EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL

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1 MATHEMATICS OF OPERATIONS RESEARCH Vol. 28, No. 3, August 2003, Printed in U.S.A. EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL AHARON BEN-TAL, ARKADI NEMIROVSKI, and CORNELIS ROOS In this aer, we study semi-infinite systems of Linear Matrix Inequalities whih are generially NP-hard. For these systems, we introdue omutationally tratable aroximations and derive quantitative guarantees of their quality. As aliations, we disuss the roblem of maximizing a Hermitian quadrati form over the omlex unit ube and the roblem of bounding the omlex strutured singular value. With the hel of our omlex Matrix Cube Theorem we demonstrate that the standard saling uer bound on M is a tight uer bound on the largest level of strutured erturbations of the matrix M for whih all erturbed matries share a ommon Lyaunov ertifiate for the (disrete time) stability. 1. Introdution. Numerous aliations of semidefinite rogramming, eseially those in robust otimization (see, e.g., Ben-Tal and Nemirovski 1998, El-Ghaoui and Lebret 1997, El-Ghaoui et al. 1998, Ben-Tal and Nemirovski 2002b, Ben-Tal et al. 2000, and referenes therein) require roessing of semi-infinite systems of Linear Matrix Inequalities (LMIs) of the form (1) x 0 where x is the vetor of design variables, R N reresents erturbations of the data, x is a symmetri m m matrix whih is bi-affine, i.e., affine in x for fixed, and affine in for x fixed, R N is the set of data erturbations of magnitude not exeeding 1, and 0 is the unertainty level. As a simle and instrutive examle of this tye, onsider the following Lyaunov Stability Analysis Problem. We are given a nominal linear dynamial system (2) ż t = S z t The m m matrix S of the system is artitioned into retangular bloks S, = 1 k, of sizes l r. In real life, the bloks are affeted by erturbations S S + whih we assume to be norm-bounded:, where is the standard matrix norm (maximal singular value). Exet for this norm-boundedness assumtion, the erturbations are omletely free and may even deend on time. With this unertainty model, the atual desrition of the dynamial system beomes (3) ż t = S + L T t R ]z t (the matries R R r m, L R l r m are readily given by the ositions of the bloks). All we know about the erturbations t R l r is that they are measurable funtions of t suh that t for all t. Reeived November 6, 2001; revised January 13, MSC2000 subjet lassifiation. Primary: 90C34. OR/MS subjet lassifiation. Primary: Programming/nonlinear. Key words. LMI, semi-infinite system, aroximation X/03/2803/0497/$ eletroni ISSN, 2003, INFORMS

2 498 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS A basi question ertaining to a dynamial system is whether it is stable, i.e., whether all its trajetories tend to 0 as t. The standard suffiient stability ondition is that all matries S we an get from S by the erturbations in question share a ommon Lyaunov stability ertifiate, whih is a ositive definite matrix X suh that SX + XS T 0 By homogeneity reasons, the existene of suh a ommon Lyaunov stability ertifiate is equivalent to the existene of a ositive definite solution to the semi-infinite LMI (4) X I S X + XS T + L H R X + R X T T L 0 R l r = 1 k whih is of the generi form (1). The Lyaunov Stability Analysis examle is instrutive in two ways: it demonstrates the imortane of semi-infinite LMIs and suggests seifi ways of reresenting the erturbations, the set, and the matrix-valued funtion x aearing in (1). Namely, in this examle: 1) A erturbation is a olletion of erturbation bloks matries of given sizes l r, = 1 k. 2) The maing x is of the form (5) x = A x + L T R x + R T x T L ] where A x is a symmetri m m matrix, L R l m, R x R r m, and A x, R x are affine in x. 3) The set of erturbations of magnitude 1 is omrised of all olletions 1 k suh that R l r, 1, and, besides this, matries for resribed values of, are restrited to be salar (i.e., of the form I r ; of ourse, l = r for indies in question). In fat, in our motivating examle there was no need for salar erturbations; nevertheless, there are many reasons to introdue them rather than to allow all erturbations to be of full size. The simlest of these reasons is that with salar erturbations, the outlined erturbation model allows to reresent in the form of (5) every (affine in x and in ) funtion x with symmetri matrix values; to this end it suffies to treat every entry in R N as a salar matrix erturbation I m. We see that when salar erturbations are allowed, items 1) and 2) above do not restrit the exressive abilities of the erturbation model (rovided that we restrit ourselves to affine erturbations). What does restrit generality, is the art of item 3) whih says that the only restrition on = 1 k, exet for the requirement for some of the erturbation bloks to be salar matries, is the ommon norm bound 1 on all erturbation bloks. This assumtion rovides (1) with a seifi struture whih, as we shall see, allows for a rodutive roessing of (1). It makes sense to assume one for ever that the matries are square. This does not restrit generality, beause we an always enfore l = r by adding to L or to R a number of zero rows; it is easily seen that this modifiation does not affet anything exet for simlifying notation. From now on, we denote the ommon value of l and r by d. Tyial roblems assoiated with a semi-infinite LMI of the form (1) are to find a oint in the feasible set of the LMI and to minimize a linear objetive over this feasible set. These are onvex roblems with an imliitly defined feasible set; basially all we need to solve suh a roblem effiiently is a feasibility orale aable to solve effiiently the analysis

3 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 499 roblem as follows: Given x, hek whether x is feasible for 1 (for details on relations between analysis and synthesis in onvex otimization, see Grötshel et al or Ben- Tal and Nemirovski 2001, Chater 5). Note that with model 1) 3), the analysis roblem for (1) is the Matrix Cube Problem as follows: Given a symmetri m m matrix A, d m matries L R, = 1 k, and 0, hek whether all matries of the form A + L T R + R T T L where 1 for all and = I d for resribed values of, are ositive semidefinite. Unfortunately, the Matrix Cube Problem, same as the majority of other semi-infinite LMIs known from the literature, in general is NP-hard. However, it was found in Ben-Tal and Nemirovski (2002a) that when all erturbations are salar, the roblem admits a omutationally tratable aroximation whih is tight within a fator of = O 1 max d. Seifially, Given a Matrix Cube Problem with salar erturbation bloks, one an build an exliit system of LMIs in variables u of size olynomial in m and d with the following roerty: if is feasible, the answer in the Matrix Cube Problem is affirmative; if is infeasible, then the answer in the Matrix Cube Problem with the erturbation level relaed by is negative. Besides this, if the data A, R 1 R k in the Matrix Cube Problem deend affinely on a deision vetor x, while L 1 L k are indeendent of x (f. (5)), then is a system of LMIs in x u. As it is shown in Ben-Tal and Nemirovski (2002a), this result allows to build tight aroximations of several imortant NP-hard roblems of the form (1). The goal of this aer is to extend the aroah and the results of Ben-Tal and Nemirovski (2002a) from the ase of urely salar erturbations onto the more general erturbation model 1) 3). We onsider both the outlined model of real erturbations and its omlexvalued ounterart (whih is imortant for some of the ontrol aliations); in both real and omlex ases, we build omutationally tratable aroximations of the resetive Matrix Cube Problems and demonstrate that these aroximations are tight within a fator O 1 d s, where d s is the maximum of sizes d of salar erturbation bloks in. Surrisingly, the full size erturbation bloks, however large they are, do not affet the quality of the aroximation. The rest of this aer is organized as follows. In the remaining art of the Introdution, we fix the notation to be used. Setion 2 deals with the tehnially slightly more diffiult omlex ase version of the Matrix Cube Problem; the real ase of the roblem is onsidered in 3. In 4 and 5we illustrate our main results by their aliations to the roblem of maximizing a ositive definite quadrati form over the omlex ube z C m z 1, = 1 m, and to the roblem of bounding from above an imortant ontrol entity the omlex strutured singular value. Notation. C m n, R m n stand for the saes of omlex, resetively, real m n matries. As always, we write C n and R n as shorthands for C n 1, R n 1, resetively. For A C m n, A T stands for the transose, and A H for the onjugate transose of A: where z is the onjugate of z C. A H rs = A sr

4 500 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS Both C m n, R m n are equied with the inner rodut A B =Tr AB H = r s A rs B rs The norm assoiated with this inner rodut is denoted by 2. We use the notation I m, O m n for the unit m m, resetively, the zero m n matries. H m, S m are real vetor saes of m m Hermitian, resetively, real symmetri matries. Both are Eulidean saes w.r.t. the inner rodut. For a Hermitian/real symmetri m m matrix A, A is the vetor of eigenvalues r A of A taken with their multiliities in the nonasending order: and 1 A 2 A m A For an m n matrix A, A = 1 A n A T is the vetor of singular values of A: r A = 1/2 r A H A A = 1 A = max Ax 2 x C n x 2 1 (by evident reasons, when A is real, one an relae C n in the right-hand side with R n ). For Hermitian/real symmetri matries A, B, we write A B (A B) to exress that A B is ositive semidefinite (res., ositive definite). We denote by H+ n (Sn + ) the ones of ositive semidefinite Hermitian (res., ositive semidefinite real symmetri) n n matries. For X 0, X 1/2 denotes the ositive semidefinite square root of X (uniquely defined by the relations X 1/2 0, X 1/2 2 = X). On many oasions in this aer we use the term effiient omutability of various quantities. An aroriate definition of this notion does exist (for a definition that best fits the ontents of this aer, see Ben-Tal and Nemirovski 2001, Chater 5), but for our uroses here it suffies to agree that all LMI-reresentable quantities those whih an be reresented as otimal values in semidefinite rograms A i S K or generalized eigenvalue roblems A i B i S K { min T x A x A 0 + x { min A x A 0 + x N i=1 } x i A i 0 N N x i A i 0 B x B 0 + i=1 i=1 } x i B i A x are effiiently omutable funtions of the data A 0 A N, res., A 0 A N, B 0, B N. 2. Matrix Cube Theorem, omlex ase. The Comlex Matrix Cube (CMC) Problem is as follows: Let m, d 1 d k be ositive integers, and A H+ m, L R C d m be given matries, L 0. Let also a artition 1 2 k = Is r I s I f of the index set 1 k into three nonoverlaing sets be given. With these data, we assoiate a arametri family of matrix boxes (6) { = A + L H R + R H H L } 1 H m = 1 k

5 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 501 where 0 is the arameter and I d R Is r real salar erturbations ] (7) = I d C Is omlex salar erturbations ] C d d I f full size omlex erturbations ] Given 0, hek whether I H m + Remark 2.1. In the sequel, we always assume that d > 1 for Is. Indeed, onedimensional omlex salar erturbations an always be regarded as full size omlex erturbations. It is well known that the CMC Problem is, in general, NP-hard. Our goal is to build a omutationally tratable suffiient ondition for the validity of I and to understand how onservative is this ondition. Consider, along with rediate I, the rediate Y H m, = 1 k suh that II a Y L H R + R H H L 1 = 1 k b A Y 0 Our main result is as follows: Theorem 2.1 (The Comlex Matrix Cube Theorem). One has: (i) Prediate II is stronger than I the validity of the former rediate imlies the validity of the latter one. (ii) II is omutationally tratable the validity of the rediate is equivalent to the solvability of the system of LMIs (8) s R Y ± L H R + R H L 0 Is r Y V L H s C R ] R H L 0 Is V Y L H f C L ] R H 0 If I d A R Y 0 in the matrix variables Y H m, = 1 k, V H m, I s, and the real variables, I f. (iii) The ga between I and II an be bounded solely in terms of the maximal size (9) d s = max d I r s I s of the salar erturbations (here the maximum over an emty set by definition is 0). Seifially, there exists a universal funtion C suh that (10) and C 4 1 (11) if II is not valid, then I C d s is not valid

6 502 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS Corollary 2.1. The effiiently omutable suremum ˆ of those 0 for whih the system of LMIs (8) is solvable is a lower bound on the suremum of those 0 for whih H+ m, and this lower bound is tight within the fator C d s : (12) ˆ C d s ˆ Remark 2.2. From the roof of Theorem 2.1, it follows that C 0 = 4/, C 1 = 2. Thus, When there are no salar erturbations: Is r = I s =, the fator in the imliation (13) II I an be set to 4/ = When there are no omlex salar erturbations (f. Remark 2.1) and all real salar erturbations are nonreeated (Is =, d = 1 for all Is r ), the fator in (13) an be set to 2. Remark 2.3. From the roof of the Matrix Cube Theorem 2.1, it follows that its statement remains intat when in definition (6) of the matrix box, the restritions 1, If, on the norms of full size erturbations are relaed with the restritions 1, where are norms on C d d suh that for all C d d and = whenever is a rank 1 matrix (e.g., one an set to be the Frobenius norm 2 of a matrix). The following simle observation is ruial when alying Theorem 2.1 in the ontext of semi-infinite bi-affine LMIs of the form (1). Remark 2.4. Assume that the data A, R 1 R k of the Matrix Cube Problem are affine in a vetor of arameters x, while the data L 1 L k are indeendent of x (f. (5)). Then (8) is a system of LMIs in the variables Y, V,, and x Proof of Theorem 2.1. Item (i) is evident. We rove item (ii) and item (iii) is roved in Proof of Theorem 2.1(ii). The equivalene between the validity of II and the solvability of (8) is readily given by the following fats (the first of them is erhas new): Lemma 2.1. Let B C m m and Y H m. Then, the relation (14) Y B + B H C 1 is satisfied if and only if (15) V H m Y V B H B V ] 0 Lemma 2.2 (See Boyd et al. 1994). Let L C l m and R C r m. (i) Assume that L R are nonzero. A matrix Y H m satisfies the relation (16) Y L H UR+ R H U H L U C l r U 1 if and only if there exists a ositive real suh that (17) Y L H L + 1 R H R (ii) Assume that L is nonzero. A matrix Y H m satisfies (16) if and only if there exists R suh that ] Y L H L R H (18) 0 R I r

7 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 503 Lemmas 2.1, 2.2 Theorem 2.1(ii). All we need to rove is that a olletion of matries Y satisfies the onstraints in II if and only if it an be extended by roerly hosen V, If, and, Is, to a feasible solution of (8). This is immediate, beause the matries Y, If, satisfy the orresonding onstraints II a if and only if these matries along with some matries V satisfy (8.s.C (Lemma 2.1), while matries Y, Is, satisfy the orresonding onstraints II a if and only if these matries along with some reals satisfy (8.f.C) (Lemma 2.2(ii)). Proof of Lemma 2.1. If art: Assume that V is suh that ] Y V B H 0 B V Then, for every C n and every C, =1, we have ] H ] ] Y V B H 0 B V = H Y V + H V H B + B H so that Y B + B H for all C, =1, whih, by evident onvexity reasons, imlies (14). Only if art: Let Y H m satisfy (14). Assume, on the ontrary to what should be roved, that there does not exist V H m suh that ] Y V B H 0 B V and let us lead this assumtion to a ontradition. Observe that our assumtion means that the otimization rogram (19) { min t t V tim + Y V B H B V ] } 0 has no feasible solutions with t 0; beause roblem (19) is learly solvable, its otimal value is therefore ositive. Now, our roblem is a oni roblem on the (self-dual) one of ositive semidefinite Hermitian matries. (For bakground on oni roblems and oni duality, see Nesterov 1994, Chater 4 or Ben-Tal and Nemirovski 2001, Chater 2.) Beause the roblem learly is stritly feasible, the Coni Duality Theorem says that dual roblem (20) max Z H m W C m m 2R Tr W H B Tr ZY ] Z W H 0 W Z Tr Z = 1 a b is solvable with the same ositive otimal value as the one of (19). In (20), we an easily eliminate the W -variable; indeed, onstraint (20a), as it is well known, is equivalent to the fat that Z 0 and W = Z 1/2 XZ 1/2 with X C m m, X 1. With this arameterization of W, the W -term in the objetive of (20) beomes 2R Tr X H Z 1/2 BZ 1/2 ; as it is well known, the maximum of the latter exression in X, X 1, equals to 2 Z 1/2 BZ 1/2 1. Beause the otimal value in (20) is ositive, we arrive at the following intermediate onlusion: There exists Z H m, Z 0, suh that (21) 2 Z 1/2 BZ 1/2 1 > Tr ZY = Tr Z 1/2 YZ 1/2 The desired ontradition is now readily given by the following simle observation:

8 504 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS Lemma 2.3. Let S H m, C C m m be suh that (22) S C + C H C =1 Then, 2 C 1 Tr S. To see that Lemma 2.3 yields the desired ontradition, note that the matries S = Z 1/2 YZ 1/2, C = Z 1/2 BZ 1/2 satisfy the remise of the lemma by (14), and for these matries the onlusion of the lemma ontradits (21). Proof of Lemma 2.3. As it was already mentioned, { C 1 = max R Tr XC H X 1 } X Beause the extreme oints of the set X C m m X 1 are unitary matries, the maximizer X in the right-hand side an be hosen to be unitary: X H = X 1 ; thus, X is a unitary similarity transformation of a diagonal unitary matrix. Alying aroriate unitary rotation A U H AU, U H = U 1, to all matries involved, we may assume that X itself is diagonal. Now we are in the situation as follows: we are given matries C S satisfying (22) and a diagonal unitary matrix X suh that C 1 =R Tr X C H. In other words, (23) { m C 1 =R X ll C ll l=1 } m C ll l=1 (the onluding inequality omes from the fat that X is unitary). On the other hand, let e l be the standard basi orths in C m. By (22), we have C ll + C ll = eh l C + C H e l e H l Se l = S ll C =1 whene, maximizing in, 2 C ll S ll, l = 1 m, whih ombines with (23) to imly that 2 C 1 Tr S. Proof of Lemma 2.2. (i) if art: Let (17) be valid for ertain >0. Then, for every C m one has H Y H L H L + 1 H R H R 2 H L H L H R H R = 2 L 2 R 2 U U 1 H Y 2 L H U R 2R L H U R = H L H UR+ R H U H L as laimed. (i) Only if art: Assume that Y satisfies (16) and L R are nonzero. We rove that then there exists >0 suh that (17) holds true. First, observe that w.l.o.g. we may assume that L and R are of the same sizes r n (to redue the general ase to this artiular one, it suffies to add several zero rows either to L (when l<r), or to R (when l>r)). We have

9 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 505 the following hain of equivalenes: 16 (24) C m H Y 2 L 2 R 2 C n C r 2 L 2 H Y H R H R H 0 C m C r H L H L H 0 H Y H R H R H 0 ] Y R H L H L 0 R I r Y L H L R H a R I r ] 0 ] 0 S-Lemma] Reall that the -Lemma we have referred to is the following extremely useful statement: Let P Q be real symmetri matries of the same size suh that x T P x >0 for ertain x. Then, the imliation x T Px 0 x T Qx 0 holds true if and only if there exists 0 suh that Q P. From this real symmetri ase statement one an immediately derive its Hermitian analogy (the one we have atually used), beause Hermitian quadrati forms on C m an be treated as real quadrati forms on R 2m. Condition (24a), in view of R 0, learly imlies that >0. Therefore, by the Shur Comlement Lemma (SCL), (24a) is equivalent to Y L H L 1 R H R 0 as laimed. (ii). When R 0, (ii) is learly equivalent to (i) and thus is already roved. When R = 0, it is evident that (18) an be satisfied by roerly hosen R if and only if Y 0, whih is exatly what is stated by (16) when R = Proof of Theorem 2.1(iii). To rove (iii), it suffies to rove the following statement: Claim 2.1. Assume that 0 is suh that the rediate II is not valid. Then, the rediate I C d s, with aroriately defined funtion C satisfying (10), is also not valid. We are about to rove Claim 2.1. The ase of = 0 is trivial, so from now on we assume that >0 and that all matries L, R are nonzero (the latter assumtion, of ourse, does not restrit generality). From nowuntil the end of 2.1, we assume that we are under the remise of Claim 2 1, i.e., the rediate II is not valid.

10 506 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS First ste: duality. Consider the otimization rogram Y ± L H R + R H L 0 Is r }{{} 2A A =A H U R H L ] L H R 0 Is V ] 1 (25) min t 0 I t Y H m I r f s 1 U V H m I s R I f ti + A Y + U + V Is r Is + ] L H L + R H R 0 If a b d Introduing bounds Y = U + V for Is and Y L H L + R H R for If and then eliminating the variables U, Is,, If, we onvert (25) into the equivalent roblem Y ± L H R + R H L 0 Is r Y V R H L ] L H R 0 Is V t Y L H L ] R H 0 If I d min t Y H m k V H m I s R I f R ti + A Y 0 By (already roved) item (ii) of Theorem 2.1, rediate II is valid if and only if the latter roblem, and thus roblem (25), admits a feasible solution with t 0. We are in the situation when II is not valid; onsequently, (25) does not admit feasible solutions with t 0. Beause the roblem learly is solvable, it means that the otimal value in the roblem is ositive. Problem (25) is a oni roblem on the rodut of ones of Hermitian and real symmetri ositive semidefinite matries. Beause (25) is stritly feasible and bounded below, the Coni Duality Theorem imlies that the oni dual roblem of (25) is solvable with the same ositive otimal value. Taking into aount that the ones assoiated with (25) are self-dual, the dual roblem, after straightforward simlifiations, beomes the oni roblem maximize 2 Tr P Q A + R Tr S R H L + ] w Tr ZA Is r Is If s.t. a 1 P Q 0 Is r a 2 P + Q = Z Is r (26) Z S H ] b 0 Is S Z Tr L ZL H w ] w Tr R ZR H 0 If d Z 0 Tr Z = 1

11 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 507 in matrix variables Z H+ m, P Q H m, Is r, S C m m, Is, and real variables w, If. Using (26), we an eliminate the variables w, thus obtaining the following equivalent reformulation of the dual roblem: (27) maximize 2 Tr P Q A R Tr S R H L Is r Is + I f Tr L ZL H }{{} L Z 1/2 2 s.t. a 1 P Q 0 Is r a 2 P + Q = Z Is r Z S H ] b 0 Is S Z Z 0 Tr Z = 1 Tr R ZR H }{{} R Z 1/2 2 ] Tr ZA Next, we eliminate the variables S, Q, and R. It is lear that (1) (27a) is equivalent to the fat that P = Z 1/2 P Z 1/2, Q = Z 1/2 Q Z 1/2 with P Q 0, P + Q = I m. With this arameterization of P Q, the orresonding terms in the objetive beome 2 Tr P Q Z 1/2 A Z 1/2. Note that the matries A are Hermitian (see (25)), and observe that if A H m, then max P Q H m { Tr P Q A 0 P Q P + Q = Im } = A 1 l l A (w.l.o.g., we may assume that A is Hermitian and diagonal, in whih ase the relation beomes evident). In view of this observation, artial otimization in P, Q in (27) allows to relae in the objetive of the roblem the terms 2 Tr P Q A with 2 Z 1/2 A Z 1/2 1 and to eliminate the onstraints (27a). (2) Same as in the roof of Lemma 2.1, onstraints (27b) are equivalent to the fat that S = Z 1/2 U Z 1/2 with U 1. With this arameterization, the orresonding terms in the objetive beome 2 R Tr U Z 1/2 R H L Z 1/2, and the maximum of this exression in U, U 1, equals 2 Z 1/2 R H L Z 1/2 1. With this observation, artial otimization in S in (27) allows to relae in the objetive of the roblem the terms 2 R Tr S R H L with 2 Z 1/2 R H L Z 1/2 1 and to eliminate the onstraints (27b). After the above redutions, roblem (27) beomes (28) maximize 2 Z 1/2 A Z 1/2 1 + Z 1/2 R H L Z 1/2 1 Is r Is + ] L Z 1/2 2 R Z 1/2 2 Tr ZA If s.t. Z 0 Tr Z = 1 Reall that we are in the situation when the otimal value in roblem (26), and thus in roblem (28), is ositive. Thus, we arrive at an intermediate onlusion as follows.

12 508 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS (29) Lemma 2.4. Under the remise of Claim 1, there exists Z H m, Z 0, suh that 2 Z 1/2 A Z 1/2 1 + Z 1/2 R H L Z 1/2 1 Is r Is Here, the Hermitian matries A are given by + ] L Z 1/2 2 R Z 1/2 2 > Tr Z 1/2 AZ 1/2 If (30) 2A = L H R + R H L I r s Seond ste: Probabilisti interretation of (29). The major ste in omleting the roof of Theorem 2.1(iii) is based on a robabilisti interretation of (29). This ste is desribed next. Preliminaries. Let us define a standard Gaussian vetor in R n (notation: R n)as a real Gaussian random n-dimensional vetor with zero mean and unit ovariane matrix; in other words, l are indeendent Gaussian random variables with zero mean and unit variane, l = 1 n. Similarly, we define a standard Gaussian vetor in C n (notation: C n ) as a omlex Gaussian random n-dimensional vetor with zero mean and unit (omlex) ovariane matrix. In other words, l = l + i n+l, where 1 2n are indeendent real Gaussian random variables with zero means and varianes 1/2, and i is the imaginary unit. We shall use the fats established in the next three roositions. Proosition 2.1. Let be a ositive integer, and let S, H be given by the relations { { } } 1 S = min E l 2 l R 1 = 1 R l=1 (31) { { } 1 H = min E l l } 2 R 1 = 1 C Then, (i) Both S, H are nondereasing funtions suh that l=1 (32) a 1 S 1 = 1 S 2 = /2 a 2 S /2 1 b 1 H 1 = 1 H 2 = 2 b 2 H S 2 /2 1 (ii) For every A S n, one has (33) E T A A 1 1 S Rank A n R and for every A H n one has (34) E T A A 1 1 H Rank A n C Proof. The funtion S was introdued in Ben-Tal and Nemirovski (2002a), where (32a) and (33) were roved as well. From the definition of H, it is lear that this funtion is nondereasing. To establish (34), note that the distribution of a random omlex Gaussian vetor is invariant under unitary transformations of C n, hene it suffies to verify (34) in the artiular ase when A is a diagonal Hermitian matrix with Rank A nonzero diagonal

13 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 509 entries (the nonzero eigenvalues of A), and the remaining diagonal entries equal to 0. But in this ase (34) is readily given by the definition of H. It remains to verify (32b). The relation H 1 = 1 is evident. Further, we learly have 1 H 2 = min = E C 0 1 The funtion is onvex in 0 1 and is symmetri: 1 =. It follows that its minimum is ahieved at = 1/2; diret omutation demonstrates that 1/2 = 1/2, whih omletes the roof of (32b.1). It remains to rove the first inequality in (32b.2). Given R, 1 = 1, let = T T T R 2.Now,if = + i is a standard Gaussian vetor in C, then the vetor = 2 1/2 T T T is a standard Gaussian vetor in R 2.Wenowhave { } { } E l l 2 = E l 2 l + 2 l = 1 { } 2 2 E l 2 l S 2 l=1 l=1 = 1 S 2 whene H 1 1 S 2, and the desired inequality follows. Proosition 2.2. For every A C n n, one has (35) E H 1 A A 1 2 Rank A n C ] A Proof. Let A = A H 4 1 H l=1, so that A H 2n, Rank A = 2 Rank A, and the eigenvalues of A are ± l A, l = 1 n. Let also = T T T be a standard Gaussian vetor in C 2n artitioned into two n-dimensional bloks, so that are indeendent standard Gaussian vetors in C n.wehave (36) H A = 2R H A =R + H A + H A H A + i i H A i H A H A olarization identity] E H A E + H A + + E i H A i + 2E H A + 2E H A. Beause are indeendent standard Gaussian vetors in C n, the vetors 2 1/2 + and 2 1/2 i also are standard Gaussian. Therefore, (36) imlies that (37) E H A 8E H A Beause A is a Hermitian matrix of rank 2 Rank A and A 1 = 2 A 1, the left-hand side in (37) is at least 2 A 1 H 1 2 Rank A, and (37) imlies (35). Proosition 2.3. (i) Let L R C d n, and let be a standard Gaussian vetor in C n. Then, (38) E L 2 R 2 4 L 2 R 2 (ii) Let L R R d n, and let be a standard Gaussian vetor in R n. Then, (39) E L 2 R 2 2 L 2 R 2

14 510 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS Proof. (i) There is nothing to rove when L or R are zero matries; thus, assume that both L and R are nonzero. Let us demonstrate first that it suffies to verify (38) in the ase when both L and R are rank 1 matries. Let L H L = U H Diag U be the eigenvalue deomosition of L H L, so that U is a unitary matrix and 0. We have (40) E L 2 R 2 = E H L H L R 2 { } = E U H Diag U }{{} 1/2 RU H }{{} 2 0 { } n = E l l 2 = l=1 { n x = E x l l } 2 l=1 The funtion x of x R+ n is onave; therefore its minimum on the simlex { S = x R n + x l = } l l l is ahieved at a vertex, let it be e. Now let L C d n be suh that L H L = U H Diag e U. Note that L is a rank 1 matrix (beause e is a vertex of S) and that L 2 2 = Tr L H L = l e l = l l = Tr L H L = L 2 2 Beause the unitary fator in the eigenvalue deomosition of L H L is U, (40) holds true when L is relaed with L and with e, so that E L 2 R 2 = e = E L 2 R 2 Alying the same reasoning to the quantity E L 2 R 2 with R laying the role of L, we onlude that there exists a rank 1 matrix R suh that and R 2 = R 2 E L 2 R 2 E L 2 R 2 Thus, relaing L and R with the rank 1 matries L, R, we do not inrease the left-hand side in (38) and do not vary the right-hand side, so that it indeed suffies to establish (38) in the ase when L, R are rank 1 matries. Note that so far our reasoning did not use the fat that is standard Gaussian. Now let us look what inequality (38) says in the ase of rank 1 matries L, R. By homogeneity, we an further assume that L 2 = R 2 = 1. With this normalization, for rank 1 matries L, R, we learly have L = zl and R = wr for unit deterministi vetors l r and a Gaussian random vetor z w C 2 = R 4 suh that E z 2 = E w 2 = 1 (both z and w are just linear ombinations, with aroriate deterministi oeffiients, of the entries in ). Beause E z 2 = E w 2 = 1, we an exress z w in terms of a standard Gaussian vetor C 2 as z =, w = os + sin, where 0 /2 is suh

15 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 511 that os is the absolute value of the orrelation E zw between z and w. With this reresentation, inequality (38) beomes os + sin dg ( ) (41) C C 4 2 where G is the distribution of. We should rove (41) in the range 0 /2 of values of ; in fat we shall rove this inequality in the larger range 0.Given 0,weset u = os /2 + sin /2 v = sin /2 + os /2 it is immediately seen that the distribution of u v is exatly G. At the same time, = os /2 u sin /2 v os + sin = os /2 u + sin /2 v whene We see that = = C C C C os /2 u sin /2 v os /2 u + sin /2 v dg u v os 2 /2 u 2 sin 2 /2 v 2 dg u v min = min = u C C 2 1 v 2 dg u v The funtion learly is onvex and 1 = (beause the distribution of u v is symmetri in u v). Consequently, attains its minimum when = 1/2, and attains its minimum when os 2 /2 = 1/2, i.e., when = /2, whih is exatly what is stated in (41). (ii) Alying exatly the same reasoning as in the roof of (i), we onlude that it suffies to verify (39) in the ase when L R are real rank 1 matries. In this ase, the same argument as above demonstrates that (39) is equivalent to the fat that if are indeendent real standard Gaussian variables and G is the distribution of, then the funtion (42) = os + sin dg R R of 0 ahieves its minimum when = /2. To rove this statement, one an reeat word by word, with evident modifiations, the reasoning we have used in the omlex ase Comleting the roof of Theorem 2.1(iii). We are now in a osition to omlete the roof of Theorem 2.1(iii). Let us set (43) dr s = 2 max d Is r dc s = 2 max d Is = max H dr s 4 H dc s 4/ ] Here, by definition, the maximum over an emty set is 0, and H 0 = 0. Note that by (32) one has 4 d s (f. (9), (10)).

16 512 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS Let be a standard Gaussian vetor in C n. Invoking Proositions , we have (for notation, see Lemma 2.4) Z 1/2 A Z 1/2 1 H Rank Z 1/2 A Z 1/2 E H Z 1/2 A Z 1/2 E H Z 1/2 A Z 1/2 I r s by Proosition 2.1 beause A = A H and Rank A = Rank L H R + R H L 2d ] Z 1/2 R H L Z 1/2 1 4 H 2 Rank Z 1/2 R H L Z 1/2 E H Z 1/2 R H L Z 1/2 E H Z 1/2 R H L Z 1/2 I s by Proosition 2.2 beause Rank R H L d ] L Z 1/2 2 R Z 1/2 2 4 E L Z 1/2 2 R Z 1/2 2 E L Z 1/2 2 R Z 1/2 2 by Proosition 2.3(i)], and, of ourse, E H Z 1/2 AZ 1/2 = Tr Z 1/2 AZ 1/2 In view of these observations, (29) imlies that E H Z 1/2 L H R + R H L Z 1/2 + E 2 H Z 1/2 R H L Z 1/2 Is r Is + ] E 2 L Z 1/2 2 R Z 1/2 2 > E H Z 1/2 AZ 1/2 If (we have substituted the exressions for A ; see (30)). It follows that there exists a realization ˆ of suh that with = Z 1/2 ˆ one has H L H R + R H L + 2 H R H L + ] (44) 2 L 2 R 2 > H A Is r Is If Observe that The quantities H L H R + R H L are real; we therefore an hoose =±1, Is r, in suh a way that with = I d one has H L H R + R H H L = H L H R + R H L I r s For Is, we an hoose C, =1, in suh a way that with = I d one has H L H R + R H H L = 2 H R H L I s For I f, we an hoose C d d, 1, in suh a way that H L H R + R H H L = 2 L 2 R 2 I f With s we have defined, (44) reads H A ] L H R + R H H L <0 } {{ } C

17 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 513 so that C is not ositive semidefinite; on the other hand, by onstrution C. Thus, the rediate I is not valid; realling the definition of, this omletes the roof of Claim 2.1 and thus the roof of Theorem 2.1(iii). 3. Matrix Cube Theorem, real ase. The Real Matrix Cube (RMC) Problem is as follows: Let m, d 1 d k be ositive integers, and A S m +, L R R d m be given matries, L 0. Let also a artition 1 2 k = Is r I f r of the index set 1 k into two nonoverlaing sets be given. With these data, we assoiate a arametri family of matrix boxes { = A + L T R + R T T L R } 1 (45) S m = 1 k where 0 is the arameter and (46) { R = Id R Is r salar erturbations ] R d d I r f full size erturbations ] Given 0, hek whether I R S m + Remark 3.1. In the sequel, we always assume that d > 1 for Is r. Indeed, nonreeated (d = 1) salar erturbations always an be regarded as full size erturbations. The RMC roblem, same as the CMC one, is, in general, NP-hard; similar to the omlex ase, we intend to build a omutationally tratable suffiient ondition for the validity of I R and to understand how onservative is this ondition. Consider, along with rediate I R, the rediate II R Y S m = 1 k a Y L T R + R T T L R 1 = 1 k b A Y 0 The real ase version of Theorem 2.1 is as follows: Theorem 3.1 (The Real Matrix Cube Theorem). One has: (i) Prediate II R is stronger than I R the validity of the former rediate imlies the validity of the latter one. (ii) II R is omutationally tratable the validity of the rediate is equivalent to the solvability of the system of LMIs (47) s Y ± L T R + R T L 0 Is r Y L T f L ] R T 0 If r I d R A Y 0 in matrix variables Y S m, = 1 k, and real variables, I r f.

18 514 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS (iii) The ga between I R and II R an be bounded solely in terms of the maximal size d s = max d I r s of the salar erturbations (here the maximum over an emty set by definition is 0). Seifially, there exists a universal funtion R /2, 1, suh that (48) if II R is not valid, then I R R d s is not valid Corollary 3.1. The effiiently omutable suremum ˆ of those 0 for whih the system of LMIs (47) is solvable is a lower bound on the suremum of those 0 for whih S m +, and this lower bound is tight within the fator R d s : (49) ˆ R d s ˆ Remark 3.2. From the roof of Theorem 3.1 it follows that R 0 = /2, R 2 = 2. Thus, When there are no salar erturbations: Is r = (f. Remark 3.1), the fator in the imliation (50) II R I R an be set to /2 = When all salar erturbations are reeated twie (d s = 2), the fator in (50) an be set to 2. The roof of the Real Matrix Cube Theorem reeats word by word, with evident simlifiations, the roof of its omlex ase ounterart and is therefore omitted. Note that the differene in absolute onstant fators in bounds on in Theorems 2.1 and 3.1 (whih is in favour of the real ase) omes mainly from the absolute onstants in (32) whih are different for the real and the omlex ases. The differene between the absolute onstants in Remarks 2.2 and 3.2, whih is in favour of the omlex ase, omes from the differene between (38) and (39). Note also that Remarks 2.3 and 2.4 remain valid in the real ase. Matrix Cube Theorems and known results on tratable aroximations of semiinfinite LMIs. The results we have established omare favourably with known results on the quality of tratable aroximations of semi-infinite LMIs (1). Aside of the Matrix Cube Theorem of Ben-Tal and Nemirovski (2002a) (this result, whih is the rototye of all our develoments here, was already disussed in Introdution), there is, to the best of our knowledge, a single relevant result, seifially, Theorem in Ben-Tal et al. (2000). This theorem is of the same sirit as Theorem 3.1. Seifially, it deals with a bi-affine real LMI (1) where real erturbations are diagonal matries, and the set in (1) is (51) = { Diag 1 k R d 2 1 = 1 k } (the struture d 1 d k is fixed). The theorem assoiates with (1) an exliit system of LMIs in x and additional variables u in suh a way that (a) If x an be extended to a feasible solution of, then x is feasible for the semiinfinite LMI (1), the level of erturbations being ; (b) If x annot be extended to a feasible solution to, then x is not feasible for (1) when the erturbation level is inreased from to. The tightness fator (whih lays the same role as the fator in the Matrix Cube Theorems) is shown to be ] mk (52) = min d

19 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL 515 (m is the row size of the LMI (1), k is the number of bloks, and d = dim ). When omaring this result with those given by Theorem 3.1, it makes sense to restrit ourselves with the ase when d = 1, = 1 k this is the only ase where the statements under onsideration seak about the same erturbation set. Note that in the ase in question (1) beomes the semi-infinite LMI (53) A x + A x 0 = 1 k R k where A x, A x are symmetri matries affinely deending on x. On a losest insetion, it turns out that in the ase of d = 1, = 1 k, both tratable aroximations of (1) the one built in Ben-Tal et al and the one given in Theorem 3 1 are idential to eah other and are given by the system of LMIs (54) Y ±A x = 1 k A x Y 0 in the original variables x and additional matrix variables Y 1 Y. Although both Theorem from Ben-Tal et al. (2000) and our Theorem 3.1 seak about the same air of entities (53), (54), the tightness fators rovided by these two statements are different: In the ase of d 1 = =d k = 1, (52) results in the tightness fator = k, while Theorem 3.1 says that the fator is at most max Rank A x /2 max 1 k x Formally seaking, these two uer bounds are in general osition no one of them dominates the other one. However, in tyial aliations (e.g., those onsidered in Ben-Tal and Nemirovski 2002a or to be onsidered below) the seond bound is by far better than the first one. We are about to illustrate the use of the Matrix Cube Theorems by two aliation examles. The first examle ( 4) is a omlex-ase version of the /2 -Theorem of Nesterov (1997). The seond examle ( 5) deals with an imortant ontrol entity the strutured singular value. 4. Maximizing Hermitian quadrati form over the omlex unit ube. Let S H m, S 0. Consider the roblem of maximizing the quadrati form x H Sx over the omlex unit ube: (55) { S = max z H Sz z max r } z r 1 It is well known that the real ase version of the roblem (S is real symmetri, and the omlex unit ube is relaed with the real one) is NP-hard; the same an be shown to be true for the omlex ase (55). It is also known that in the real ase the standard semidefinite relaxation bound { m } (56) ˆ S = min r R m + Diag S r=1 is an uer bound on S tight within the fator /2( /2-Theorem of Nesterov 1997). It is immediate to see that ˆ S is an uer bound on S in the omlex ase as well. Indeed, if R+ m is suh that Diag S, and z 1, then z H Sz z H Diag z = r r z r 2 r r

20 516 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS We are about to demonstrate that in the omlex ase ˆ S oinides with S within the fator 4/ : (57) S ˆ S 4 S The roof follows the lines of an alternative roof of the /2-Theorem given in Ben-Tal and Nemirovski (2002). Observe that S is the minimum of those R for whih the ellisoid z C m z H Sz ontains the omlex unit ube z C m z 1. This inlusion is equivalent to the fat that the olar of the ellisoid (whih is the ellisoid C m H S 1 1 ) is ontained in the olar of the unit ube (whih is the set C m 1 r z r 1 ). In turn, the inlusion C m H S 1 1 C m 1 r z r 1 is, by homogeneity, equivalent to the fat that H S for all C m. Combining our observations, we arrive at the equality Further, by evident reasons, one has 1 S = max{ R + H S } 2 1 = max{ H B B H m B rs 1 1 r s m } Indeed, when B = B H, B rs 1, the quantity H B learly is 2 1 and is equal to 2 1 when H = ˆ ˆ H, where ˆ r = r / r (when r = 0, one an set ˆ r = 0 as well). Combining our observations, we arrive at the relation (58) 1 S = max{ R + H S 1 H B B H m B rs 1 } = max { R + S 1 B 0 B H m B rs 1 } Denoting by e r the standard basi orths in C m and seifying the data of a Comlex Matrix Cube Problem as (59) d rs = 1 1 r s m R rs = er H { 1 r s m 1/2 e H L rs = r r = s es H 1 r s m r <s A = S 1 If = r s 1 r s m Is r = I s = (here it is onvenient to index the erturbations by airs of integers rather than by single integer), we see that (58) says exatly that 1 S is the largest = for whih the omlex matrix box given by the data (59) is ontained in H+ m. Alying the omlex ase Matrix Cube Theorem, we arrive at an exliit Generalized Eigenvalue Problem suh that its otimal value, let it be ˆ, is a lower bound, tight within the fator 4/, for = 1 S (note that we are in the ase when there are no salar erturbations). Consequently, ˆ 1 is an uer bound, tight within the same fator 4/, for S. Exatly in the same way as in the real ase (see Ben-Tal and Nemirovski 2002a, 4), it an be further verified that ˆ 1 is nothing but the semidefinite relaxation bound ˆ S.

21 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL Lyaunov stability radius and strutured singular value Preliminaries. When a roblem of the form CMC omes from aliations, its struture m, k, d 1 d k, Is r, I s, I f is usually fixed, while the data A L R k may deend on a vetor of design variables x of a ertain master roblem of the generi form = max x 0 x and I x is valid where I x is the rediate I assoiated with the data A = A x, L = L x, and R = R x (f. the Lyaunov Stability Analysis examle resented in Introdution). In these ases, the following observation allows us to build a tratable aroximation of the master roblem : Proosition 5.1. Assume that A x, R x are affine in x and L are indeendent of x, and let the set in be semidefinite-reresentable: (60) x u x u 0 where x u is a symmetri matrix affinely deending on x u. Under these assumtions, the roblem x u 0 Y ± L H R x + R H x L 0 Is r Y V L H R ] x R H x L 0 Is V ˆ = max 0 Y L H L R H x ] 0 If R x I d A x Y 0 in matrix variables Y H m, = 1 k, V H m, Is, R, If, x u is an exliit Generalized Eigenvalue Problem whih is a onservative aroximation of : Whenever x u an be extended to a feasible solution of, then x u is feasible for. The quality of this aroximation an be quantified as follows: if is feasible, then so is, and (61) ˆ C d s ˆ (f. Theorem 2 1). Note that ˆ is the otimal value in an exliit Generalized Eigenvalue Problem and is therefore effiiently omutable. This statement is readily given by Theorem 2.1 and admits a straightforward real ase analogy imlied by Theorem 3.1. We are about to onsider an instrutive aliation examle for Proosition Estimating Lyaunov stability radius for an unertain disrete-time dynamial system. Consider a disrete-time dynamial system with states z t C m obeying the dynamis (62) z t+1 = A t z t t = 0 1

22 518 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS We assume that the system is unertain, in the sense that the matries A t are not known in advane; all we know is that they vary in a given unertainty set: (63) { t A t = A = A + C H D } where is the erturbation level and are given by (7). We are interested to ertify the stability of (62) the fat that all trajetories z t of all realizations A t t=0 of the system onverge to 0 as t. The standard suffiient ondition for stability is that all matries A t share a ommon disrete-time Lyaunov stability ertifiate Y, i.e., there exists Y 0 suh that A H YA Y for all A. Setting X = Y 1 and alying the SCL, it is easily seen that the existene of a ommon Lyaunov stability ertifiate is equivalent to the solvability of the following semi-infinite system of LMIs in matrix variable X: ] X XA H 0 A AX X X I Realling the desrition of, we an rewrite this system equivalently as (64) ] X XA H XD H + H C ] A X X C H }{{} 0 D X }{{} A X L H R X +R H X H L ] L = 0 d m C R X = D X 0 d m k The Lyaunov stability radius of unertain system (62) (63) is the suremum of those unertainty levels for whih the system admits a Lyaunov stability ertifiate, or, whih is the same, for whih the semi-infinite system of LMIs (64) is solvable. Assuming that the nominal matrix A defines a stable system (i.e., the setral radius A of A is less than 1), it is immediately seen that (65) X I = su >0 X A X + ( ) L H R X +R H X H L 0 =1 k We see that the roblem of omuting the Lyaunov stability radius is of the generi form (with X laying the role of x and = X X I ). Alying Proosition 5.1, we arrive at the following onlusion. Corollary 5.1. The Lyaunov stability radius of (62) (63) (whih by itself is, in general, NP-hard to omute) admits an effiiently omutable lower bound ˆ whih oinsides with u to a fator not exeeding O 1 d s, where d s is the maximum of row sizes of salar erturbation bloks. When there are no salar erturbation bloks, one has / ˆ 4/. The results similar to Corollary 5.1 hold true for the Lyaunov stability radius of the ontinuous-time unertain system (see the Introdution); besides this, we ould onsider the ase of real systems and erturbations (in both the disrete- and the ontinuous-time settings). The artiular setu we dealt with is motivated by the desire to link our onsiderations with an imortant ontrol entity the omlex strutured singular value.

23 EXTENDED MATRIX CUBE THEOREMS WITH APPLICATIONS TO -THEORY IN CONTROL Bounding the omlex strutured singular value. The omlex strutured singular value of a matrix is an imortant ontrol entity (for an overview of the orresonding -theory, see, e.g., Pakard and Doyle 1993) whih is defined as follows. A (omlex) blok struture on C m is an ordered olletion of ositive integers d 1 d k, k d k = m, along with a artitioning of the index set 1 k into two nonoverlaing sets Is, I f. Same as in the desrition of the CMC roblem, suh a struture defines the sets C d d of salar omlex and full omlex erturbation bloks, = 1 k.weset = { = Diag 1 k = 1 k } C m m Given a blok struture, the orresonding strutured singular value M of a matrix M C m m is defined as (66) M = max M 1 where S is the setral radius of a square matrix S. Realling that the setral radius of a matrix A is <1 if and only if A admits a disrete-time Lyaunov stability ertifiate X XA H ] (i.e., there exists X 0 suh that AX X 0), an equivalent definition of M is as follows: (!) The quantity 1/ M is the suremum of those 0 for whih every one of the matries M,, admits a disrete-time Lyaunov stability ertifiate. In general, it is NP-hard to omute the quantity ; this is why an imortant role in -theory is layed by omutable bounds on. As far as uer bounds are onerned, the standard one (and, to the best of our knowledge, the only one) is the saling uer bound ˆ M defined as follows. Let D be the set of all Hermitian m m matries D I m whih ommute with all matries from ; in other words, D D if and only if D = Diag D 1 D k, where D H d are Id and, besides this, D = I d for If. The bound ˆ M is defined as (67) ˆ M = 1ˆ where (68) ˆ = su 0 D D { D DM H MD D ] } 0 (note that by homogeneity reasons, the otimal value in the latter roblem remains unhanged when the normalization onstraint D I in the definition of D is relaxed to D 0). The fat that ˆ M is an uer bound on ˆ M is immediate. Indeed, taking into aount the definitions of and ˆ, we observe that to rove that ˆ M M is the same as to verify that if and D D are suh that ] D DM H 0 MD D (or, whih is the same, 2 M H D 1 M D 1 ), then t M < 1 for all and 0 t<. To verify the latter laim, note that 2 M H D 1 M D 1 t 2 M H D 1 M t/ 2 H D 1 t/ 2 D 1 D 1 }{{}

24 520 A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS where is readily given by the fat that ommutes with D D and 1. We see that the matrix tm admits a disrete-time Lyaunov stability ertifiate, or, whih is exatly the same, t M = tm < 1, as required. As defined above (and as arising in the ontrol literature), the saling uer bound ˆ M looks as a very useful ad ho invention. Could we build a omutable uer bound on M in a more systemati way? The answer is affirmative. Indeed, onsider the set of matries (69) = M note that this set is of the form (63) with A = 0 m m. By (!), 1/ M is the suremum of those for whih every A admits disrete-time Lyaunov stability ertifiate. This roerty, of ourse, is weaker than the existene a ommon Lyaunov stability ertifiate for all matries from ; it follows that 1 M ˆ where is the Lyaunov stability radius of an unertain dynamial system with unertainty set (69), and ˆ is the omutable lower bound for this radius mentioned in Corollary 5.1. We have arrived at a omutable uer bound on M, seifially, the quantity ˆ M = ˆ 1. An exliit desrition of ˆ M is given by the otimization roblem ( ), the data of the roblem oming from the desrition (69) of the underlying unertainty set. Skiing the urely mehanial derivation, here is the resulting desrition of ˆ M : (70) 1 ˆ M = max R X H m Y V H 2m R X I m a Y V L H R ] X R H X L 0 Is b V Y L H L R H X ] 0 If R X I d A X Y 0 P = 0 d d 1 + +d 1 I d 0 d d d k C d m X R X = P MX 0 d m L = 0 d m P A X = d ] X Now a natural question arises: There are two effiiently omutable uer bounds on the omutationally intratable strutured singular value M the usual saling bound ˆ M (see (67) (68)) and the bound ˆ M given by (70). What is the relation between these bounds? Here is the answer: (71) Proosition 5.2. One always has ˆ M =ˆ M Proof. The roof deomoses into two arts. 1. Let us first rove that ˆ M ˆ M. Realling the origins of ˆ M and ˆ M, to verify this inequality we should rove that if > 0 D is feasible for (68), then an be extended to a feasible solution of roblem (70). Let > 0 D be feasible for (68), so that D = Diag D 1 D k with ositive definite Hermitian d d bloks D whih