Proceedings of the 37th IEEE FM05 12:lO Conference on Decision & Control Tampa, Florida USA December 1998

Matrix-Based Bounding vs. Element-Wise Bounding for the MPEP Global Optimization

Yuji YAMADA~ Shinji H A R A ~ Department of Computational Intelligence and Systems Science

Tokyo Institute of Technology 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan

Abstract

In this paper, we compare tlie matrix-based bounding and the element-wise bounding concerning the glohal optiiiiizatioii for tlie matrix product eigenvalue prob- lem (MPEP) , which addresses lots of typical bilinear matrix inequality (BMI) problems for control syiitlie- sis. I t is shown that using tlie matrix-based lminding has sonie advantages to consider tlie global optiniixa- tion for the M P E P over tlie element-wise boiinding. Numerical experiments illustrate that the algorithm tis- ing tlie matrix-based boiinding is better than that of the element- wise b oundiiig in the t o t a1 conipii t a t ional time.

1 Introduction

In this paper, we consider the matrix procliict eigen- valiie problem (MPEP) introdiiced in [12, 131, which is a problem between tlie linear matrix ineqiiality (LMI) problem and the bilinear matrix ineqiiality (BMI [9]) problem, i.e., LMI c MPEP c BMI. Tlie motivation is siininiarized as follows:

Tlie LMI problem is known to be highly tractable (e.g., [I]), but it is restrictive to formilate control synthesis problems. The BMI problem is a fairly general framework to formulate control synthesis problems, bi i t it is very difficiilt to solve [4]. Therefore, we iieed to define a siibclass of tlie BMI problem to solve control synthesis problems niore efficiently.

Tlie M P E P is formulated as follows: Minimite the .spec- tral rad,ius of th,e product of trrio hlock-diagonal po,sit%.ue d sym,m,etr%c m.ntrices Although tlie MPEP defines a subclass of the BMI problem as stated above, it can formulate most of typi- cal BMI prohlenis for control synthesis characterized by the Lyapuiiov ineqiialities. In other words, the MPEP is also a general framework for control synthesis.

E-mail: yuj iPcs . dis . t i t e c h . ac . j p 2E-niail: haraQcs .d is . t i t e c h . ac . j p

In all corrcsporidc~icc. contact. t.hc second aiit,hor. S. Hara tiy E-riiail or Fax: +81-45-924-5442

Tlie objective of this paper is to compare oiir pro- posed algorithm [12, 131 for solving the M P E P with BMI lxtiirh and boiiiid algorithms [2, 4, 101 proposed for solving tlie BMI problem, which are respectively characterized by the matrix-based l~oiinding and the element-wise boiiiiding. We show that iisiiig tlie matrix-based boiinding has soiiie advantages over the element-wise boiinding to solve the MPEP. e.g.. we can coiistriict a glohal algorithm siich that tlie itera- tion nuinher is giiaranteed to he l~oiindecl by ail itera- tion iipper boiind. Moreover, tlie algorithm iisiiig tlie matrix-based boiinding solves each convex siibproblem (relaxatioii problem) niore efficiently. In tlie seqiiel. we will clarify tlie following facts:

Tlie matrix size of LMIs and tlie niiniber of vari- ahles in one convex siibprohlem for the niatrix- based boiinding is niiicli less than that for tlie element-wise lioiuiding, wliicli leads to tlie speed- tip to solve one siihprohleni.

A niiniber of iiiiinerical experiments illiistrates that tlie performance of tlie algorithm iisiiig tlie matrix-based boiinding is better even in tlie ac- tual case.

This paper is organized as follows: We state tlie proh- leni foriiiiilation and disciiss its property in Section 2 . In Sectinn 3, we compare two relaxation prohleiiis, the element-wise boiinding and tlie matrix-hased boiind- iiig. In Section 4, we rlisciiss tlie total iteration niinilxr and compiitational time in tlie worst and tlie average cases, where tlie average case iteration i i i~ni l~er is il- liistratecl by ntimerical experinieiits for the constantly scaled 71m control synthesis. Section 5 offers sonie con- cli d i n g remarks.

2 Problem description

2.1 The matrix product eigenvalue problem We first define a set of block-diagonal iiiatxices V by

0-7803-4394-8198 $1 0.00 0 1998 IEEE 3861

where

s,, := { c I c E !RFx, c = ET > 0 } (2.2) Let 2, he a closed lmiiiicled coiivex subset of 2, x D. Then the MPEP is foriiiiilated as follows [12, 131:

iiiiiiiiilize A:,!:~ (EA) siihjec; to (E, A ) E 2,. c V x D MPEP 1

(2 .3) where ~t~!:~ (.) denotes the sqiiare root of tlie maxiiiiitiii eigeiivaliiel (or the square root of the spectral rachis). The coiivex set 2, is iiormally defined by LMIs for coii- trol synthesis prohleiiis, so tlie reader inay associate the coiivex constraint (E, A ) E 2, in (2 .3) with LMIs on (E, A ) E V x V .

We make the following assiiiiiptioii, which is not re- strictive when control synthesis prohleiiis are consid- ered [12, 131:

Assumption 2.1 (Monotonicity on 2c) Th.e follo.iiiiirg relation h,olds for airy given! (2, A) E V x 2):

(2 , A) E 2, + { ( E , A ) E D x D I C > Q , A > A } C ~ ~

2.2 6-global optimization and BMI condition The scope of glohal optimization in this paper is to find a siil,-optiiiial soliition within any specified rela- tive tolerance c: E (0, 1) from tlie globally optimal so- liitioii, namely 6-glohal optimization proldem definecl as follows:

(2 .6)

The MPEP can be recast as a prohlem of iiiiiiiiiiiziiig a linear objective fimctioii iiiider iioiicoiivex constraints, i.e., tlie followiiig eqiialities hold:

yr,l)t = iiif { y > 0 I A:~!~~(EA) I 7. (E. A ) E 2, }

Here tlie secoiid equality is ohtaiiierl by a siiiiple pa- rameter change, and the third eqiiality is ohtaiiied by observiiig that

Notnice that the optiniizatioii problem (2.G) is a BMI problem, becaiise all tlie coiistraiiits are cliaracterized as BMIs. This iiiiplies that the M P E P defines a sub- class of BMI problems, and hence, global optimization sclieiiies based on the hraiicli and boiiiid met,hod pro- posed for solving the BMI prohlem [2, 4, 101 are avail- able after rewriting the iioiicoiivex coiistraiiits in (2.G) as a liaffiiie fiiiiction.

On the other hand, we have proposed a global op- tiiiiizatioii scheme to solve the M P E P directly. T h e main difference of the two approaches lies in their relaxation problems, which are respectively based on the matrix-hased lmiindiiig aiid the element-wise boiiiidiiig. Next section, we will compare these two relaxation problems aiid show that i.ising the iiiatrix based hoiiiidiiig has soiiie advantages over tlie eleiiient- wise lmiindiiig for solving oiie coiivex siihprohleiii.

Remark 2.1 Th.e M P E P is con,s%dered as n gen,eral- %zo.t%on, to th,e problem, fo r th,e po.sit%.iie definite m,atrix case from, th.e linear mzrltiplicat%rie prograni,nr%n,g ( L M P ) problem, [5], ,irih,ich. h,as heen, explored ,in, th,e f ie ld of n,tr- rn,erical opt%nr%tat%on, on. th,e Euclidean, sp~,ce. Th.e L M P prolilem is 0, con,carw m.in,%miration prohlem. form,,ulated 0, s

iiiiniiiiize f l ( z ) f i ( x )

fl(.) > 0, ti(.) > 0 LMP 1 sithject to z E C c 8

3862

3 Matrix-based bounding vs. element-wise bounding

As both tlie BMI problem aiid tlie MPEP are 11011- coiivex whicli may liave pliiral local minima, we have to search over a soliition on all tlie entire doiiiaiii di- rectly or secondhand in general. This is tlie difference between coiivex aiid iioiicoiivex optimization proldeiiis, aiid it makes tlie iioiicoiivex optimization prohleiii ex- treiiiely difficult. A general and perhaps aii efficient way to do this process is the so-called lxaiicli aiid I~oiiiid method (see e.g.. [7]). Roiiglily speaking, tlie branch and boiiiid method consists of the following t,wo oper- ations [7]:

Branching: Siibdivide the boiiiided doiiiaiii to be searched into a finite niiiiiber of smaller s i i h - spaces, e.g. hyper rectangles.

Boiiiidiiig: For each siibspace, fiiid iipper aiid lower boiiiids on tlie objective fiiiictioii Iiy iisiiig iipper lmiiiid and lower lmiiid fiiiictioiis. If the lower lmiiid on certain siilxpace is larger than the ciir- rent optimal valiie. priiiie the s i i l qace from the tloniaiii.

If we iiiteiid to apply the lxmcli aiid l~oiiiid process to a iioiicoiivex optimization p r o l h i i , we iieed to coii- striict relaxation prohleiiis which give iipper and lower boiiiids respectively on a given siilxpace. These relax- ation proldeiiis slioiild be defiiied by coiivex constraints which approximates iioiicoiivex coiistraiiits on the siih- space. Tlie perforiiiaiice of tlie branch aiid l~oiiiid type algorithm is clearly depeiideiit of tlie relaxation proli- lems.

In this sectioii. we disciiss relaxation prohleiiis which enable 11s to apply a branch aiid Imiiid process for the BMI proldeiii and tlie MPEP. which are respectively cliaracterized as tlie b*eleiiient-wise hoiiiidiiig" aiid the bbiiiatrix-based boiiiidiiig."

3.1 Element-wise bounding Let 11s define a lkffiiie fiiiiction CP: for given symmetric iiiatrices @ i j =

P " , r x !R7'u -+ !R"1~X7rL E R7r1,xTf1~:

i= 1 j = 1 i=l j=1

(3 .1) Then the BMI eigeiivaliie problem is foriiiiilated as fol- lows [4]: Given @ i j , minimize p (CP (2, y)) over (x. y) E Q D c R7".r x R7',v. where Q D is a given closed boiiiitled hyper rectangle.

Go11 et al. [4] proposed a relaxation prohleiii in order to apply tlie liraiich aiid hoiiiid techiiiqiie to the BMI eigeiivaliie prol1leii1, by element-wise replaceiiieiits of liaffiiie teriiis :i;.iyj by 7uij with additional l,oiiiid coii- straiiits on n:i, y j aiid w i j . element-wisely. Tliey showed tliat tlie relaxation probleiii provides an lower Imiiid

of tlie objective fiiiictioii in 011 a given hyper rectangle. Tlie relaxation prol~leiii has been iiiiproved by Fiijioka aiid Hosliijima [2] and Takaiio et al. [lo]. They pro- posed a iiietliod to obtain a tighter lower boiiiid, which coiistriicts tlie coiivex hi111 on tlie iioiicoiivex constraints 7 4 j = z iy j on a given hyper rectangle. A coiiiiiioii point of these relaxation problems is that each 11011- coiivex constraint 7 q j = :i;iyj is partially approxiiiiated in terms of a coiivex region, aiid tlie approximation is doiie eleiiieiit-wisely, i.e., the eleiiieiit-wise 1,oiiiiding.

However, tlie precision of the approxiiiiatioii Ixcoiiies worse for the elenient-wise Imiiitliiig as the iiiiiiil~er of iioiicoiivex constraints gets increase. and fiirtlier- iiiore, additional linear constraints are needed per oiic replacement of iioiicoiivex constraint. e.g.. two addi- tional linear constraints for [4] aiid foiir for [2. lo]. Coiiseqiieiit,ly, tlie iipper and lower hoiiiids ol,taiiied by tlie element-wise hoiiiidiiig teiid to l x conservative. aiid tlie speed to coiiipiite tlie lower Imiiiid lxcoiiies slower. Moreover, the iiiiiiil,er of iioiicoiivex constraints increases rapidly coiiiparet-l with tlie sizes of iiiatrices. For instance. consider tlie simplest matrix case, i.e.. XU, (X. Y ) E S2 x S2. Tlie condition iiicliides "seven biaffiiie teriiis," siiice X Y is written as

J 211y11 + x121J12 n:11!/1z + n:lzV22 n:122/11 + 3:22y12 3:12?J12 + 3:zzyz This implies that we have to search over seven iioiicoii- vex areas in spite of the siiiiplest case. More generally. in tlie case of 71. x n, positrive tlefiiiite iiiatrices, the iiiiiii- her of iioiicoiivex constraints is given l,y

713 - 71,(?1. - 1 ) / 2

which increases rapidly with respect to n,.

3.2 Matrix-based bounding On the other liaiid, tlie relaxation prol,leiii [ l a , 131 for the MPEP is characterized by the iiiatrix-1,aset-l I,oiiiic-l- iiig. i.g.. the parameter boiiiids of the siihspacc for the prohleiii are determined hy block-diagonal positive t-lef- iiiite iiiatxices. Tlie idea is hased on tlie eqiialities in (2.G) and tlie following relation: For any given C E V and A E V, it holds that

AA!2x(EA) 5 y

Hence. we liave

3(1) E V ; C+OAO 5 278 (3.2)

7,yoI,t, = inf{ y > 0 I C + O A 8 5 270 8 E V. (E, A ) E 2, } (3.3)

Tlie prol~lcm (3.3) is not coiivex either since it lias a iioiicoiivex comt,ra.int, C + O A O 5 2 7 0 , hiit it l ~ e c o i i i ~ s coiivex if 0 E V is fixed.

3863

Consider tlie paraiiie:t.er space with respect to t h last eqiiality in (2.G). Let F ( y ) be a set of feasible with a given level y > 0 defined by

0 relaxation

3 ( y ) := { C E V 1 A E D s.t.

E W M B

element-wise matrix base

For a given norm-boiiii~led siibspace of V, deiioted by B c V. let 11s defiiie r ( B ) as

(3.5)

If B 3 F(T) and F(+) # 8 liold for soiiie + > 0. we see that Y,,,,~, = r ( B ) holds. We fiirtlier define Be.K by

for given 0 E D aiid K, E (1, m). With these def- initions, t,he following leiiiiiia provides a coiivex opti- mization problem to find iipper aiid lower boiiiids on r(Be,,) for any given 0 aiid K, E (0, 1)[12, 131:

Lemma 3.1 Let

In Lemma 3.1, Fe in (3 .G) caii be compiited by solving a coiivex optimization prol,leiii, or the semidefinite pro- gramiiiiiig prohleiii if 2, is defiiied by LMIs. Note that probleiii (3.15) caii be considered as a relaxation Ixoh- lem. Iwcaiise tlie iioiicoiivex coiistraiiit A:,::~ (CA) 5 y in the original problem is relaxed. This relaxation problem has tlie followiiig properties:

0 The lmiiiidiiig region is determined by tlie iiiter- val of positive definite matrices.

0 It gives iipper and lower boiiiids siiiiiiltaneoiisly by solving proldeiii (3.G) once at each step.

0 Tlie matrix size and t,he iiiiiiilier of variables for the corresponding LMI problem do not increase compared with tlie original proldeiii.

0 Tlie gap between tlie iipper aiid lower baiiiid can be estimated by h: which deteriiiiiies the size of tlie lmiiiidiiig region.

C E V, aiid k depends on the way to constnick re- laxation problems aiid is given by k = 2 for [4] and k = 4 for [2, 101. I t is readily seen that, the relax- ation problem iisiiig tlie matrix-based hoiiiidiiig can be solved iiiore efficiently than the ones iisiiig tlie elenleiit- wise lmiiiding, hecaiise the miinher of variables aiid tlie matrix size of LMIs are Imth iiiiicli less than those for the element-wise hoiiiidiiig. Coiiseqiieiitly, we conchide that tlie relaxation iisiiig tlie matrix-based lmiiiding is better than the element-wise lmiiiidiiig at least for solv- iiig m e coiivex siibprol~lem.

increase of @ iiiatrix size / / k (113 - w) I 0

4 Total computational cost

We iiext compare tlie total iteration iiiiiiil)er for tlie two global optimization algorithms of [2] aiid [ 131 which are respectively lmsed oil the eleiiieiit-wise l~oiiiidiiig aiid tlie matrix-based hoiiiidiiig. We especially fociis on the algorithm in [2] aiiiong the algorithms [2. 4, 101 iisiiig the eleiiient-wise hoiiiidiiig for the BMI proldeiii, from tlie following reasons: Using coiivex hiill seems the tightest, to approxiniate the iioiicoiivex region. aiid the algorithm in [2] is at least better than the nile in [4]. Also note that, tlie algorithm in [lo] iises tlie saiiie coiivex hi111 as that in [2].

4.1 Worst case In [12, 131, we have proposed aii algorithm to solve tlie 6-global optimization problem for the M P E P ancl sliowii that solving coiivex optimization prohleiii (3.G) for different valiies of 0 at iiiost N (< m) tiiiies yields a siihoptiiiial soliition y- satisfying (2.4). Moreover, we have derived the worst case order as sliowii in tlie followiiig theorem:

3.3 Discussions Theorem 4.1 N 5 N,,,,, hmlds, ,trth.cre Table 3.1 compares tlie matrix sizes of LMIs and the iiiinil>er of varialdes for tlie relaxation problems by 11s- iiig the element-wise (EW) boiiiidiiig aiid tlie iiiatrix- based (MB) boiiiidiiig, where the case V = S,, is coii- sidered for simplicity. n, correspoiids to tlie size of

N,,,,, := 0 (( l/t)) [ := 1 of scalar ,rrar%ablcs in, C E V

3864

In tlie above theorem, N,,,,, provides tlie worst case iteration number, and note that the actual iteration iiiiniber and order may be much less than those of the worst case.

Avg. Iteration tf

On the other hand, we have not obtained the worst case iteration iiiiiiiber for tlie BMI branch and lioiind algorithnis [2, 4, 101 using the element-wise boiinding in general. Altlioiigli the siiiallest size of rectangles for tlie worst case has been shown in [2], their resiilts are not guaranteed that the iteration niiiiiber is bounded by an exact worst case iteration nuniber. Consequently, it may liappeii that we need a fairly large compiitational effort even if the problem size or the number of lion- coiivex constraints is small. Numerical examples in the next subsection will siiggest that the actiial iteration nuniber of the algorithm in [2] iisiiig the element-wise bounding niay vary widely and soiiietiiiies can be large even if tlie simplest case is considered.

Avg. CPU-time (s) - - - - - - - - - - - - total/one iteration

4.2 Average case In this siihsection. we make numerical experiiiients and discuss the actiial case computational complexity. We consider the constantly scaled 'tim control problem for two cases. All the compiitations in this paper were carried out by using Matlab.

FH PI YH [13]

4.2.1 Randomly generated systems: Coii- sider tlie constantly scaled 31, problem with one scaler scaling parameter, i.e., V = { diag(aI, I) I CT > O }. In this case, the problem contains one hiaffine constraint (TA, (a E E, X E 3). We siippose that tlie entire do- main is restricted to (cr, A ) E [O.O5, 201 x [&OS, 20).

We have solved raiidoiiily generated 100 problems of plant order four, by using the algorithm proposed in Fiijioka and Hosliijima (FH [2]) and Yaiiiada and Hara (YH [13]). Note that, for a k e d scaling matrix, i.e., tlie standard Zrn problem, the matrix size of LMIs to be solved is 32, which contains two Lyapiiiiov matrices of size 4 x 4 as variables. Table 4.1 shows oiir iiiiiiier- ical results. The first coliiniii compares the average number of iterations reqiiired to find a global soliition within tolerance c = 1 x 1W2. For the FH algorithm, we oiily counted the iiiiiiiber of branches correspoiiding to tlie number of lower bound problems, so the ntini- ber of convex siibprobleiiis gets increase if we coiiiit the number of upper boiiiid problems. On the other hand, the niiniber of iterations is exactly correspond- iiig to the niiniber of convex siibprohlems for the YH algorithm, becaiise the convex subproblem in the YH algorithm computes iipper and lower h i n d s siniiilta- neoiisly by solving oiie convex optimization problem at each step. In spite of these siiiiplest scalar case, we see that there is a large difference of tlie average niiiiiber of iterations between tlie two:, the iteration miniher of the YH algorithm is less tliaii oiie third of that for the FH algorithm.

43.3 269.G/G.23

13.5 18.9G/1.40

The second coliiiiin compares average CPU- time. We first notice that there is a large difference I3etween the two. Especially, we see that the average CPU-time per oiie iteration for the YH algorithm is less than qiiarter of that for the FH algorithm. Note that the iiiatrix size of LMI to be solved in the FH algoritliiii is 38 and that in tlie YH algorithm is given by 33.

Table 4.1: 100 raiidoiiily generated prohlems

Table 4.1 is a histogram which shows a relation 1x:tween the iteration niiiiiber and the iiiiiiil>er of pro1)leiiis ob- tained by the F H algorithiii for the same probleiii da t a as in Table 4.1. Ahoiit 10 o/I of prohlenis are iiiore than 70 iterations, altlioiigli tlie average is less than 50. Moreover, two prolileiiis are more than 90 which is iiiore than t,wice the average iteration iiiimlier. On the other hand, for tlie YH algoritlini, every iteration iiuiiiber was less tlian 15 and it was almost constant. Froiii this niiiiierical experiment, we coiicliitle that the iteration numher of tlie F H algorithm niay vary widely compared with that of the YH algorithm.

< fl of Probleiiis >

- 35 ! 45 55

65 75 85 95

< Iteration tf > Fig. 4.1: Iteration tf hy the F H algorithm

4.2.2 ACC-benchmark problem: We next solve the constantly scaled 3t, prol~lem for a nieclian- ical system consisting of two iiiasses connected by a spring [8], where the block-diagonal matrix set is given

V = { block diag(C, 1 2 ) 1 C E PSZ } Note that tlie order of the generalized plant. is foiir, and hence tlie iiiatrix size of LMIs to he solved in tlie

3865

standard U , problem is 32, coiitaiiiiiig two Lyapuiiov matrices of size 4 x 4; as variables. Siiiiilarly to tlie previoiis siibsectioii, we compare tlie FH aiid the YH algorithms.

Algorithm

F H

N CPU-time ( s )

Avg. CPU-time per oiie iteration ( s )

11.7

l / E 1 / E Fig. 4.2: Iteration # Fig. 4.3: CPU-time

Fig. 4.2 coiiipares the number of convex siibprobleiiis N vs. the inverse of the tolerance l / ~ for the two algo- rithms. Similarly to the previoiis case, we only counted the iiiiiiiher of brandies correspoiiding to tlie iiiiiiiber of lower hoiiiid prohleiiis for tlie FH algorithm. More- over. the giveii size of doiiiaiii to be soiight for the F H algorithm is siiialler than that for tlie YH algorithm in this numerical example in order to iiiake a fair coiii- parisoii. Nevert,lieless, the iteration iiiiiiiber of the YH algoritliiii is fewer than that for tlie F H algorithm for each tolerance E .

Altlioiigli the difference of tlie iteration iiiiiiiher he- tween t,lie t,wo algorit,hiiis is not big, there is a sigiiifi- cant difference of the CPU-time shown in Fig. 4.3 aiid Table 4.2 I d o w , which compare the total CPU-time vs. I/E aiid the average CPU-time per oiie iteration. re- spectively. The difference is caiised by tlie iiiatrix size of LMIs aiid tlie iiiiiiiher of variables in each convex siibprobleiii, i.e., we have to join four additional linear constrailits per oiie iioiicoiivex constraint to solve oiie lower hoiiiid problem for the F H algorithm. Siiice tlie iiiiiiiher of iioiicoiivex constraints is seven for this ex- ample, the iiiatrix size of LMI becoiiies G4 for tlie F H algorithm, althoiigh that in the YH algorithm is given hy 34.

YH II 1.77 Table 4.2: Avg. CPU-Time

5 Conclusion

In this paper, we have compared tlie iiiat,rix-hased boiiiidiiig aiid tlie element-wise h ouiicliiig coiiceriiiiig tlie global optiiiiizatioii for the MPEP. In tlie seqiiel, we have obtained tlie followiiig facts:

0 Tlie iiiatrix size of LMIs aiid tlie iiuiiiber of vari- ables in oiie convex sitbprohleiii for the matrix- based boiiiidiiig is much less than that for the eleiiieiit-wise bounding. Coiisequeiitly, the speed to solve oiie siihprohlem for the iiiatrix- based boiiiidiiig is much faster than that for the element-wise boiiiicliiig.

e A iiiiiiiber of numerical experiiiients ilhistrates that tlie perforiiiaiice of the algorithm using tlie matrix-based boiiiidiiig perform is better even in tlie actual case.

Acknowledgments: Tlie aiitliors woiild like to thank Prof. Fiijioka of Kyoto Uiiiv. for his helpfiil disciissioiis aiid valiiahle comments.

References [l] S.P. Boyd et al., Linwar Matriz ln,eq~u,al%tie.s in System, an,d Con,trol Th.eory, SIAM, 1994. [2] H. Fiijioka aiid K. Hosliijiiiia, Tram. SICE, 33(7):GlG 621, 1997. [3] Hisaya Fiijioka and Koliji Yaiiiasliita, Proc. IEEE

[4] K.G. Goli, M.G. Safoiiov, aiid G.P. Papavas- silopoiilos, Proc. IEEE CDC, 2009 -2014, 1994. [ 5 ] H. Koiiiio aiid T . Kiiiio, Mathematical Program- ming, 56: 51 -64. 1992. [GI T. Kiiiio aiid H. Koiiiio, Journal of Global Opti- mization, 1: 2G7--286, 1991. [7] G.L. Neiiiliaiiser, A.H.G. Riiiiiooy Kaii aiid M.J. Todd (Eds.) , Opti?lr%7~t%0?1,: Hmdhooks in, Operat%on,.s

[8] A. Packard et al., Proc. IEEE CDC, 1245-1250, 1991. [9] M.G. Safoiiov, K.G. Goli, aiid J.H. Ly, Proc. ACC, 45/49, 1994. [lo] Y. Takaiio, T. Wataiiahe aiid K. Yasuda, Trms. SICE, 33(7):701-708, 1997 (in Japanese). [ll] H.D. Tiiaii aiid S. Hosoe, Proc. ACC, 350--355, 1997. [12] Y. Yaiiiada, Global Optiiiiizatioii for Robiist Control Synthesis based oii tlie Matrix Prodiict Eigeii- valiie Problem, Pli. D. dissertation, Tokyo Inst. of Tecli., 1998. [13] Y. Yaiiiada and S. Hara, Proc. IEEE CDC, 492G- 4931, 1997. [14] Y. Yaiiiada aiid S. Hara, IEEE Trans. AC., AC-

[15] Y. Yaiiiada, S. Hara, aiid H. Fiijioka, Proc. ACC,

CDC, 440--445, 199G.

g e m e n t S c i m c e , 1, Elsevier, 1989.

43(2):191-203, 1998.

427 ~ 430, 1995. 3866