Old Dominion University ODU Digital Commons
Mathematics & Statistics Faculty Publications Mathematics & Statistics
1997 Hoffman’s Error Bounds and Uniform Lipschitz Continuity of Best l(p) -Approximations H. Berens
M. Finzel
W. Li Old Dominion University
Y. Xu
Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_fac_pubs Part of the Applied Mathematics Commons
Repository Citation Berens, H.; Finzel, M.; Li, W.; and Xu, Y., "Hoffman’s Error Bounds and Uniform Lipschitz Continuity of Best l(p) -Approximations" (1997). Mathematics & Statistics Faculty Publications. 86. https://digitalcommons.odu.edu/mathstat_fac_pubs/86 Original Publication Citation Berens, H., Finzel, M., Li, W., & Xu, Y. (1997). Hoffman's error bounds and uniform lipschitz continuity of best lp-approximations. Journal of Mathematical Analysis and Applications, 213(1), 183-201. doi:10.1006/jmaa.1997.5521
This Article is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 213, 183᎐201Ž. 1997 ARTICLE NO. AY975521
Hoffman’s Error Bounds and Uniform Lipschitz
Continuity of Best l p-Approximations
H. BerensU and M. Finzel†
Mathematical Institute, Uni¨ersity of Erlangen-Nuremberg, D-91054, Erlangen, Germany
W. Li‡
Department of Mathematics and Statistics, Old Dominion Uni¨ersity, Norfolk, Virginia 23529-0077
and
Y. Xu§
Department of Mathematics, Uni¨ersity of Oregon, Eugene, Oregon 97403-1222
Submitted by Joseph D. Ward
Received October 24, 1996
In a central paper on smoothness of best approximation in 1968 R. Holmes and n B. Kripke proved among others that on ޒ , endowed with the l p-norm, 1 - p - ϱ, the metric projection onto a given linear subspace is Lipschitz continuous where the Lipschitz constant depended on the parameter p. Using Hoffman’s Error
Bounds as a principal tool we prove uniform Lipschitz continuity of best l p-ap- proximations. As a consequence, we reprove and prove, respectively, Lipschitz continuity of the strict best approximation Ž.sba, p s ϱ and of the natural best approximation Ž.nba, p s 1. ᮊ1997 Academic Press
U E-mail address: [email protected]. † E-mail address: [email protected]. ‡ E-mail address: [email protected]. § E-mail address: [email protected].
183
0022-247Xr97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 184 BERENS ET AL.
1. INTRODUCTION
In the second week of July the four authors sat together in the Mathematical Institute in Erlangen to discuss various problems on approx- imation and got stuck on a problem of finite dimensional discrete linear approximation. For the second named author it is one of her research areas, and a recent exchange with the third named author fostered the discussion. But before we get into detail, we have to introduce the necessary notation. Let ޒ n be the n-dimensionalŽ. Euclidean space of column vectors. The ޒ n 55 ŽÝn < U 5 xyu555ppsminÄ4x y u : u g U .1Ž. Ž. Let ⌸ U, ppx be the set of all best l -approximants of x in U; i.e., U5U555 ⌸U,pŽ.x[½5ugU:xyuppsmin x y u . ugU n Then ⌸ U, p is called the metric projection from ޒ to U with respect to the p-norm. For 1 - p - ϱ, ⌸ U, p defines a continuous point-valued mapping be- cause of the strict convexity of the norm, while for p s 1orpsϱ,itisin n Ž. general set-valued; i.e., for x g ޒ , ⌸ U, p x might contain infinitely many Ž. elements. In fact, ⌸ U, p x is a closed convex polytope. In this case, the metric projection ⌸ U, p is Lipschitz continuous as a set-valued mapping with respect to the Hausdorff metric; i.e., there exists a positive constant ␥ Ž.Ždepending only on U such that cf.wx 17 . n HŽ.⌸U,pŽ.x,⌸U,p Ž.yF␥и55xyyfor x, y g ޒ , p s 1orϱ;2Ž. here HSŽ.12,S [max½5 max distŽ.x, S2, max dist Ž.y, S1 xgSy12gS denotes the Hausdorff distance of the sets S12and S and distŽ.x, Si[ Ä 4 min 55x y ¨ : ¨ g Siithe distance from the point x to the set S . Conse- Ž.Ž Ž.. quently, if ⌸ U,1 x or ⌸ U,ϱ x contains exactly one element for every x n Ž. in ޒ , then the metric projection ⌸ U,1 or ⌸ U,ϱ is point-valued and Lipschitz continuousŽ as a mapping from ޒ nnto ޒ .. HOFFMAN’S ERROR BOUNDS 185 Ž.Ž . Since ⌸ U,p x with p s 1orϱ is a compact convex set for each x,it follows fromŽ. 2 that one can use the Steiner point selector to get a Ž . Lipschitz continuous selection for ⌸ U, p cf.wx 2 and references therein . Ž n Recall that a mapping : ޒ ª U is called a selection for ⌸ U, p if Ž. Ž. n. xg⌸U, p xfor x g ޒ . However, the Steiner point selector requires Ž. information on the whole set ⌸ U, p x ; thus it is not a practical way to construct a Lipschitz continuous selection for ⌸ U, p. The most intensively investigated selection for ⌸ U, ϱ is the strict best approximation, denoted by sba, that can be defined as the limit of ⌸ U, p as p ª ϱ: n sbaŽ.x [ lim ⌸ U,pŽ.x for x g ޒ .3Ž. pªϱ The strict best approximation was introduced by Rice as the best of best approximations. The limitŽ. 3 was first proved by Descloux and was rediscovered later by Mityaginwx 19 . For the computation of the strict best approximation, seewx 9, 1, 3 . Even though sba was believed to be a continuous mapping, a formal proof based on generalized inverses first appeared inwx 10 ; another proof can be found in wx 5 . The limitŽ. 3 gives an easy description of sbaŽ.x , but it does not reveal much about the proper- ties and structure of strict best approximation. Recently, Finzelwx 6 used Plucker-Grassmann¨ coordinates to give a complete structural description of sba. As a consequence, she proved that ޒ n can be subdivided into finitely many polyhedral cones where on each of them sba is a linear mapping and, hence, sba is Lipschitz continuous. For further properties and references on strict best approximation, seewx 6, 13 . The selection for ⌸ U,1 corresponding to sba is the natural best approxi- q mation, denoted by nba, which is defined by the limit of ⌸ U, p as p ª 1 wx15 : n nbaŽ.x [ lim ⌸ U,pŽ.x for x g ޒ .4Ž. pª1q Landers and Rogge 15 proved that for each x, lim q⌸ Ž.x exists wx pª 1 U, p ŽŽ.i.e., nba x is well-defined . . Independently, Fischerwx 8 also proved the existence of this limit. In addition, Landers and Roggewx 15 characterized nbaŽ.x as the unique solution of the following minimization problem: n min <<< Our approach to establish the Lipschitz continuity of sba and nba, without studying the structure of the mappings directly, is to prove the uniform Lipschitz continuity of ⌸ U, p with respect to 1 - p - ϱ, which is the main aim of the paper. The result is summarized in the following theorem. n THEOREM 1. Let U be a linear subspace of ޒ . Then there exists a positi¨e constant such that n ⌸U,pŽ.xy⌸U,p Ž.yFи55xyy,for x, y g ޒ ,1-p-ϱ.6Ž. As a consequence, the strict best approximation sba and the natural best approximation nba are Lipschitz continuous selections for ⌸ U, ϱ and ⌸ U,1, respecti¨ely. The theorem was conjectured inwx 15 in connection with a general discussion on best approximation in polyhedral spaces, which was moti- vated by the paper of Holmes and Kripkewx 12 on smoothness of best approximation in Banach spaces. Inwx 12 , Holmes and Kripke proved that Ž. ⌸U, p is Lipschitz continuous for each p g 1, ϱ . Their proof was given for LTpŽ.,with a nonatomic measure on a compact Hausdorff space T, and because of that, they had to restrict their arguments to 2 - p - ϱ. But for ޒ n endowed with p-norms, their proof holds true for 1 - p - 2as well. That is, for 1 - p - ϱ, n ⌸U,pŽ.xy⌸U,pp Ž.yFи55xyy, for x, y g ޒ ,7Ž. where p is some positive constant depending on U and p only. Technically,Ž. 6 is an improvement ofŽ. 7 . The improvement comes from the analysis of the Jacobian of ⌸ U, p, which led us to the following theorem about matrix inequalities. THEOREM 2. LetBbeann=n matrix. Then there exists a positi¨e constant such that q m 55WBy F и 5BB WBy5 for y g ޒ and W s diagŽ.w1,...,wn with wiG 0,Ž. 8 q Ž. where B is the pseudo-in¨erse of B and diag w1,...,wn the diagonal matrix whose ith diagonal entry is wi. m If we define U to be the column space of B; i.e., U [ Ä By: y g ޒ 4, thenŽ. 8 can be interpreted as follows: the norm of a scaled vector in U is uniformly bounded by the norm of the orthogonal projection of the scaled vector to U. Hoffman’s error bound, seewx 11 , for approximate solutions of HOFFMAN’S ERROR BOUNDS 187 linear inequalities and equalities turns out to be the key in establishingŽ. 8 . The reason is that the set generated by scaling vectors in U is a union of finitely many polyhedral sets. In addition, we can even give an explicit estimate of inŽ. 6 in terms of submatrices of any matrix whose columns form an orthonormal basis of U. For further applications of Hoffman’s error bound in connection with best approximation in polyhedral spaces seewx 17 . The paper is organized as follows. Section 2 is devoted to the study of the matrix inequalityŽ. 8 and its ramifications. In Section 3, we establish an Ž. estimate of the Jacobian of ⌸ U, p based on 8 . Then we give two proofs of Theorem 1: one is self-contained and another is a simpler proof based on Lipschitz continuity of ⌸ U, p proved by Holmes and Kripke. To conclude this section we introduce commonly used notations in this paper. For a matrix A Ž.or a vector x and an index set J, we define A J Žor xJ .Ž.to be the matrix or the vector consisting of rows of A Žor components of x.Žwhose corresponding indices are in J. In particular, Aiior x . represents the ith rowŽ.Ž or ith component of A or x.. For an index set c J;Ä41,...,n, let J [ Ä41,...,n _J be the complement of J. The dot ޒ n ²:Ýnnޒ product in is denoted by x, y s is1 xyiifor x, y g . For an m n=mmatrix B, its spectral norm is defined by 55B [ maxÄ55By : y g ޒ 4 with 55y F 1 . The k = k identity matrix is denoted by Ik and sometimes we also use I as the identity matrix if there is no confusion about its dimension. For the relative interior of a subset K of ޒ n, we will use the symbol riŽ.K . Finally, a subset K of ޒ nis called a polyhedral set if it is an intersection of finitely many closed half-spaces. 2. HOFFMAN’S ERROR BOUND AND MATRIX INEQUALITY As preparation of the proof of Theorem 2 we formulate and prove four lemmas. LEMMA 3. LetBbeann=m matrix. Then nm ² xyBBq x, By: s 0 for x g ޒ , y g ޒ .9Ž. Proof. It is well known that Bqx is the least norm solution of the following least squares problem: min 1 55x By 2 .10Ž. m2 y ygޒ T By a characterization for an optimal solution ofŽ. 10 , we have BxŽy BBq x.s 0, which implies T nm ²:xyBBq x, By s²:BxŽ.yBBq x , y s 0 for x g ޒ , y g ޒ . 188 BERENS ET AL. LEMMA 4. Suppose that B is an n = m matrix, D a symmetric positi¨e m semidefinite matrix, and y g ޒ . If BBqDBy s 0, then DBy s 0. Proof. Let u s By and x s Du. Then BBq x s 0. ByŽ. 9 in Lemma 3, we have T uDus²:²:²Du, u s x, By s x y BBq x, By:s 0. Since D is symmetric and positive semidefinite, there exists a matrix Q T 2 T such that D s QQ. Thus, 55Qu s uDus0; i.e., Qu s 0. So DBy s T Du s QQuŽ.s0. n LEMMA 5. If X is a subset of ޒ , then the set V [ Ä4Wx: x g X and W s diagŽ.w1,...,wni with w G 011 Ž. is a union of finitely many con¨ex polyhedral sets. Ž. Ä4 Proof. For any ⑀ s ⑀1,...,⑀niwith ⑀ gy1, 0, 1 , define the octant corresponding to ⑀, denoted by OctŽ.⑀ ,as n OctŽ.⑀ [ Ä4x g ޒ : ⑀iix G 0 for ⑀i/ 0, and xis 0 for ⑀is 0. Let n IX[Ä4⑀sŽ.⑀1,...,⑀ngyÄ41,0,1 : X l ri OctŽ.⑀ / л . First we claim that V ; DOctŽ.⑀ .1Ž.2 ⑀gIX Ž. Ž. Ž Ž . Ž .. Let x g X and define ⑀ s sign x , where sign x s sign x1 , . . . , sign xn . Then OctŽ.⑀ s Ä4Wx: W s diagŽ.w1,...,wniwith w G 0. Ž. Moreover, x g X l riw Oct ⑀ x.So⑀gIX , which implies V Ä4Wx: W diagŽ.w ,...,w with w 0 OctŽ.⑀ , s DDs 1niG ; xgX ⑀gIX provingŽ. 12 . Ž. Next we prove that Oct ⑀ ; V for ⑀ g IXX. Let ⑀ g I . Then there exists x g X such that signŽ.x s ⑀. Ž Note that x g riw OctŽ.⑀ xif and only if HOFFMAN’S ERROR BOUNDS 189 signŽ.x s ⑀. . Let ¨ g Oct Ž.⑀ . Define ¨iirx if x i/ 0 wis ½0ifxi s0. Ž. Ž. Since sign ¨iis ⑀ s sign x iwhen ¨ i/ 0, we have wiG 0 for 1 F i F n. Ž. Note that ¨iiis wx for 1 F i F n; i.e., ¨ s Wx with W s diag w1,...,wn. So ¨ g V and therefore, V Ä4Wx: W diagŽ.w ,...,w with w 0 OctŽ.⑀ . s DDs 1niG > xgX ⑀gIX Since OctŽ.⑀ is a convex polyhedral set and IX finite, V is a union of finitely many convex polyhedral sets. LEMMA 6Ž. A Variation of Hoffman’s Error Bound . Let X be a polyhe- n dral subset of ޒ and S [ Ä4x g X: Ax s b . Then there exists a positi¨e constant Ž.depending only on X and A such that distŽ.x, S F и 55Ax y b for x g X.13Ž. Proof. Since X is a convex polyhedral set, there exist a matrix C and a n vector d such that X s Ä x g ޒ : Cx F d4. Then by Hoffman’s error bound for approximate solutions of systems of linear inequalitieswx 11 , there exists a positive constant such that 55 n distŽ.x, S F и Ž.Ax y b q Ž.Cx y d qfor x g ޒ ,14Ž. Ä4 where zq denotes the vector whose ith component is max zi, 0 . Note that Ž. Ž. Ž. xgXif and only if Cx y d qs 0. Thus, 13 follows from 14 . Proof of Theorem 2. We claim that for an n = m matrix B there exists a positive constant such that q m 55WBy F и 5BB WBy5for y g ޒ and W s diagŽ.w1,...,wn with wiG 0.Ž. 15 m Let X [ ÄBy: y g ޒ 4and V [ Ä4Wx: x g X and W s diagŽ.w1,...,wniwith w G 0. Then by Lemma 5 the set V is a union of finitely many convex polyhedral sets Ä4V12, V ,...,Vk; i.e., k V. VsDi is1 190 BERENS ET AL. By Lemma 6, there exist positive constants i such that q distŽ.¨ , SiiF и 5 BB ¨ 5 for ¨ g Vi,16Ž. where q Sii[ Ä4z g V : BB z s 0. m Let z g Sii. Since z g V ; V , there exist y g ޒ and a diagonal matrix Ž. Wsdiag w1,...,wniwith w G 0 such that z s WBy. Since z g S i,we have BBqWBy s BBqz s 0. Since W is symmetric and positive semidefi- Ä4 nite, it follows from Lemma 4 that z s WBys0. Therefore, Si s 0 and Ž. Ž. dist ¨, Si s 55¨ . The inequality 16 actually reads q 55¨Fiiи 5BB ¨ 5 for ¨ g V .17Ž. Ä4 Setting [ max i:1FiFk, we obtain 55¨Fи 5BBq¨ 5 for ¨ g V .18Ž. By the definition of V , we know that the inequality in Theorem 2 holds. Remark. If we allow a bigger motion of the subset X than multiplica- tion with a positive diagonal matrix, then Theorem 2 does not hold any more. To be precise, let D be a symmetric positive semidefinite matrix, m X[Ä DBy: y g ޒ 4Ä, and S [ x g X: BBq x s 04 . Then, by Lemma 5, SsÄ40 . It follows from Lemma 6 that there exists a positive constant Ž.D such that distŽ.x, S F Ž.D и 5 BBq x5 for x g X ; i.e., m 55DBy F Ž.D и 5BBq DBy5for y g ޒ .19Ž. Thus, in the light of Theorem 2, one might expect that the Ž.D in Ž. 19 are uniformly bounded for all symmetric positive semidefinite matrices. How- ever, the following example shows that the Ž.D in Ž. 19 are not uniformly bounded with respect to all symmetric positive semidefinite matrices. Let n 2, m 1, y 1 ޒ, B 1 , 4 - ␣ - 2, s s s g s ž/1 r r '2 cos ␣ ysin ␣ w 0 Q s , W s , w s , ž/ž/sin ␣ cos ␣ 00 sin ␣ y cos ␣ T and D s Q WQ. Then Q is the orthogonal matrix that represents rotation T by the angle ␣ and D s Q WQ is a symmetric positive semidefinite matrix. By matrix multiplications, we obtain cos ␣ DBy QT WQBy '2.y s s ž/sin ␣ HOFFMAN’S ERROR BOUNDS 191 q Ž.T y1 T 1 Ž. Since B s BB B s2 1 1 , we can compute 1 q sin ␣ y cos ␣ BB DBy s . '2 ž/sin ␣ y cos ␣ So 55DBy '2 Ž.D Gs .20Ž. 55BBq DBy sin ␣ y cos ␣ Therefore, Ž.D is unbounded as ␣ ª Ž.r4q . As a consequence, Theo- rem 2 fails to be true if we allow W to be an arbitrary symmetric positive semidefinite matrix. COROLLARY 7. LetQbeann=m matrix with rank m. Then there exists a positi¨e constant ␥ such that T m 55WQy F ␥ и 5Q WQy5 for y g ޒ and W s diagŽ.w1,...,wn with wiG 0.Ž. 21 Proof. Since the columns of Q are linearly independent, Qqs ŽQQT .y1 QT. Thus, by Theorem 2, there exists a positive constant such that TTy1 55WQy F и QŽ. Q Q Q WQy m for y g ޒ and W s diagŽ.w1,...,wniwith w G 0. T1 T 1 Since 5QQQŽ.y z55F QQQŽ .y555и z, the above inequality implies T 55WQy F ␥ и 5Q WQy5 m for y g ޒ and W s diagŽ.w1,...,wniwith w G 0, T1 where ␥ s 5QQQŽ.y5. Remark. By using explicit estimates of Lipschitz constants for feasible solutions of a system of linear equalities and inequalities, we can get explicit estimates of in Theorem 2 and ␥ in Corollary 7. For illustration purpose, we derive the following estimate for ␥, T 55WQy F ␥ и 5Q WQy 5 m for y g ޒ and W s diagŽ.w1,...,wniwith w G 0, Ž. 22 192 BERENS ET AL. where y1 In mJJyQQc ␥[maxy : Q is a nonsingular m = m matrix . y1 J J½5ž/0QJ Ä m4 Ž. Let X [ Qy: y g ޒ and let V , Oct ⑀ , IX be defined as in the proof Ž. of Lemma 5. Let ⑀ s ⑀1,...,⑀nXgI and consider the following system of equalities and inequalities, T Q x s b and Ax F 0,Ž. 23 m where b g ޒ and I y Jq IJ A y , s I J0 I 0yJ0 Ä4Ä⑀ ⑀ 4 Iis the n = n identity matrix, Jq[ i: i s 1, Jy[ i: i sy1 , and Ä4ŽŽ..Ž. J0 [i:⑀i s0 . Note that Ax F 0 if and only if x g Oct ⑀ . Let Sb be the solution set ofŽ. 23 . By the remark on page 34 inwx 18 , we have X X X m HSbŽ.Ž.,Sb Ž .F␥˜и5 byb5 for b, b g ޒ ,24Ž. where y1 AAJJ ␥˜ [maxTT : is a nonsingular n = n matrix . J ½5ž/QQ ž/ T X X For any x g OctŽ.⑀ , let b s Qxand b s 0. Then SbŽ .sÄ40Ž by Lemma T 1 T 4 and Qqs ŽQQ.y Q.Ž.Ž.and x g Sb. So 24 implies X X T 55xFHSbŽ.Ž.,Sb Ž .F␥˜˜и5byb55s␥иQx5 for x g OctŽ.⑀ , ⑀ g IX.25 Ž . Ž. Since the rows of A J are rows of I or yI, let D s diag d1,...,dn be such that di s "1 and A J IJ c D s , ž/QT ž/QT HOFFMAN’S ERROR BOUNDS 193 where J is some index subset ofÄ4 1, . . . , n . Since D is an orthogonal matrix, we have y1 y1 AAJJ sD ž/QQTTž/ ž/ y1 y1 IJc Inym0 ssT TT ž/Q ž/QQJJc 1 y y1 c IQQc IQnymJ nymJJy ss1, ž/0QJ 0Qy ž/J where the third equality holds since we only exchange columns of the matrix y1 IJ c , ž/QT T the fourth one follows from 55B s 5B 5, and the last equality is derived IQc from the calculation of the inverse ofnymJ . Thus,Ž. 25 is equivalent to ž/0 QJ Ž.22 . 3. UNIFORM LIPSCHITZ CONTINUITY OF BEST l p-APPROXIMATION n For x g ޒ , consider the best l p-approximation problem of finding Ž. n ⌸U, p xin a subspace U of ޒ such that 55 xy⌸U,pŽ.xpsminÄ4x y u p: u g U ,1-p-ϱ. As we pointed out in the Introduction, Holmes and Kripke actually proved Ž. that for each p g 1, ϱ , there exists a positive constant p such that 55 n ⌸U,pŽ.xy⌸U,pp Ž.ypFиxyyp for x, y g ޒ .26Ž. We want to show that p inŽ. 26 is uniformly bounded for 1 - p - ϱ.We start with an estimate on the Jacobian of the metric projection ⌸ U, p. n THEOREM 8. Let U be a subspace of ޒ . Then there exists a positi¨e constant ␥ such that .⌸x U,ppŽ.x F␥ for 1 - p - ϱ, x g D ,27Žٌ 194 BERENS ET AL. where x ޒ n x ⌸ x for i n Dp[ ½5g : Ž.y U,pŽ.i/ 0 1 F F .28Ž. Proof. Let U be a matrix whose columns Äu1,...,um4form an or- n Ž. thonormal basis of a subspace U of ޒ . Since ⌸ U, p x is unique for each 1 - p - ϱ, there exists a unique vector tx1, pŽ. . txpŽ.s . tx 0m,pŽ. such that m k ⌸ U,pkŽ.x s Ýtxu,pp Ž.sUt Ž. x .29Ž. ks1 n m Let the function fp: ޒ = ޒ ª ޒ be defined by 1 m p k fxpkŽ.,t[ xyÝtu .30Ž. p ks1p Then for each fixed x, the gradient of fp with respect to t has to be zero Ž. Ž Ž.. Ž. at txpt;i.e., ٌ fxpp,txs0. Thus, for any given x, txpis the unique solution of the following system of nonlinear equations: nmpy1 m xtukksign x tu ui0, for 1 i m. ÝÝrkrrkrryž/y Ý s F F rs1 ks1 ks1 Ž.31 Ž. Let x g Dpt. Then ٌ fxp,tis continuously differentiable with respect to Ž.x,tin a neighborhood of Žx, txp Ž... In fact, Ѩ2 fxŽ.,t nmpy2 p kij sŽ.py1ÝÝxrkrrrytu uu, for 1 F i, j F m, ѨtѨt ij rs1 ks1 Ž.32 and Ѩ 2 fxŽ.,t m py2 p ki sŽ.1ypxjkjjyÝtu u, for 1 F i F m,1FjFn. ѨtѨx ij ks1 Ž.33 HOFFMAN’S ERROR BOUNDS 195 Equivalently, we have the following matrix representations of the Jaco- 2 2 ,bians ٌttfx pŽ.,tand ٌtxfx pŽ.,t, respectively 2 T m=m 3Ž.4, ޒ ttfx pŽ.Ž,tspy1 .UWU inٌ 2 T m=n 3Ž.5, ޒ txfx pŽ.Ž,ts1ypU .W inٌ Ž. <Ýmk 22y1 .xptxŽ.sy ٌ ttppfxŽ.Ž.,txŽ. ٌ txppfx,txŽٌ TTTTy1 y1 sy Ž.py1 UWU Ž.1ypU WswxUWU UW.36Ž. Ž. Ž. The gradient is an m = n matrix whose i, j th entry is Ѩ txi, pjrѨx. Thus, we obtain T T ٌxptxŽ. z xptxŽ.sٌxptx Ž. ssupٌ z/0 55z T T xptŽ. x U WUy55 WUyٌ ssup TTs sup F ␥ ,37Ž. y/0 55U WUyy/0 55 U WUy where the first equality follows from the definition of the norm of an adjoint matrix, the second one is by the definition of the spectral norm, the third one follows from the nonsingularity of U T WU, the fourth is derived fromŽ. 36 , and finally, the last inequality follows from Corollary 7. Since Ž. Ž. Ž. Ž. ⌸U, ppxsUt x , by the chain rule, we get ٌ⌸ xU,pxpx sUٌtxfor xgOŽ.x. Since 55U s 1, ,␥⌸x U,pxpxpxpŽ.x s Uٌtx Ž.F55UиٌtxŽ.sٌtx Ž.Fٌ Ž. where ␥ is independent of x g Dp and p g 1, ϱ . Remark. It follows from the remark after Corollary 7 andŽ. 37 that if U is an n = m matrix whose columns form an orthonormal basis of U, then .⌸x U,ppŽ.x F␥ for x g D ,38Žٌ where y1 In mJJyUUc ␥[maxy : U is a nonsingular m = m matrix . y1 J J½5ž/0UJ 196 BERENS ET AL. Proof of Theorem 1. Now we are ready to prove Theorem 1: If U is a subspace of ޒ n, then there exists a positive constant such that n ⌸U,pŽ.xy⌸U,p Ž.yFи55xyyfor x, y g ޒ ,1-p-ϱ.39Ž. We prove the theorem by induction. Obviously, if dim U s 0, thenŽ. 39 holds. We make the inductive hypothesis thatŽ. 39 holds for any n and any subspace U of dimension less than m. Now assume that U is an m-dimen- sional subspace of ޒ n. Ä4 Let J [ j: there exists u g U such that uj / 0 and let u g U. Then Ä4 uiJs0 for i f J.If U [ uJ:ugU denotes the restriction of U to the J components, ⌸ Ž.x ⌸ Ž.x . Therefore, if there exists a positive wU, pJxs UJ,pJ constant such that ⌸Ž.x⌸ Ž.yи55xyfor x, y ޒ n ,1-p-ϱ, UJJ,pJyU,pJF Jy J g thenŽ. 39 holds. Thus we may replace U by UJ and assume that there exists u g U such that ui / 0 for 1 F i F n. Ži. Ä4 Ži.Ž Let U [ u g U: ui s 0, 1FiFn. Then dim U - dim U since Ži. U _ U / л.. By our inductive hypothesis, there exists a positive constant  such that n ⌸UŽi.Ž,pŽ.xy⌸Ui.,p Ž.yFи55xyyfor x, y g ޒ ,1-p-ϱ. Ž.40 Define x ޒ n x ⌸ x i n Dp[ Ä4g : y U,pŽ.i/ 0 for 1 F F , and for 1 F i F n x ޒ n x ⌸ x Dp,i[ Ä4g : y U,pŽ.is 0. By Theorem 8, there exists a positive constant ␥ such thatŽ. 27 holds. Then it follows that ⌸U,pŽ.xy⌸U,p Ž.yF␥и55xyy, whenever 1 - p - ϱ and x, y g Dppwith x q Ž.1 y y g D for 0 F F 1.Ž. 41 This proves the desired estimate locally on Dp with the global Lipschitz constant ␥. To proveŽ. 39 , i.e., to prove Lipschitz continuity of the metric projection on ޒ n uniformly with respect to 1 - p - ϱ, we first have to HOFFMAN’S ERROR BOUNDS 197 Ž. investigate the behavior of ⌸ U, ppx for x g D ,i,1FiFn. Let U be an Ž. m n=mmatrix whose columns form a basis of U and let txp gޒ be such that ⌸ U,ppŽ.x s Ut Ž. x .42Ž. Then, for x g Dp, i, we have Ut x ⌸ x x ipŽ.s U,pi Ž.is .43Ž. Since there exists u g U such that uii/ 0, U has to be a nonzero row. Let TTy1 tpiiiisUUUŽ. x and ˜tpps txŽ.yt p.44Ž. Then Utip˜ sUt ipŽ. x yUtipsx iyx is0.Ž. 45 Ži.Ži. That is, Ut˜p gU . Moreover, for any u g U , Ž.x y Utppy u ppps x y Ž.Ut q u G x y ⌸ U,pŽ.x sŽ.xyUtppy Ut˜p. Thus, ⌸ UŽi. ,pppŽ.x y Ut s Ut˜ .46Ž. It follows fromŽ. 42 , the second equality inŽ. 44 , and Ž. 46 that ⌸ U,ppŽ.x s Ut q ⌸ UŽi. ,ppŽ.x y Ut .47Ž. From the above identity and the first equality inŽ. 44 we derive the Ž. Ž. following explicit representation of ⌸ U, p x in terms of ⌸ U i, p: TTy1 TTy1 ⌸U,piiiiŽ.xsUUŽ. UU x q ⌸ UŽi.,piiiiž/x y UUŽ. UU x for x g Dp,i.48Ž. It follows fromŽ. 40 and Ž. 48 that there is a positive constant , indepen- dent of 1 - p - ϱ, such that ⌸U,pŽ.xy⌸U,pp Ž.yFи55xyyfor x, y g D ,i,1FiFn.49Ž. Let [ maxÄ4, ␥ . We claim that n ⌸U,pŽ.xy⌸U,p Ž.yFи55xyyfor x, y g ޒ ,1-p-ϱ,50Ž. which will complete the proof. 198 BERENS ET AL. n For x, y g ޒ , define x s y q Ž.1 y x and 0[max½5 :0FF1, ⌸U,pŽ.x y ⌸U,p Ž.x F и 55x y x .51Ž. Ž. If 0 s 1, then 50 holds. Assume to the contrary that 0 F 0 - 1. We discuss two cases. First assume that there is 01- F 1 such that x g Dp Ž. for 01- - . Then, by the continuity of ⌸ U,pand 41 , we obtain 01 ⌸U,pŽ.xy⌸U,p Ž.x 01 slim lim ⌸ U,pŽ.x y ⌸ U,p Ž.x qy 0011ªª Flim lim ␥ и 55x 01y x F и I x 01y x 5. qy 0011ªª Otherwise, 1 n x : 00- - ql Dp,i/лfor k s 1,2,... . ½5k ž/D is1 Therefore an index i exists such that 1 x : 00- - qlDp,i/лfor k s 1,2,... . ½5k As a consequence, there exist 1 ) 12) ) иии k) иии G 0such that k xgDp, i and lim x k s x 0 . kªϱ Ž. By 49 and the continuity of ⌸ U, p we derive that 01k1 ⌸U,pŽ.xy⌸U,p Ž.xslim ⌸ U,pŽ.x y ⌸ U,p Ž.x kªϱ Flim и 5x ky x 155F и x 0y x 15. kªϱ So, in both cases, we have 1 ⌸ U,pŽ.x y ⌸ U,p Ž.x 10 0 F⌸U,pŽ.xy⌸U,p Ž.xq⌸U,p Ž.xy⌸U,p Ž.x Fи5x1yx055qиx0yx55sиx1yx5.52Ž. By the definition of 00, we must have G 1, a contradiction to the Ž. choice of 10.So s1 and 50 holds. HOFFMAN’S ERROR BOUNDS 199 RemarkŽ. A Second Proof of Theorem 1 . Note that our proof of Theorem 1 is self-contained. There is a shorter proof of Theorem 1 by making use of the estimateŽ. 26 of Holmes and Kripke. Let U be an m-dimensional subspace of ޒ n and let x ޒ n x ⌸ x i n Dp[ ½5g : Ž.y U,pŽ.i/ 0 for 1 F F . Define H J [ Ä4i: zi/ 0 for some z g U .5Ž.3 Ä4 Note that i g 1,...,n _J if and only if ei g U. Hence we get the representation n UsÄ4ugU:uiis0 for i f J [ Ä4u g ޒ : u s 0 for i g J n and, for x g ޒ , xi for i f J, ⌸ Ž.x Ž.54 U,p is ⌸ Ž.x for i J. ½ UJ,pJi g Ä4 Thus, it suffices to prove Theorem 1 for UJJ[ u : u g U . However, by Ž.H ŽH. the definition of J, UJJs U , containing a vector z such that zi/ 0 for i g J. Replacing UJ by U, we may assume that for each i the subspace H U contains a vector z such that zi / 0. Under this assumption we will show that the complement of Dp is a set of measure zero. n Let Tp be the continuous bijection of ޒ defined by Ty yy<<1rŽ py1. y ޒ n piŽ.issign Ž . ifor g . Än4 n Let Hii[ x g ޒ : x s 0 be a coordinate hyperplane in ޒ . By the Kolmogorov characterization of best approximations, the kernel of ⌸ U, p Ž.Ž. the so-called metric complement of U , denoted by ker ⌸ U, p , is exactly H TpŽU .. H Since there exists z g U _Hi, the dimension of the linear subspace HHŽH. UlHipis less than dim U s n y m.SoT U lHiis a manifold of Ž. ŽH . dimension less than n y m and U q TpiU lH is a manifold of ŽH. dimension less than n. Therefore, U q TpiU lH has measure zero in ޒn. Note that n ޒnnDxޒ:x⌸Ž.x0 _psDÄ4gyU,pis is1 n ker ⌸ H U sDÄ4Ž.U,pil q is1 nn TŽ.UHHTH Ž. U TU H U. sDD½5ppUliqUs Ä4 piŽ.Ul qU is1 is1 n Consequently, ޒ _ Dp is a set of measure zero. 200 BERENS ET AL. By Theorem 8, the Jacobian ٌ⌸x U, p of the Lipschitz mapping ⌸ U, p is bounded by ␥ almost everywhere in ޒ n. By a well-known characterization of Lipschitz continuous functionsŽ cf.wx 20, Sect. 2.2.1, Chap. VI. , we obtain n ⌸U,pŽ.xy⌸U,p Ž.yF␥и55xyyfor x, y g ޒ ,1-p-ϱ. One advantage of the second proof is that it leads to an explicit estimate of an upper bound for p inŽ. 26 . Let J be defined as in Ž. 53 and let Q be an n = m matrix whose columns form an orthonormal basis of UJ . Then from the above proof and the remark after Theorem 8, we obtain ⌸Ž.x⌸ Ž.y ␥и55xyfor x, y ޒ n ,1-p-ϱ, UJJ,pJyU,pJ Fy g Ž.55 where y1 In mJJyQQc ␥[maxy : Q is a nonsinglar m = m matrix . y1 J J½5ž/0QJ ByŽ. 54 and Ž. 55 we obtain n ⌸U,pŽ.xy⌸U,p Ž.yF␥и55xyyfor x, y g ޒ ,1-p-ϱ.56Ž. REFERENCES 1. N. N. Abdelmalek, Computing the strict Chebyshev solution of overdetermined linear equations, Math. Comp. 31 Ž.1977 , 974᎐983. 2. F. Deutsch, W. Li, and S.-H. Park, Characterizations of continuous and Lipschitz continuous metric selections in normed linear spaces, J. Approx. Theory 58 Ž.1989 , 297᎐314. 3. C. S. Duris and M. G. Temple, A finite step algorithm for determining the ‘‘strict’’ best approximation, SIAM J. Numer. Anal. 10 Ž.1973 , 690᎐699. 4. A. Egger and R. Huotari, Polya properties in ޒ n, Numer. Funct. Anal. Optim. 13 Ž.1992 , 123᎐128. 5. M. Finzel, On the continuity of the strict approximation, Rend. Circ. Mat. Palermo() 2 Suppl. 33 Ž.1993 , 301᎐310. ϱ 6. M. Finzel, Linear approximation in ln, J. Approx. Theory 76 Ž.1994 , 326᎐350. 7. M. Finzel, Best linear approximation in ln1Ž.,East J. Approx. 2 Ž.1996 , 1᎐30. 8. J. Fischer, The convergence of the best discrete linear Lp approximation as p ª 1, J. Approx. Theory 39 Ž.1983 , 374᎐385. 9. R. Fletcher, J. A. Grant, and M. D. Hebden, Linear minimax approximation as the limit of best Lp-approximation, SIAM J. Numer. Anal. 11 Ž.1974 , 123᎐136. 10. D. Flores de Chela, Approximations in the lϱ norm and the generalized inverse, Linear Algebra Appl. 42 Ž.1982 , 3᎐28. HOFFMAN’S ERROR BOUNDS 201 11. A. J. Hoffman, Approximate solutions of systems of linear inequalities, J. Res. Nat. Bur. Standards 49 Ž.1952 , 263᎐265. 12. R. Holmes and B. Kripke, Smoothness of approximation, Michigan Math. J. 15 Ž.1968 , 225᎐248. 13. R. Huotari and W. Li, The continuity of metric projection in lnϱŽ., the Polya´ algorithm, the strict best approximation, and tubularity of convex sets, J. Math. Anal. Appl. 182 Ž.1994 , 836᎐856. 14. D. Jackson, Note on the median of a set of numbers, Bull. Amer. Math. Soc. 27 Ž.1921 , 160᎐164. 15. D. Landers and L. Rogge, Natural choice of L1-approximants, J. Approx. Theory 33 Ž.1981 , 268᎐280. 16. W. Li, Best approximations in polyhedral spaces and linear programs, in ‘‘Approximation Theory’’Ž. G. Anastassiou, Ed. , pp. 393᎐400, Lecture Notes in Pure and Applied Mathe- matics, Vol. 138, Dekker, New York, 1992. 17. W. Li, A. J. Hoffman’s theorem and metric projections in polyhedral spaces, J. Approx. Theory 75 Ž.1993 , 107᎐111. 18. W. Li, The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program, Linear Algebra Appl. 187 Ž.1993 , 15᎐40. 19. B. S. Mityagin, Extremal points of a family of convex functions, Siberian Math. J. 6 Ž.1965 , 556᎐563. 20. E. M. Stein, Singular integrals and differentiability properties of functions, in ‘‘Princeton Math. Ser.,’’ Vol. 30, Princeton Univ. Press, Princeton, NJ, 1970.