THE STRUCTURED DISTANCE TO ILL-POSEDNESS FOR CONIC SYSTEMS. A.S. Lewis. April 18, 2003

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1 THE STRUCTURED DISTANCE TO ILL-POSEDNESS FOR CONIC SYSTEMS A.S. Lews Aprl 18, 2003 Key words: condton number, conc system, dstance to nfeasblty, structured sngular values, sublnear maps, surjectvty AMS 2000 Subject Classfcaton: Prmary: 15A12, 90C31 Secondary: 65F35, 93B35 Abstract An mportant measure of condtonng of a conc lnear system s the sze of the smallest structured perturbaton makng the system ll-posed. We show that ths measure s unchanged f we restrct to perturbatons of low rank. We thereby derve a broad generalzaton of the classcal Eckart-Young result characterzng the dstance to llposedness for a lnear map. 1 Introducton Consder two fnte-dmensonal normed spaces X and Y, a fxed convex cone K X, and a lnear mappng A : X Y. We call A well-posed f AK = Y. In partcular, n the purely lnear case K = X, well-posedness concdes wth surjectvty. Our nterest s n the dstance to ll-posedness : that s, we seek the smallest structured lnear perturbaton A : X Y such that the perturbed mappng A + A s not well-posed. When K = X and the Department of Mathematcs, Smon Fraser Unversty, Burnaby, BC V5A 1S6, Canada. aslews@sfu.ca, aslews. Research supported by NSERC. 1

2 structure of perturbatons s unrestrcted, the classcal Eckart-Young theorem dentfes the dstance to ll-posedness as the smallest sngular value of A. For more general convex cones K, and unstructured perturbatons, semnal work of Renegar [8, 9] relates the dstance to ll-posedness to the complexty of solvng assocated lnear programs. Imposng structure on the allowable perturbatons (n order, for example, to mantan a sparsty pattern n the map A) leads to a consderably more nvolved theory. In the purely lnear case K = X, such questons arse as structured sngular value calculatons n the area of control theory, poneered by Doyle, known as µ-analyss [3]. In ths artcle we follow qute closely the approach of Peña [7] n consderng structured perturbatons to general conc systems. We depend heavly on the same rank-one reducton technque used n [7] and ntroduced n [5, 6]. Our approach dffers n several respects. Frst, we develop the theory n the concse and elegant language of sublnear set-valued mappngs (n other words, mappngs whose graphs are convex cones). Ths noton substantally generalzes the dea of a conc convex system: well-posedness becomes the noton of surjectvty of the mappng. (In ths framework, the unstructured case was developed n [4], and generalzed n [2].) Secondly, the structured perturbatons we consder are rather general, beng of the form P T Q for lnear mappngs T (where the lnear mappngs P and Q are fxed at the outset). Thrdly, we allow arbtrary norms on the underlyng spaces. Lastly, our proofs consst of drect dualty arguments, avodng the necessty of lftng problems nto hgher dmensonal spaces. In ths manner we hope to llumnate the structural smplcty of the key results. The man result s as follows. We consder fnte-dmensonal normed spaces X, Y, U, V, lnear mappngs P : V Y and Q : X U (for = 1, 2,..., k), and a surjectve set-valued mappng F : X Y wth graph a closed convex cone. Then, denotng dual spaces and adjont mappngs by, the followng four quanttes are equal: mn max T : F + lnear T P T Q nonsurjectve } ; mn max T : F + rank-one lnear T P T Q nonsurjectve } ; mn max u U, z 0, 0 y Y mn v V, v 1 z P y : sup mn x X, w >0 w z Q u F (y ), u 1 } ; Q x : w P v F (x) }. 2

3 2 Rank-one perturbaton As observed by Peña [5, 6], the dea of rank-one pertubaton s fundamental to the theory of the dstance to ll-posedness. Our frst, elementary result tres to capture the underlyng dea n a way that extends to structured perturbatons. Throughout ths artcle we follow the termnology of [12]. We call a setvalued mappng F : X Y postvely-homogeneous f ts graph gph F = (x, y) X Y : y F (x)} s a cone (whch s to say, nonempty and closed under nonnegatve scalar multplcaton). To recapture the theory of conc lnear systems we typcally consder examples of the form Ax} (x K) F (x) = (x K), where the mappng A : X Y s lnear and K X s a convex cone. The nverse of a set-valued mappng F s the mappng F 1 : Y X defned by x F 1 (y) y F (x). We call F sngular f F 1 (0) 0}. We typcally denote the norm on a normed space X by (or by X f we wsh to be specfc) and the closed unt ball n X by B X, and we denote the space of lnear mappngs from X to Y by L(X, Y ). In partcular, for a mappng A L(X, Y ), we denote the usual operator norm by A. We denote the dual space of X by X, and we wrte the acton of a lnear functonal x X on an element x X as x, x. We are partcularly nterested n rank-one mappngs n L(X, Y ), whch are those mappngs of the form x X x, x y for some gven elements x X and y Y : we denote the set of such mappngs by L 1 (X, Y ). The norm of ths mappng s just x y. In what follows, we nterpret 1/0 = + and 1/+ = 0. Theorem 2.1 (rank-one reducton) Consder fnte-dmensonal normed spaces X, Y, U, V, a postvely-homogeneous set-valued mappng F : X Y, 3

4 and lnear mappngs P : V Y and Q : X U. Then the quantty n [0, + ] defned by α = } nf T : F + P T Q sngular T L(U,V ) s unchanged f we further restrct the nfmum to be over mappngs T of rank one. Furthermore, f we assume 0 F (x) and x 0 Qx 0 (as holds n partcular f Q s njectve or F s nonsngular), then 1 α = sup } Qx : P v F (x). x X, v B V Note We address the queston of the attanment n the above nfmum and supremum n the next secton. Proof Denote the rght hand sde of the last equaton by β. Consder frst the case where F s sngular. In ths case, clearly α = 0, and s attaned by choosng the rank-one mappng T = 0. Choose any nonzero x 1 F 1 (0), so by assumpton, Qx 1 0. Now by choosng x = λx 1 wth λ R + and v = 0 n the defnton of β, and lettng λ grow, we see β = +, so the result holds. We can therefore assume F s nonsngular. We next show α 1/β. Consder any feasble mappng T n the defnton of α, so there exsts a nonzero vector x (F + P T Q) 1 (0). Hence we have P T Qx F (x), so snce F 1 (0) = 0}, we deduce T Qx 0. Postve homogenety now mples so by defnton, P ( 1 T Qx T Qx) F ( 1 T Qx x), β Q T Qx 1 x 1 T. Thus all feasble T satsfy T β, and we deduce α 1/β. Next we defne the quantty γ = nf T : (F + P T Q) 1 (0) 0} }. T L 1 (U,V ) 4

5 Clearly we have the nequalty γ α, so t now suffces to prove γ 1/β. If β = 0 there s nothng to prove, so we can assume β > 0. Consder any feasble vectors x and v n the defnton of β. Snce β > 0 we can assume Qx 0. There exsts a norm-one lnear functonal u U satsfyng u, Qx = Qx. Now we have 0 F (x) P v = F (x) P T Qx where T : U V s the rank-one lnear map defned by T u = u, u Qx v. Snce we know u = 1 and v 1, we deduce γ T 1 Qx, so 1/γ Qx. Fnally, takng the supremum over all feasble vectors x and v n the defnton of β shows 1/γ β, as requred. Notce that, f X = Y, the mappng F s sngle-valued and lnear, and the mappngs P and Q are just the dentty, then we recover the classcal Eckart- Young theorem. We next generalze to perturbatons wth a composte structure. In conformty wth our prevous usage, for z R + we defne z 0 = + (z > 0) 0 (z = 0). Corollary 2.2 (rank-one reducton for sums) Gven fnte-dmensonal normed spaces X, Y, U, V, a postvely-homogeneous set-valued mappng F : X Y, and lnear mappngs P : V Y and Q : X U (for = 1, 2,..., k), the quantty α := nf max T : F + P T Q sngular } T L(U,V ) s unchanged f we further restrct the nfmum to be over mappngs T of rank one. Consequently we have the followng: z α = nf max v B V, z R +, 0 x X Q x : z P v F (x) } 5

6 = nf max v : v V, u B U, 0 x X u, Q x P v F ( x), u, Q x 0 }. Note As before, we address the queston of the attanment n the above nfma n the next secton. Proof Fx any real ɛ > 0 and consder any feasble mappngs T n the above nfmum. By applyng the precedng theorem we see there exsts a mappng ˆT k L 1 (U k, V k ) satsfyng ˆT k < T k + ɛ and ( F + k 1 =1 P T Q + P k ˆTk Q k ) 1(0) 0}. We can contnue n ths fashon, arrvng at mappngs ˆT L 1 (U, V ) satsfyng ˆT < T + ɛ (for = 1, 2,..., k) and ( ) 1(0) F + P ˆT Q 0}. Snce ɛ > 0 was arbtrary, the rank-one reducton now follows. Consequently, we have α = α 1, where where α 1 := nf max v u v V, u U, 0 x X : α 2, nf max v u v B V, u U, 0 x X : u, Q x P v F (x) } u, Q x P v F (x) } α 2 := nf max u v B V, u U, 0 x X : u, Q x P v F (x) }. On the other hand, suppose the vectors v, u and x are feasble n the nfmum defnng α 1. If we defne, for each ndex, (ˆv, û ( v ) = 1 v, v u ) (v 0) (0, 0) (v = 0), then the vectors ˆv, û and x are feasble n the nfmum defnng α 2, and û = v u for each. Ths proves α 2 α 1, so n fact α = α 1 = α 2. 6

7 A completely analogous argument shows α = nf max v : v V, u B U, 0 x X u, Q x P v F ( x) }. The fnal expresson for α clamed n the theorem now follows, snce the addtonal condtons u, Q x 0 mpose no essental restrcton: for any ndex we can always replace the par of vectors (v, u ) wth ( v, u ) wthout changng feasblty or the objectve value. Consderng the defnton of α 2, we observe, for any vectors v, nf max u u U, 0 x X : u, Q x P v F (x) } = nf max u u U, 0 x X, z R + : z P v F (x), u, Q x = z, } snce a feasble choce of the varables on the rght hand sde mmedately gves a feasble choce on the left hand sde wth the same objectve value, whle for any feasble choce of vectors u and x on the left hand sde, settng û = (sgn u, Q x )u and ẑ = u, Q x for each ndex gves a feasble choce on the rght hand sde wth the same objectve value. By observng that, for any vector x X and scalar z R +, we have } nf u u U : u z, Q x = z = Q x, the result now follows. Note It s not hard to see that the case k = 1 gves back Theorem Dualty and surjectvty We return to our motvatng example of the well-posedness of a lnear mappng A : X Y relatve to a convex cone K X (by whch we mean AK = Y ). If, as before, we defne an assocated set-valued mappng F : X Y by (3.1) Ax} (x K) F (x) = (x K), then well-posedness holds exactly when F (X) = Y. 7

8 We call a general set-valued mappng F : X Y surjectve f F (X) = Y, closed f ts graph s closed, and sublnear f ts graph s a convex cone. Sublnear set-valued mappngs are also known as convex processes. The notons of sngularty and surjectveness are ntmately connected va dualty: the adjont of F s the set-valued mappng F : Y X defned by x F (y ) y, y x, x whenever y F (x). The adjont s easly seen to be closed and sublnear, and concdes wth the classcal noton for sngle-valued lnear mappngs. More generally, drect calculaton shows that for any lnear mappng G : X Y we have (F +G) = F +G. It s smple to check that the adjont of the set-valued mappng (3.1) s defned by F (y ) = A y + K, where K X s the usual (negatve) polar cone for K. The relatonshp between surjectveness and sngularty s descrbed by the followng concse result, a specal case of an nfnte-dmensonal verson of the open mappng theorem [1]. Theorem 3.2 (open mappng) For fnte-dmensonal normed spaces X and Y, a closed sublnear set-valued mappng F : X Y s surjectve f and only f ts adjont mappng F s nonsngular. Note 3.3 If the closed sublnear set-valued mappng F s surjectve, then so s the mappng F + G for all small lnear mappngs G, and the analogous result also holds for nonsngularty [10]. Hence wth ths assumpton on F n Theorem 2.1 (rank-one reducton), the nfmum } nf T : F + P T Q sngular T L(U,V ) s attaned whenever fnte, snce t seeks the norm of the smallest element n a nonempty closed set. In ths case, followng the proof shows both the same nfmum over the rank-one mappngs T and the supremum } sup Qx : P v F (x) x X, v B V are also attaned. 8

9 Note 3.4 Usng the precedng note, f the closed sublnear set-valued mappng F s surjectve n Corollary 2.2 (rank-one reducton for sums), then the nfmum nf T L(U,V ) max T : F + P T Q sngular } s attaned whenever fnte, whether over general or rank-one lnear mappngs T, and n ths case the nfmum s also attaned. nf max v B V, z R +, 0 x X z Q x : z P v F (x) } Usng the open mappng theorem (3.2), we can quckly derve a verson of Corollary 2.2 (rank-one reducton for sums) for nonsurjectvty rather than sngularty. Theorem 3.5 (rank reducton and surjectvty) For any fnte-dmensonal normed spaces X, Y, U, V, closed sublnear set-valued mappng F : X Y, and lnear mappngs P : V Y and Q : X U (for = 1, 2,..., k), the quantty α := nf max T : F + P T Q nonsurjectve } T L(U,V ) s unchanged f we further restrct the nfmum to be over mappngs T of rank one, and n fact z α = nf max u B U, z R +, 0 y Y P y : = nf max u v B V, u U, 0 y Y : Furthermore, all four nfma are attaned f α s fnte. z Q u F (y ) } y, P v Q u F ( y ), y, P v 0 }. Proof By the open mappng theorem, we have α = nf max T : ( F + ) } P T Q sngular T L(U,V ) = nf max T T L(U,V ) : F + Q T P sngular }, 9

10 snce the adjont transformaton : L(U, V ) L(V, U ) leaves the norm fxed. Ths transformaton s n fact a bjecton, whch also preserves the classes of rank-one mappngs. Corollary 2.2 ensures the nfmum s unchanged f we restrct to mappngs T for whch T s rank-one, or n other words to rank-one T, as requred. The fnal expressons follow drectly from Corollary 2.2. The fnal clam concernng attanment follows from Note Dualty Our ultmate am s to express the structured dstance to nonsurjectvty n terms nvolvng the mappng F rather than ts adjont. For ths purpose, the followng result s crucal. Theorem 4.1 (theorem of the alternatve) For any fnte-dmensonal normed spaces X, Y, U, surjectve closed sublnear set-valued mappng F : X Y, lnear mappngs Q : X U, and vectors y Y (for = 1, 2,..., k), exactly one of the followng two systems has a soluton: ( ) ( ) w y F (x), Q x < w R for each, x X; y, y Q u F ( y ), 0 y Y, y, y 0 and u B U for each. Proof Suppose frst that both systems have solutons. By the defnton of the adjont, we deduce the nequalty y, w y y, y Q u, x or equvalently 0 y, y ( w + u, Q x ). Now each term n the sum on the rght hand sde s a product of two factors, the frst of whch s nonnegatve and the second of whch s strctly postve. Hence ths nequalty can only hold f y, y = 0 for each ndex, and n ths case we deduce 0 F ( y ). But the mappng F s surjectve, so by the open mappng theorem (3.2) ts adjont F s nonsngular, and ths s a contradcton. Hence at most one of the two systems has a soluton. 10

11 Suppose now that system () has no soluton. Then the two convex subsets of X R k (x, w) : w y F (x) } and (x, w) : Q x < w for each } are dsjont. Both sets are clearly nonempty, so there exsts a separatng hyperplane: there exsts a nonzero vector (x, w ) X R k and a real µ such that the two mplcatons w y F (x) x, x Q x < w for each x, x w w µ w w µ. Consderng the frst mplcaton, by the postve homogenety of F, we deduce (4.2) w y F (x) x, x w w 0. and µ 0. Ths, n conjuncton wth the second mplcaton, shows (4.3) w 0 for each, and x, x w Q x for all x X. Ths nequalty expresses the fact that the vector x s a subgradent at the orgn for the convex functon x w Q x, so by standard convex analyss we deduce (4.4) x w Q B U. We now apply a rather standard dualty argument to the mplcaton (4.2). We defne a functon f : Y [, + ] by f(y) = nf x, x w w : y + x X, w R 11 w y F (x) }.

12 Implcaton (4.2) shows f(0) = 0, and a standard elementary argument usng the convexty of the graph of F shows f s convex. Snce the mappng F s surjectve, the functon f never takes the value +. Consequently (see [11]), f has a subgradent y Y at the orgn, or n other words, y + w y F (x) y, y x, x w w. Settng x = 0 and y = w y shows ( w w y, y ) 0 for all w R k, so (4.5) w = y, y for each. Furthermore, settng each w = 0 shows or n other words, (4.6) y F (x) y, y x, x, x F ( y ). Fnally, puttng together the relatonshps (4.3), (4.4), (4.5), and (4.6), shows we have constructed a soluton to system () n the theorem statement, as requred. A helpful restatement of the above theorem s contaned n the followng dualty result. Recall our conventon z/0 = + for real z > 0. Theorem 4.7 (dualty) Consder fnte-dmensonal normed spaces X, Y, U, a surjectve closed sublnear set-valued mappng F : X Y, lnear mappngs Q : X U, and vectors y Y (for = 1, 2,..., k). Then the functon Φ : Y k [0, + ] defned by (4.8) Φ ( (y ) ) = s lower semcontnuous, and sup mn x X, 0<w R w Q x : w y F (x) }. Φ ( (y ) ) = nf max u u U, 0 y Y : y, y Q u F ( y ), y, y 0 }. Furthermore, the nfmum on the rght hand sde s attaned whenever fnte. 12

13 Proof We frst prove the lower semcontnuty. For each ndex consder a sequence of vectors y r y n the space Y, and consder a sequence of reals s r s as r satsfyng s r Φ((y r )), or n other words (4.9) x X, 0 < w R and w y r F (x) s r mn w Q x. Consder reals w > 0 (for each ) satsfng w y F ( x). We want to show the nequalty w s mn Q. x To see ths, we frst note that, snce F s surjectve, t s everywhere open: the mage under F of any open set s open. In partcular, for any real δ > 0, the set F ( x + nt δb X ) s an open neghbourhood of the vector w y, so for large r must contan the pont w y r. Usng ths tool, we see there exsts a subsequence R of the natural numbers such that Applyng property (4.9) shows w y r F (x r ) for all r R lm r, r R xr = x. s r mn w Q x r for all r R. Hence there exsts an ndex j 1, 2,..., k} and a further subsequence R of R such that s r w j Q j x r for all r R. Takng the lmt as r shows s w j Q j x mn w Q, x as requred. Thus the functon Φ s ndeed lower semcontnuous. Denote the rght hand sde of the second clamed expresson for Φ by Ψ((y )): we next want to prove that ths nfmum s attaned whenever Ψ((y )) s fnte. Notce that the nfmum s unchanged f we add the condton y = 1, usng postve homogenety. Now suppose that the nfmum s 13

14 fnte, so there exst feasble vectors ū then we can rewrte the nfmum as and ȳ. If we defne β = max ū, nf max u u U, y Y : y, y Q u F ( y ), y = 1, y, y 0, u β }. Ths s the nfumum of a contnuous functon over a nonempty compact set, so s attaned. It remans to prove that the two functons Φ and Ψ are dentcal. Consder any real ψ > 0. Usng the attanment property we have just proved for Ψ, the statement Ψ((y )) ψ s equvalent to the solvablty of the system y, y Q u F ( y ), 0 y Y y, y 0, u ψb U for each, or equvalently, to the solvablty of the system y, ψ 1 y Q u F ( y ), 0 y Y y, ψ 1 y 0, u B U for each. Usng the theorem of the alternatve (4.1), ths s equvalent to the unsolvablty of the system w ψ 1 y F (x), Q x < w R for each, x X, or equvalently (snce F s postvely homogeneous), to the unsolvablty of the system w y F (x), ψ < w Q x, 0 < w R for each, x X. But ths n turn s equvalent to the statement Φ((y )) ψ. To summarze, we have shown, for all real ψ > 0, Ψ ( (y ) ) ψ Φ ( (y ) ) ψ. The result now follows. 14

15 5 The man result We now have all the tools we need to derve our man result. Theorem 5.1 (dstance to nonsurjectvty) For any fnte-dmensonal normed spaces X, Y, U, V, closed sublnear surjectve set-valued mappng F : X Y, and lnear mappngs P : V Y and Q : X U (for = 1, 2,..., k), the followng four quanttes are equal: nf max T : F + T L(U,V ) P T Q nonsurjectve } ; nf rank-one T L(U,V ) max T : F + P T Q nonsurjectve } ; nf max u B U, z 0, 0 y Y z P y : z Q u F (y ) } ; nf v B V sup mn x X, w >0 w Q x : w P v F (x) }. Furthermore, f these quanttes are fnte, each nfmum above s attaned. Proof The equalty of the frst three expressons follows mmedately from Theorem 3.5 (rank reducton and surjectvty). The last expresson also follows from the same result, after applyng the dualty theorem (4.7). References [1] J.M. Borwen. Norm dualty for convex processes and applcatons. Journal of Optmzaton Theory and Applcatons, 48:53 64, [2] A.L. Dontchev, A.S. Lews, and R.T. Rockafellar. The radus of metrc regularty. Transactons of the Amercan Mathematcal Socety, 355: , [3] J. Doyle. Analyss of feedback systems wth structured uncertanty. IEEE Preceedngs, 129: , [4] A.S. Lews. Ill-condtoned convex processes and lnear nequaltes. Mathematcs of Operatons Research, 24: ,

16 [5] J. Peña. Condton numbers for lnear programmng. PhD thess, Cornell Unversty, [6] J. Peña. Understandng the geometry of nfeasble perturbatons of a conc lnear system. SIAM Journal on Optmzaton, 10: , [7] J. Peña. A characterzaton of the dstance to nfeasblty under structured perturbatons. Lnear Algebra and ts Applcatons, To appear. [8] J. Renegar. Incorporatng condton measures nto the complexty theory of lnear programmng. SIAM Journal on Optmzaton, 5: , [9] J. Renegar. Lnear programmng, complexty theory and elementary functonal analyss. Mathematcal Programmng, 70: , [10] S.M. Robnson. Regularty and stablty for convex multvalued functons. Mathematcs of Operatons Research, 1: , [11] R.T. Rockafellar. Convex Analyss. Prnceton Unversty Press, Prnceton, N.J., [12] R.T. Rockafellar and R.J.-B. Wets. Varatonal Analyss. Sprnger, Berln,