sgn (gl (z2))- f2 (x) - f l ( x ) .

Then - 1 ___ f ( x ) _< 1, since the suppor ts of f l and f2 are disjoint. I t is easy to verify tha t

P c ( f ) = {0}. Moreover, if x • Vi and [f(x)l = 1, then f i(x) = 1 and it follows from (41) t ha t

gl (z) = gl (xi) . Therefore,

( - 1 ) i f ( x ) • gl (x) > 0 (42)

and

gl(x) : gl (xi) (43)

for x • V /wi th I f (x) l = 1.

Let ¢(x) = sgn(gl(x2)) , f2(x) q- fl(X). Then I1¢11 = 1. By Lemma 10, there exist two points x~

* such tha t I f(z*)l = 1, and x 2

p( f , ¢)(x) = sgn (gl (x~)) • ¢ (x~) -q- sgn (gl (x~)) • ¢ (x~) gl (x ) ,

Igl (x~)[ + 161 (X~)I

f (X~)" gl (X~)" f (X~) ' g l (X~) < O.

(44)

(45)

If bo th x~ and x~ are in the same Vi, then (45) contradicts (42). Wi thou t loss of generality, we

may assume tha t x* • Vi. Since f ( x ) = ( - 1 ) i ¢ ( x ) for x • Vi, it follows from (45) and I f(x*)l = 1

t ha t

Isgn (gl (x~)) • ¢ (x~) + sgn (gl (x~)) • ¢ (x~) I = 2.

Thus, by (44) and (43), we get

2 rlgl(Z)H IlP(f,¢)l] = Igl (xl)l + Igl (x2)l" II

R E F E R E N C E S

1. B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Nonlinear Parametric Optimization, Birk- h~iuser Verlag, Basel, (1983).

2. A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, (1983).

3. T. Gal, Postoptimal Analyses, Parametric Programming and Related Topics, McGraw-Hill, (1979). 4. L. Cromme, Strong uniqueness: A far reaching criterion for convergence analysis of iterative processes,

Numer. Math. 29, 179-193 (1978). 5. M.C. Ferris, Finite termination of the proximal point algorithm, Math. Programming 50, 359-366 (1990). 6. M.C. Ferris and O.L. Mangasarian, Error bounds and strong upper semicontinuity for monotone affine

variational inequalities, Technical Report: TR 1056a, Computer Sciences Department, University of Wis- consin-Madison, (November 1991) (Revised February 1992).

7. P.T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity prob- lems: A survey of theory, algorithms and applications, Math. Programming Series B 48, 161-220 (1991).

8. J.L. Goffin, The relaxation method for solving systems of linear inequalities, Math. Oper. Res. 5, 388-414 (1980).

9. K. Jittorntrum and M.R. Osborne, Strong uniqueness and second order convergence in nonlinear discrete approximation, Numer. Math. 34, 439-455 (1980).

10. W. Li, Error bounds for piecewise convex quadratic programs and applications, Technical Report TR93-1, Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, Jan., 1993, SIAM J. Con- trol Optim. (to appear).

11. W. Li, Linearly convergent descent methods for unconstrained minimization of a convex quadratic spline, Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, March 1993, J. Optim. Theory Appl. (to appear).

Error Estimates 267

12. W. Li, P. Pardalos and C.-G. Han, Gauss-Seidel method for least distance problems, J. Optim. Theory Appl. 75, 487-500 (1992).

13. W. Li and J. Swetits, A new algorithm for solving strictly convex quadratic programs, Technical Report TR92-14, Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, 1992, SIAM J. Optim. (to appear).

14. Z.-Q. Luo, Convergence analysis of primal-dual interior point algorithms for convex quadratic programs, Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada, Manuscript dated June 20, 1992.

15. Z.-Q. Luo and P. Tseng, On the convergence rate of dual ascent methods for strictly convex minimization, Math. Oper. Res. (to appear).

16. Z.-Q. Luo and P. Tseng, On the convergence of the coordinate descent method for convex differentiable minimization, J. Optim. Theory Appl. 72, 7-36 (1992).

17. Z.-Q. Luo and P. Tseng, Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM J. Optimiz. 2, 43-54 (1992).

18. Z.-Q. Luo and P. Tseng, On the linear convergence of descent methods for convex essentially smooth minimization, SIAM J. Control Optim. 30, 408-425 (1992).

19. Z.-Q. Luo and P. Tseng, Error bound and reduced gradient projection algorithms for convex minimization over a polyhedral, SIAM J. Optim. 3 (1) (1993).

20. O.L. Mangasarian and R. De Leone, Error bounds for strongly convex programs and (super)linearly con- vergent iterative schemes for the least 2-norm solution of linear programs, Ann. Math. Oper. 17, 1-14 (1988).

21. M.R. Osborne and R.S. Womersley, Strong uniqueness in sequential linear programming, J. Austral. Math. Soc. Set. B 31, 379-384 (1990).

22. A. Auslender and J.-P. Crouzeix, Global regularity theorems, Math. Oper. Res. 13, 243-253 (1988). 23. C. Bergthaller and I. Singer, The distance to a polyhedron, Linear Alg. Appl. 169, 111-129 (1992). 24. J.V. Burke and P. Tseng, A unified analysis of Hoffman's bound via Fenchel duality, Department of Math-

ematics, University of Washington, March 1993, SIAM J. Control Optim. (to appear). 25. W. Cook, A.M.H. Gerards, A. Schrijver and l~. Tardos, Sensitivity theorems in integer linear programming,

Math. Programming 34, 251-264 (1986). 26. A.J. Hoffman, Approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards 49,

263-265 (1952). 27. O. Giiler, Distance to a convex polyhedron and Hoffman's lemma, Working Paper 91-3, University of Iowa,

Department of Management Sciences, (January 1991). 28. O. Giiler, A.J. Hoffman and U.R. Rothblum, Approximations to solutions to systems of linear inequalities,

Working Paper, University of Maryland at Baltimore County, Department of Mathematics and Statistics, (September 1992).

29. W. Li, The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program, Linear Algebra and Applications 187, 15-40 (1993).

30. W. Li, Error bounds for solutions of parametric convex-concave minimax problems, In Parametric Opti- mization and Related Topics III, Approximation ~4 Optimization, (Edited by J. Guddat, H.Th. Jongen, B. Kummer and F. Nozicka), Vol. 3, pp. 373-394, Verlag Peter Lang, (1993).

31. W. Li, Sharp Lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs, SIAM J. Control Optim. 32, 140-153 (1994).

32. X.-D. Luo and Z.-Q. Luo, Extension of Hoffman's error bound to polynomial systems, Communication Research Laboratory, McMaster University, Hamilton, Ontario, Canada, May 15, 1992 (preprint).

33. Z.-Q. Luo and J.-S. Pang, Error bounds for analytic systems and their applications, Department of Mathe- matical Sciences, The Johns Hopkins University, Baltimore, MD, January 1993 (preprint).

34. Z.-Q. Luo and P. Tseng, Perturbation analysis of a condition number for linear systems, Department of Mathematics, University of Washington, Seattle, WA, December 27, 1991.

35. Z.-Q. Luo and P. Tseng, On global error bound for a class of monotone AVI problems, Operations Research Letters 11. 159-165 (1992).

36. O.L. Mangasarian, A condition number of linear inequalities and equalities, In Methods of Operations Research 43, Proceedings of 6 th Symposium iiber Operatzons Research, Universit~it Augsburg, (Edited by G. Bamberg and O. Opitz), September 7-9, 1981, pp. 3-15, Verlagsgruppe Athenn~ium/Hain/Scriptor/Han- stein, KSnigstein, (1981).

37. O.L. Mangasarian, Error bounds for nondegenerate monotone linear complementarity problems, Math. Programming 48, 437-445 (1990).

38. O.L. Mangasarian, A condition number for differentiable convex inequalities, Math. Oper. Res. 10, 175-179 (1985).

39. O.L. Mangasarmn, Global error bounds for monotone affine variational inequality problems, Linear Al9. Appl. 174, 153-164 (1992).

40. O.L. Mangasarian and R.R. Meyer, Nonlinear perturbation of linear programs, SIAM J. Control Opt~m. 17, 745-752 (1979).

41. O.L. Mangasarian and T.-H. Shiau, Error bounds for monotone linear complementarity problems, Math. Programming 36, 81-89 (1986).

268 M. BARTELT AND W. LI

42. O.L. Mangasarian and T.-H. Shiau, Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems, SIAM J. Control Optim. 25, 583-595 (1987).

43. R. Mathias and J.S. Pang, Error bounds for the linear complementarity problem with a P-matrix, Linear Alg. Appl. 132, 123.-136 (1990).

44. S.M. Robinson, Bounds for error in the solution set of a perturbed linear program, Linear Alg. Appl. 6, 69-81 (1973).

45. S.M. Robinson, An application of error bounds for convex programming in a linear space, SIAM J. Control Optim. 13, 271-273 (1975).

46. J.W. Demmel and N.J. Higham, Improved error bounds for underdetermined system solvers, SIAM J. Matrix Anal. Appl. 14 (1) (1993).

47. J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control Optim. 31 (5) (1993).

48. M.C. Ferris and O.L. Mangasarian, Minimum principle sufficiency, Math. Programming 57, 1-14 (1992). 49. W. Li, A.J. Hoffman's Theorem and metric projections in polyhedral spaces, J. Approx. Theory 75, 107-111

(1993). 50. B.T. Polyak, Introductwn to Optimization, Optimization Software, Inc., New York, (1987). 51. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkh~iuser, Boston, (1990). 52. W. Li, Strong uniqueness and Lipschitz continuity of metric projections: A generalization of the classical

Haar Theory, J. Approx. Theory 56, 164-184 (1989). 53. E.W. Cheney, Introduction to Approximation Theory, 2nd edition, Chelsea Pub. Co., New York, (1982). 54. D.J. Newman and H.S. Shapiro, Some theorems on Chebyshev approximation, Duke Math. J. 30, 673-684

(1963). 55. G. Niirnberger, Strong unicity constants in Chebyshev approximation, In Numerical Methods of Approxi-

mation Theory, (Edited by L. Collatz, G. Meinardus and G. Nfirnberger), Vol. 8, ISNM 81, pages 144-168, Birkh~user Verlag, Basel, (1987).

56. M.R. Osborne, Finite Algorithms in Optimization and Data Analysis, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, (1985).

57. H. Sp~ith, Mathematical Algorithms for Linear Regression, Academic Press, New York, (1992). 58. W. Li, Best approximations in polyhedral spaces and linear programs, In Approximation Theory: Proceedings

of the Sixth Southeastern International Conference, (Edited by G. Anastassiou), pages 393-400, Marcel Dekker, New York.

59. W. Li, Lipschitz continuous metric selections in Co(T), SIAM J. Math Anal. 21, 205-220 (1990). 60. S.-H. Park, Uniform Hausdorff strong uniqueness, J. Approx. Theory 58, 78-89 (1989). 61. A.V. Kolushov, Differentiability of the best approximation operator, Math. Notes Acad. Sci. USSR 29,

295-306 (1981). 62. A. KroO, Differential properties of the operator of best approximation, Acta Math. Hungar. 34, 185-203

(1977). 63. M.W. Bartelt and J.J. Swetits, Characterizations of the local Lipschitz constant, J. Approx. Theory 77,

249-265 (1994). 64. S.M. Robinson, Some continuity properties of polyhedral multifunctions, Math. Programming Study 14,

206-214 (1981).