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Complementarity problems over matrix cones in systems and control theory

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Complementarity problems over matrix cones in systems and control theory

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Content COMPLEMENTARITY PROBLEMS OVER MATRIX CONES IN SYSTEMS AND CONTROL THEORY by Mehran Mesbahi Advisor: Professor B. Rozovskii A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree M ASTER OF SCIENCE (Applied M athematics) May 1995 UNIVERSITY O F SO U T H E R N CALIFORNIA TH E GRADUATE SC H O O L U N IV ER SITY PARK LOS A N G ELES. C A L IFO R N IA 9 0 0 0 7 This thesis, written by . Z L & ktteU br. _________________________________________ under the direction of hJ^„„.Thesis Committee, and approved by all its members, has been pre sented to and accepted by the Dean of The Graduate School, in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE / ............................................ Dtan THESIS COMMITTEE Contents L ist O f F ig u res iii A b str a c t iv 1 In tr o d u ctio n 1 1.1 N o ta tio n ..................................................................................................................... 2 2 L C P s and C o m p le m e n ta r ity P r o b lem s over C o n es 4 2.1 W hat is th e com plem entarity problem ......................................................... 4 3 T h e L inear and B ilin ea r M a tr ix In e q u a litie s 8 3.1 T he P r o b le m s .......................................................................................................... 8 3.2 T he LMI and the B M I ........................................................................................ 9 4 C o n e-L C P s an d th e B ilin ea r M a trix In e q u a litie s 14 4.1 In tro d u c tio n .............................................................................................................. 14 4.2 P r e lim in a r ie s ......................................................................................................... 17 4.2.1 T h e Cone-LCP and the E F P ................................................................ 17 4.3 Cone-LCP Form ulation of th e B M I ............................................................... 20 4.3.1 Few Initial S t e p s ................................................................... 20 4.3.2 Two M atrix C o n e s .................................................................................. 22 4.3.3 E F P Form ulation of the B M I ............................................................. 26 4.3.4 Cone-LCP Form ulation of the E F P S(M ) ( B M I ) ........................... 28 5 C o n clu sio n 3 4 u List Of Figures 3.1 Stability of Interconnections.......................................................................... 12 4.1 Embeddings and Cone Generalizations of Various Problem s................. 16 in Abstract We discuss an approach for solving the Bilinear M atrix Inequality (BM I) based on form ulating the problem as a Linear Com plem entarity Problem (LCP) over a certain m atrix cone. Specifically, we show th at a BMI has a solution if and only if there is a corresponding com plem entarity problem th at testifies to this. Since the BMI may be regarded as the central problem in robust control, the results reported in the thesis opens up new avenues for the com putational procedures for solving the BMIs and many (perhaps most) robust control problems. A d v iso r: Professor B. Rozovslcii. Chapter 1 Introduction Traditionally, system s and control theory intersected m athem atical program m ing, for the m ost p art, via th e optim al control theory and th e calculus of variations. This has been achieved by reducing the optim al control problem , posed in an infi nite dim ensional function space to a finite dimensional nonlinear program m ing by discretization. T he reduction is also invaluable for the finding com putational pro cedures for the infinite dimensional problem on the digital com puters. Accordingly, the connection between systems and control theory to the com plem entarity prob lems has been established through the necessary and sufficient conditions for the corresponding finite dimensional nonlinear program m ing problem. However, the above state of affairs has changed recently. Specifically, it has been realized th a t m any problems in systems and control theory can naturally be for m ulated in th e context of convex programming. These problem s include, among m any others, th e stability studies of differential inclusions, control synthesis prob lems, realization theory, etc. T he non-traditional elem ent in these form ulations is the dom ain on which these optim ization problems are form ulated, namely, the cone of positive semi-definite m atrices. Although one can in principle transform these cone problem s to a convex program m ing problem form ulated in the Euclidean space via a n atu ral isom orphism , the resulting form ulation is generally a non-sm ooth opti m ization problem . Nevertheless, it turns out th at the recent advances in th e interior point m ethods for the general convex program m ing problem allows one to develop provably efficient algorithm s for these non-smooth problem s by designing the algo rithm to work directly on the cone of positive semi-definite m atrices. These advances 1 have both practical and theoretical significance. On the practical side, the formula tion allows one to develop polynomial tim e algorithm s for non-sm ooth optim ization problems which, on the first encounter, seem to be very difficult. On th e theoretical side, using the general m achinery of the interior point m ethods, one can view all convex program m ing problems, a t least computationally, as a generalization of the well understood linear program m ing problem. The linear com plem entarity problem (LCP) can also be considered as the gener alization of the linear program m ing, but the LCP, in general, corresponds to a non-convex optim ization problem , and therefore, is a very difficult com putational problem. Consequently, th e cone generalization of th e LCP, which is defined in this thesis, is a priori a difficult problem. Nevertheless, it has been observed th at certain classes of the LCPs, and the cone generalizations of them , can be solved efficiently using th e interior point m ethods. In fact it has been realized th a t efficient algorithm s can be developed for com plem entarity problems which do not necessarily, correspond to a convex optim ization problem . Thereby, it is of im portance to reduce a difficult, non-convex problem to a com plem entarity problem when it is possible to do so. This is in light of the fact th at a com plem entarity problem has m uch m ore structure than an arbitrary global optim ization form ulation. T he m ain contribution of this thesis, is th a t certain, not necessarily convex, optim iza tion problems th a t arise in system s and robust control theory can be transform ed to a linear com plem entarity problem over a certain m atrix cone. This form ulation thus, opens up new avenues for designing com putational procedures for m any robust control problems. 1.1 Notation In this section, we catalog the notations th a t are used in th e thesis. A >- B : A — B is sym m etric positive definite. A > : B : A — B is sym m etric positive semi-definite. A • B : Trace A B \ the inner product used for the space of m atrices. 2 5 ftp : T he p-dimcnsional Euclidean Space. 3£p : The positive orthant. in p-ditnensional Euclidean Space. Sfjpxp ; T he space of p x p m atrices with real entries. 3?+xp : The space of p x p positive semi-definite m atrices w ith real entries (not necessarily sym m etric). : p x p positive definite m atrices with real entries (not necessarily sym m etric). S^E^.Xp : Sym m etric p x p positive semi-definite m atrices with real entries. : Sym m etric p x p positive definite m atrices w ith real entries, v e c M : The vector obtained by stacking up the columns of the m atrix M. Bpxp ' ■ A m atrix in 5 R pxp. C ” : T he Dual of the set C , A ® B : T he Kronecker product of m atrices A and B. A 1 : Transpose of the m atrix A. Aij : The elem ent of m atrix A located in the z-th row and j- th column, int A : The interior of the set A. diag(y) : The m atrix of appropriate dimensions with vector y on its diagonal and all other elem ents equal to zero. 3 Chapter 2 LCPs and Complementarity Problems over Cones In this chapter, we will define the general com plem entarity problem over cones. Our discussion of these problems is inevitably concise. The reader is referred to [4] and [10] for much m ore on the com plem entarity problems. 2.1 W hat is the complementarity problem C om plem entarity problems are a unifying form ulation for m any problems in en gineering and economics. T he m ain notion th at leads to complementarity is the existence of an equilibrium . An equilibria has to obey a certain necessary condi tions, which in tu rn , facilitate the design of com putational procedures to find the equilibria. T he conditions in the case of unconstraint optim ization is the familiar statem ent from calculus, th a t at an equilibria, the gradient vanishes. In the case of constraint optim ization, the necessary conditions are m ore com plicated, and are essentially those referred to as the K arush-Kuhn-Tucker (KI<T) conditions, which are also connected to the im portant notion of duality. In fact, the K K T conditions and duality in th e constraint optim ization, can be con sidered as the m ain sources of w hat ones calls the th e com plem entarity conditions. O ther sources also exists; for exam ple, the discretization of a variational inequality problem , posed on an infinite dimensional H ilbert space, leads to a com plem entarity problem . As it is welt known, the original variational inequality does not need to be 4 originated from an optimization problem. Lets us now define the general com plem entarity problem (CP). Consider the cone fC in an arbitrary Hilbert space 'H, with an inner product 7 -L x 1-t 3?. Consider a m ap / : H 'H. Let K.“ denote the dual cone of K,, i.e., fC' = {y < = n \< x ,y > > 0; V ie fC } Given "H, and / , the com plem entarity problem is to find an elem ent x~ € 'H . (if it exists), such that: xmefC (2.1.1) f(x ’) e tC* (2.1.2) < x ‘ J (x * ) >= 0 (2.1.3) The above instance of th e com plem entarity problem is referred to as C P (/C ,/). W hen / is an affine m ap, the above problem is referred to as the general linear complemen tarity problem over cones. We shall formally define the affine case over the finite dim ensional H ilbert spaces in chapter 4. A nother specifications in th e formulation of the CP, besides the affinity of the map / , can be m ade on the space "H and the cone fC. T he linear com plem entarity prob lem (LCP), is the com plem entarity problem with 1-i = 3?, and K, being the positive orthant. T he LCP and it various finite dimensional generalizations, have numerous applications in engineering, economics, game theory, etc. If one restrict the CP to th e case where H is the space of sym m etric n x n m atrices, 5 R nXn, K > a certain m atrix cone in this space (e.g. the cone of sym m etric positive semi-definite m atrices (PSD )), and / an affine m ap on 5'5in><n, th e Cone-LCP for m ulation is obtained, which we shall formally define in chapter 4. As it will be shown, the Cone-LCP is a very nice form ulation for m any problems in system s and control theory. More specifically, in chapters 3 and 4, it will be shown 5 that a host of important problems in control theory are reducible to the Cone-LCP. B ut why should one be interested in form ulating the system s and control prob lems as a Cone-LCP? Is it not true th a t it m ight be easier to come up with an algorithm for the original problem rather th a t transform ing it (and sometimes in a rather peculiar m anner) to a Cone-LCP over some finite dimensional cone? The answer is not so clear. This is so in the light of the fact th a t even the LCP, a special case of the Cone-LCP, is provably a hard com putational problem in general. So what is the point of transform ing one hard problem into another?! We note th at this issue loses its relevance in the context of the linear program m ing (which can be reduced to an LCP via duality theory) and it generalization to the Semi-Definite program m ing (SDP) (which can be reduce to a Cone-LCP over the PSD cone). This is the case, since both LP and SDP are convex optim ization problems and can be solved by the interior point m ethods in a provably efficient m anner. In fact, this is true for all convex program m ing problems. So there is a big stake in transform ing a given problem to a convex program m ing problem , since one can claim th at having done this, the problem has been essentially solved. B ut this is not true, for the general non-convex program m ing problems; and the LCP, and its generalization in term s of cones, corresponds to a non-convex optim ization problem. So why should we be concerned w ith transform ing a problem to a Cone-LCP. T he reason is as follows: “ C ertain classes of Cone-LCPs can be solved efficiently (at least theoretically) by employing th e interior point m ethods and its various extensions.” This is in fact the m ain m otivation for the present research. The LCP, and subse quently a Cone-LCP, form ulation of a non-convex problem, gives one a finer insight in to the com putational tractibility of th e problem . By transform ing a problem to a Cone-LCP, one has a b etter chance of coming up with an efficient algorithm for its solution, th an transform ing the sam e problem to a global optim ization problem. Some com m ents on the interior point m ethods is justified at this point. The in terior point m ethods for solving the constraint optim ization problems goes back to at least to the work of Fiacco and M cCormick in the 1960s. But it was not until 6 the work of Karmarkar in 19S4., and subsequently Renegar [1G ] and Gonzaga [6] in 198G and 1987, respectively, th at it was realized th at these m ethods can be modi fied to yield a polynomial time algorithm for the linear programming problem. The work of Nesterov and Nemirovskii has also been a milestone in the development of the interior point methods, where these methods are extended to yield polynomial tim e algorithms for an arbitrary convex programming problem. Extensions of these m ethods to monotone linear com plem entarity problems and monotone Cone-LCPs have recently been studied [12]. Currently, it is one of the main research directions to extend these m ethods to certain non-convex program m ing problems. Some re sults along this direction are those of K ojim a et al. [11], Ye and Pardalos [22], and Mesbahi and Papavassilopoufos [13]. 7 Chapter 3 The Linear and Bilinear Matrix Inequalities In this chapter vve shall discuss various problems th a t can be expressed in term s of certain m atrix inequalities. The types of the inequalities th a t we consider are the linear and th e bilinear ones, which have recently found various applications, ranging from graph theory to systems and control theory. The organization of the chapter is as follows. Initially, we shall present the problems th a t are the m ain focus of th e thesis. We then list some of th e applications of these problem s, along w ith a discussion on the reducibility among the various problems. 3.1 The Problems We shall now catalog the problem s th at are occasionally referred to in th e thesis. The reader is referred to the section 1.1 for the glossary of notation which is used hereafter. T h e L in ear M a trix In e q u a lity (LM I): Given sym m etric m atrices Fi = F{ € S$tnXn (i — 1, • • * m ), find x E Sim, if it exists, such th at F {x ) = Fa + Z)f!Li is positive definite. T h e B ilin ea r M a tr ix In eq u a lity (B M I): Given sym m etric m atrices Hij E < 5 '9 ftpX p (i = 1 ,..., n\ j = 1,... m ), find x E 3?n, and y E 9?m (if the}' exist), such th at ]C?=i S iL i is positive definite. 8 T h e (P rim a l) P o sitiv e S em i-D efinite P ro g ra m m in g (S D P ): Given /‘ Vs as in the description of LMI above, and F(x) = /'b + Z!ZLi find the solution to the following constraint optimization problem, or show th at it has no solution: minc'.-r, subject to the constraint: F (x )^ 0. T h e D ual o f th e S D P : Given F,-’s as in the description of the LMI above, find the solution to the following constraint optimization problem, or show that it has no solution: - max subject to the constraints: = c,-, i = 1,..., L, and 0. 3.2 The LMI and the BMI If one examines the description of the dual formulation of the SDP, one notices its close analogy with the usual linear programming problem (LP), except th at the in ner products in in the linear programming formulation is replaced by the inner product in FJE1 1 *". Therefore, it is hardly surprising that there is a close relationship between the LP and the SDP, both conceptually and algorithmically. In fact, it can shown th at the LP is a special instance of the SDP. The SDP, the Dual of the SDP, and the LMI are all equivalent, in the sense that one can reduce an instance of one to an instance of another. This is similar to the well known fact that the primal and the dual formulations of the LP are equivalent to a system of linear inequalities. Among the numerous applications of the LMI, we list the following in the systems and control theory, • M atrix scaling problems. • Construction of quadratic Lyapunov functions for stability and performance anal ysis of linear differential inclusions. • Synthesis of Lur’e-type Lyapunov functions for nonlinear systems. • O ptim al system realization. • M ulticriterion LQG/LQR. 9 We shall only consider one of these examples in a greater detail, namely the stabil ity question for the differential inclusions and show how it can be reduced to an LMI. Given m atrices A\ . .. A l , and an initial point x(0) € 5 ftp, consider finding an in variant ellipsoid for the differential inclusion, dx/dt ~ C o {j4i ..., A l} x(t) (3.2.1) where C o denotes the convex hull of a set of m atrices. The invariant ellipsoid E, is an ellipsoid such th a t if x(0) € E , x(t) E E , for all t > 0. If we let E = {x : x 'P x < 1}, for some positive definite m atrix P , we can reform ulate the problem of finding the invariant ellipsoid to examining the consistency of the following system of linear m atrix inequalities, — {A'iP + P A i)h 0, (i = 1,. - ., L) (3.2.2) P>- 0 (3.2.3) The above system of inequalities is clearly an LMI in the entries of the m atrix P. Moreover, one can form ulate the same problem as the following SDP, with an additional normalizing constraint, m in t (3.2.4) subject to: — {A\P + PAi) 0 (f = 1,..., L) (3.2.5) P + t l h 0 (3.2.6) / - P h 0 (3-2.7) Therefore the stability of a linear differential inclusion can be reform ulated as an LMI, as well as an SDP. 10 We now briefly describe how an SDP, and therefore an LMI, is reducible to a Cone- LCP. This result is mainly a generalization of the fact th a t the optim ality conditions for the linear program m ing leads to an LCP, Using the result of Nesterov and Ne- m irosvkii [14], the Prim al and the Dual of SDP can be w ritten as: min C * X (3.2.S) subject to: X € L + D (3.2.9) X>z 0 (3.2.10) and, rainD *Y (3.2.11) subject to: Y € 4- C (3.2.12) Y h 0 (3.2.13) where L is a given subspace in S3?nXn and ZA is its orthogonal com plem ent. Under the assum ption th at th e prim al and the dual have nonem pty interiors, and letting F — (L + D ) x (ZA + C), the necessary and sufficient conditions for the solution of the prim al and the dual can be w ritten as checking the existence of the pair (A', V) F, such th a t X t: 0, Y>z 0. and X * Y = 0. Since F is an affine subspace of £'lP2nx2n, there exists a linear m ap M : Sdinxn — * ■ < 5,5£nxn, and a m atrix Q t such that: F = {(A , Y ) : Y = Q + M(X), X e SK nXn} Therefore the SDP, and consequently the LMI, is reducible to the Cone-LCP. T he LMI is clearly reducible to a BMI. This can be done by appropriately choosing the m atrices H{j in the BM I formulation. We shall now describe an application of the BM I in the control theory. T he problem th at is considered is th e nf K m-synthesis. Since a com plete description of the prob lem will take us too far off the m ain them e of the thesis, we shall confine ourselves w ith a rath er verbal description of the y./K m-synthesis problem, than a technical one. 11 y u r y u .d %A % B Figure 3.1: Stability of Interconnections T he starting point of the /i//ifm-synthesis problem is the interconnections I a and I b in Figure 3.1, where A and T are p-input, //-output, causal, stable operators on some particular Hilbert space. O perator T can be thought of as the nom inal plant transfer function and A , as an elem ent of the uncertainty set. W hen the uncertainty elem ents have a particular structure, usually block diagonal when the operator is expressed in term of m atrices, the uncertainty is called structured. The interconnec tion IA is said to be stable if and only if the induced gain from the pair (r, d) to (u,y) in the interconnection Ib is bounded. The fijK m-synthesis form ulation of the robust control synthesis problem can be described as follows: Given a nominally linear time invariant plant with an associated uncertainty set, the objective in (if K m-synthesis is to synthesis a controller for the plant that guarantees the stability of the interconnection of the plant with any clement of the uncertainty set. Recently Safonov et al. [18], Goh et al. [5] have shown th a t fifKm-synthesis, for structured uncertainty, w ith linear tim e invariant param etric and dynam ic uncer tainty blocks, can be reduced to a BMI problem . This is in addition to a wide array of other problems in control and system s theory th a t can form ulated as an BMI. Since the LMI and the SDP, with their num erous applications in graph and systems theory are special cases of the BMI, it is evident th a t finding an efficient algorithm for th e BM I is of im m ense theoretical and practical significance. We also note th at 12 the BMI problem , corresponds to a non-convex optim ization, and therefore it cannot be reduced to a convex program m ing problem. In the next chapter we shall discuss certain relationships between the BMI and the com plem entarity problem form ulated over a certain m atrix cones. This connection, in light of th e discussion of chapter 2, should open up certain avenues for solving the BMI, which is considered by m any researchers, to be the central problem in robust control. 13 Chapter 4 Cone-LCPs and the Bilinear Matrix Inequalities 4.1 Introduction The Bilinear M atrix Inequality (BMI) is considered to be the central problem in the field of robust control. The BMI feasibility problem is as follows: Given sym m etric m atrices H{j € 3£pxp (i ~ 1 ~ 1,... m ), does there exist x € 5 R n, and y 6 such th at D"=1 ZiVjHij is positive definite. As it was shown by Safonov et al. [IS] it is possible to reduce a wide array of control synthesis problems such as the fixed-order H 1 = 0 control, ^/fcm-synthesis, decentralized control, robust gain-scheduling, and simultaneous stabilization to a BMI. It is also known th a t the Linear M atrix Inequality (LMI) approach to control synthesis [3] is a special case of the BMI. Since the LMI is equivalent to the Semi-Definite Program m ing (SDP), the BMI can also be considered as a generalization of the SDP. It is therefore hardly surprising that the solution to the BMI is not only of central im portance in the context of robust control [19], but also in its connections with the SDPs and the LMIs. The BMI can be reformulated as a non-convex program m ing problem. More specifi cally, Safonov and Papavassilopoulos [19] have shown th at the BMI feasibility prob lem is equivalent to checking whether the diam eter of a certain convex set is greater than two. Since this is equivalent to a maximization of a convex function subjected to a set of convex constraints (an N P-hard problem ), no efficient algorithm is be lieved to exist for a general BMI. Moreover, Toker and Ozbay have recently shown th at the BMI feasibility is an NP-hard problem by reducing the Subset-Sum problem 14 to it [21]. In view of the above considerations, one is led to consider various special cases of the BMI in hope of recognizing efficiently solvable instances. The situation is anal ogous to what happens in the case of the Linear Complementarity Problem (LCP) [4], which has been well studied due to its im portant applications in economics and engineering. The LCP has the following formulation: Given M € U P 1 *” , and q 5 R n, find z G 3?” (if it exists), such that, z > 0 (4.1.1) q + M z > 0 (4.1.2) z '(q + M z ) = Q (4.1.3) where “s'" denotes the transpose of the vector z , and the ordering “> ” for vectors is interpreted component-wise. The above instance of the problem is referred to as the LCP(q, M ). It is known that the Linear Programming Problem (LP) can be formulated as an LCP. It is also well known that the LCP belongs to the class of NP-hard problems. Nevertheless an immense amount of insight has been provided by studying various m atrix classes th at arise in the LCP formulation. It seems conceivable that the same approach is valuable for the BMI problem. An alternative approach would be to directly transform the BMI to a generalized LCP, and use the results of the complementarity theory to classify efficiently solvable BMIs. The generalization of the LCP that has proved to be suitable for this purpose is the Cone- LCP formulation. Since the LMIs and the SDPs can be considered as Cone-LPs, one can illustrate the relationship among these problems as shown in Figure 4.1. In this figure, we have used the vertical arrows to indicate the “Cone” generalizations of the problems, and the horizontal arrows to indicate th at the problem formulation on the right side of the arrow has as an special case, the problem on the left side. In Figure 4.1, the “question mark” between the BMI and the Cone-LCP is the main topic of this paper. Specifically, our main result is th at a BMI has a solution if and only if there is a corresponding Cone-LCP that testifies to this. This in turn implies com putational procedures that might be adopted for solving the BMI. 15 ? * Cone-LP (SDP) BMI - > Cone-LCP | (NP-hard) A LP - - - - - - - - - - - - - - - - - - - > LCP (NP-hard) Figure 4.1: Em beddings and Cone G eneralizations of Various Problem s An interm ediate step in transform ing th e BM I to a Cone-LCP is th e introduction of a problem which we shall refer to as the E xtrem e Form P roblem (E F P ). Form ulated in the n -th dim ensional Euclidean space 9£n and denoting the- nonnegative o rth an t by 9?+, the E F P has the following form ulation: Given M : 5E n — y 5 R ” , find z £ 3?” (if it exists), such th at, z > 0 (4-1.4) M z > 0 (4.1.5) z an extrem e form of 9R " (4.1.6) An extrem e form of a cone is a face of th e cone which is a half-line em anating from the origin [17]. T he above instance of th e E F P is referred to as the EFP|Rn(A/). T he E F P , as we ju st defined, is not an interesting problem . In fact, th e EFPojn(M ) has a solution if and only if M has a positive colum n. O n th e o ther hand, the E FP becomes non-trivial w hen is replaced by an arb itrary cone. W e have used the cone generalization of th e E F P as an interm ediate step in form ulating th e BM I as a Cone-LCP. It can be argued th a t com putational procedures could be developed for th e E F P directly, w ithout transform ing it to a Cone-LCP. We have chosen to m ake this transform ation, since at th e present tim e, th e Cone-LCP seems to be m ore am enable for the application of the interior point m ethods th an th e EFP. It is still 16 an open question whether an interior point m ethod can be adapted for solving the EFPs directly. T he organization of this paper is as follows. In the Prelim inaries we present some basic definitions, certain m atrix cones, as well as the precise form ulation of the Cone- LCP and the EFP. In Section 4.3, the “cone” form ulations of the BMI is presented. We also discuss certain com putational im plications of this reduction. In the final section, we discuss aspects of the problem th at call for further investigations. 4.2 Preliminaries In this section we define tlie Cone-LCP and introduce the Extrem e Form Problem (E FP) over arbitrary cones. We also m ention certain m atrix classes, which when generalized appropriately, will make the transform ation of the BMI to the E FP and the Cone-LCP m ore explicit. 4.2.1 The Cone-LCP and the EFP Before defining the Cone-LCP and the EFP, few basic definitions are in order. In the following, we shall restrict ourselves to finite dimensional H ilbert spaces. Let H be a finite dimensional H ilbert space equipped w ith the inner product < •, ■ >: r H x 77 — * (e.g., the n-dim ensional Euclidean space or the space of n x n m atrices, w ith the appropriate notion of an inner product defined on them ). A set /C C 77 is a cone if for all a > 0, afC C fC. K is a convex cone, if K. is a cone and it is convex, i.e., for all a [0 1], cctC + (1 — a)fC C Af, or equivalently, if K is a cone and K, + K, C fC. An extrem e form (or extrem e ray) of a convex cone JC is a subset E = {a x : a > 0} of /C, such th at if x = a y + (1 — a )z, for 0 < a < 1, and y ,z £ fC, one can conclude th at y, z € E [8]. T he dual cone of a set S C "H, denoted by S*, is defined to be, S* = {y € 'H :< x ,y > > 0; Va: € 5} It is well known th at S m is always a convex set. In addition, S = (5*)*, if and only 17 if S is a closed convex cone. For more on cones and duality the reader is referred to Berman [2], Rockafellar [17], and Stoer and W itzgall [20]. In subsequent sections we shall be referring to a property of a linear m ap that we now define. Given a convex cone X C "H, a linear m ap M : — > T4 is called £-copositive if < X , M(X)>> 0, for all X EhZ. We are ready to form ulate the Cone-LCP: Given a cone X C 'H, a linear map M : U 71, and Q € 71, find Z E Ti (if it exists) such that: Z E X (4.2.1) Q + M{Z) E X* (4.2.2) < Z ,Q + M ( Z ) > = 0 (4.2.3) The above instance of the problem shall be referred to as the Cone-LCPjc(<2, M), with the solution set denoted by SOLgone~LCP (Q, M ). W hen X is the nonnegative orthant in the n-dimensional Euclidean space, the Cone-LCPjc(Q, Aff) (4.2.1)-(4.2.3) is equivalent to the LCP(q,M) (4.1.1)-(4.1.3). It can also be shown th at when X is the cone of positive semi-definite m atrices (PSD), the Cone-LCPjc(<2, M) embeds in itself the SDP and consequently the LMI [12], [14]. Consequently, the studying of the Cone-LCP is believed to have an im pact on the SDP and the LMI, both the oretically and computationally. A problem which serves as a bridge between the BMI and the Cone-LCP is what we have referred to as the Extreme Form Problem (EFP): Given a cone X C a linear m ap M :'H — ¥ “ H , find X E H (if it exists), such th at, X E X M(X) E int X m X an extrem e form of X (4.2.4) (4.2.5) (4.2.6) where the “int K,” denotes the interior of the cone X. The above instance of the E FP is referred to as the EFPjc(M ), with the solution set denoted by SO LjS^f A/). As we mentioned in the introduction, when K, is the nonnegative orthant in the n-dimensional Euclidean space, the EFP is a trivial problem. It should be noted th at SOLj® FP(M) is non-convex. Given the two extrem e forms of fC that solve the E F Pjc(M ), a non-trivial convex combination of them is not an extreme form of fC. It is also im portant to note that the EFP requires M(X) to lie in the interior of the dual cone. This is in light of the fact that for many linear maps, including the map th at arises in the context of the BMIs, M{X) is known to lie in, but possibly on the boundary of, the dual cone, for all X £ tC. It is still not clear how to extend the interior point methods for Semi-Definite Programming [1],[14] for solving the EFP. This is the main reason why we have chosen to transform the EFP that arises from the BMIs, to a Cone-LCP. It was hoped initially to transform the BMI problem to a Cone-LCP over the cone of positive semi-definite matrices (PSD), since it is the cone which has received much attention recently in the context of Semi-Definite Programming. Moreover, interior point methods have already been suggested for a class of Cone-LCPs over the PSD cone by Kojima et al. [12]. Subsequently, it was realized that a different kind of m atrix cone is in fact needed for the Cone-LCP reformulation of the BMI. This cone turned out to be a generalization of the cone of completely positive matrices, and its dual, the cone of copositive matrices. The former m atrix cone, which is the set of matrices with quadratic forms expressible as a sum of squares of linear forms, has been studied in the context of block design in combinatorial theory [7]. The latter m atrix cone, which is the set of matrices with quadratic forms nonnegative over the nonnegative orthant, has been studied in the context of the LCP [4], The interior point methods for Semi-Definite Programming have not yet been adapted for the above m atrix cones (and their generalizations which are defined in this pa per). Nevertheless, since in principle interior point methods can be developed for arbitrary convex and cone programming problems [14], [15], we have assumed, for the purpose of this paper, that the copositive Cone-LCPs can be solved efficiently 19 by these m ethods. These Cone-LCPs arc of the form Cone-LCP*;(<5, A/), with M being A^-copositive. The m ain obstacle for the realization of these interior point m ethods for th e cones of com pletely positive and copositive m atrices, and their gen eralizations, m ight be the N P-com pleteness of checking w hether a given m atrix is copositive. 4.3 Cone-LCP Formulation of the BM I In this section we discuss the reduction of th e BM I feasibility problem to a Cone- LCP over a suitable generalization of th e cone of com pletely positive m atrices. This is done by first reducing the BM I to an EFP, and subsequently reducing the E FP to a Cone-LCP. As it becomes evident, various em beddings of m atrices in different dim ensions are needed to m ake th e reduction as tran sp aren t as possible. For this purpose th e v e c notation, which is used in the studying of K ronecker products, has becom e specially handy. T he two volumes of Horn and Johnson [8], [9] provide a basic reference for the general m atrix theory and the K ronecker products. 4.3.1 Few Initial Steps Consider again the BMI feasibility problem : Given Iiij = & S R pXp, does there exist a:t’s (1 < i < n ), and y / s (1 < j < m ), such that: ^ 0 (4.3.1) * j Let us rew rite (4.3.1) as: £ £ y>H ‘> = £ >- o (4.3.2) i j ' t w here, a * = € «*>"' 3 20 As it becomes apparent by subsequent developm ents, it is convenient to assume th a t m = p and th at y,’s (1 < j < m ), are nonnegative. T he first assum ption is m ade to avoid defining inner products between m atrix classes of different dim en sions. T he second assumption is made to facilitate the dual cone characterization in the Cone-LCP approach. These assum ptions are w arranted for the following reason. First, note th at if we define H f = 'f2ix iHij £ 3?pxp (1 < j < m ), then B ut the last sum is a linear inequality in / / J ’s. Thus, as it is custom ary in the Linear Program m ing, one can assum e th a t m < p and th at 7/j’s are positive (by an appropriate augm entation). Now we would only need to define Hij ~ 0 (1 < i < n; m < j < p ), for the assum ption m = p to be justified. From the G ordan’s theorem of alternative [20] (over the cone of sym m etric posi tive sem i-definite m atrices), one concludes th a t the BMI (4.3.1) does not have a solution if and only if, (Vy G ^ ) (BZ G SW +xp) : Z = 0 (4.3.3) Therefore, the BMI (4.3.1) has a solution if and only if, {By G SR ?.) (VZ G SStp + Xp) : ^{HfrnZ)2 > 0 (4.3.4) t R em a rk 4 .3 .1 We note that if (f.S.f) is satisfied, and that the corresponding y is found by a certain procedure, then the B M I (f.3 .1 ) is reduced to a Linear Matrix Inequality (LM I). Let, p v p Hi = [vec Hii 0 . .. O ', 0 v ec H a . . . 0 , . . . , 0 0 . . . v ec G W ^Xp2 (4.3.5) and F —diag(y) € S&pXp. Since {Hf)1 = Hf = JZjyjH^ (recalling th a t H{j's are sym m etric m atrices), v ec H r = H{ v ec Y 21 Thereby, Hf*Z = (vec Hf')'(vec Z ) = (//,-vec K )'(v ec Z) = (vec Y ) ‘H [(yec Z) (4.3.6) Com bining (4.3.4) and (4.3.6) we conclude th a t (4.3.1) has a solution if and only if there exists Y G S W f p, y = d iag (p ), for some y € such th a t for all Z G £'9R pXp, (vec Z)' { ^ 2 H i(vec y )(v e c Y)'H-} (vec Z) > 0 (4.3.7) * Let X = (v ec y )(v e c V ’)' and, M (X ) = J2 HiXH'{ (4.3.8) i Then (4.3.7) can be interpreted as requiring M (X ) to belong to a certain m atrix class. T he m atrices in this class have quadratic forms which are positive over the v ec form of the m atrices in 5 3 fE ^ .xp. This observation justifies the introduction of two m atrix classes th at we shall discuss in th e next subsection. 4.3.2 Two Matrix Cones A t this point let us introduce certain classes of m atrices. We then delineate (4.3.7) in term s of these m atrix classes, which in turn facilitate the transition to the Cone-LCP form ulation. In what follows it is assum ed th a t all th e m atrix classes are subsets of 3 R p2xpJ. Let T^S'D denote the class of (not necessarily sym m etric) p2 x p 2 m atrices with quadratic forms nonnegative over the v ec form of th e sym m etric p x p m atrices, i.e., V SV = {A e 3?p2xp2 : (vec Z y A (vec Z) > 0; Z G S3£pxp} 22 2 2 Note th at VST) is different from 5 R P X p (the p2 x p2 positive semi-definite m atrices), since the elements of the latter class have quadratic forms which are nonnegative over the vec form of all (not necessarily symm etric) p x p m atrices. It can be shown th a t if A is symmetric, A € VST), and has rank t, then there are m atrices W{ € 5'3R pXp(0 < i < f), such that A = X^=i(vec W'fjfvec W i)'. In addition, if A has such a representation with Wi* Wj = 0 [i ^ j) , then the reverse implication also holds. Let C denote the class of PSD-copositive m atrices, i.e., C = {A E 3 R p2xp2 : (vec Z)' A (vec Z ) > 0; Z € 5'5RpXp} Finally, let B denote the class of PSD-completely positive m atrices, B = {A € 53?p2xp2 : A = X X vec ^ )( v e c ZCf, Z { € SW+ *p, t > 0} i=i R em ark 4.3.2 The matrix classes C and B are generalizations of the copositive and the completely positive matrices, respectively. Note that the matrices in class C have quadratic form s which are nonnegative over the vec form of the sym m etric positive semi-definite matrices. We now observe th at Equation (4.3.7) states whether a nonlinear com bination of m atrices H \s € 5 R p2xp2, belongs to the interior of the cone of PSD-copositive m atri ces. It should be noted that m any results in the context of copositive m atrices [4] can also be proven for the PSD-copositive m atrices. It is also clear th at VST) C C. We now establish th at m atrix classes B and C are in fact closed convex cones. In addition, we show th at they are the dual of each other. These results will facili ta te the reduction of the BMI to an EFP, and subsequently to a Cone-LCP. These 23 proofs are not significantly different from those provided by Hall [7] for the classes of com pletely positive and copositive m atrices. L em m a 4 .3 .3 B and C art closed convex cones in W> Q xp 2 , Moreover B * = C and C" = B. Proof: C is a cone since if (vec Z)'A(vec Z ) > 0, then for all a > 0, (vec Zy(aA)(vec Z ) = a (v e c Z )'A (v ec Z) > 0, i.e., olC C C. Moreover, if A, B G C, then (v ec Z)'(A + Z?)(vec Z) = (v ec Z )'A (vec Z) + (vec Z)'B(vec Z) > 0. Therefore C + C C C. B is a convex cone since if A = Z);=1(vec Z ,)(vec Z ,)' for some I > 1, then for all a > 0, a A = Y2j=i(j3vec Z,)(/3vec Z,)', where (3 = i.e., aB C B. Also from the definition of B it is clear th at, for all A , B G B , A-\~B G B. To show th a t B = C*, we proceed as follows: F irst we note th a t VST) C C and VSTT = VST), consequently, C* C VST). There fore, for all A G C*, A = 23i=i(v ec l/t)(v ec U i)\ for some < 7 ,- G iS '5 R pxp, and some q > 0. It suffices to show th a t all £/;’s in the representation of A G C*, have to be pos itive semi-definite. Suppose that there exists A G C* such th a t A = (vec 17)(vec U)' (we shall assum e th at q = 1 and U n > 0, w ithout loss of generality), and th at there exists an x G 5 R P such th a t x'U x < 0. Since A G C*, A • B > 0, for all B G C. Therefore, (vec U )'B (v e c 17) > 0, for all B G C. Let, E = [vec sa/O -'-O ] G 5 R p2xp2 Then E G C, since for all Z G 5 ftpXp, (vec Z )rE { v e c Z ) = (v ec Z )'(v e c x x ,)(Z 11) > 0 On the other hand, (vec f7 )'F (v ec U) = (vec U )'(v e c x x r)U u = (x'U x)U u < 0 24 Hence B • (vec U )(vec U )' < 0 , E G C, i.e., A £ C*, which is a contradiction. To show th at B * = C, we proceed as follows: Let A G 0*, then for all Z G S E pXp, A• (vec Z )(v ec Z )' = (vec Z )'A (v ec Z) > 0 and therefore A G C. Hence 0 ' C C. On the other hand, if A €. C then, A* (vec Z )(v ec Z ) f > 0; (VZ G SW+ Xp) and thereby A • E > 0, for all E G B. Hence A G B* and consequently, C = 0*. T he closure of B and C follows from th e fact th at the dual cone of any set is closed. □ Since we will later need an explicit expression for the extrem e forms of 0 , the fol lowing lem m a is of im portance. L e m m a 4 .3 .4 The extreme form s o f B are matrices (vec Z )(v ec Z)', Z G 53ftpXp. Proof: Every form in B can be w ritten as £ - =1(v ec Z,-)(vec Z;)'; (Z; G (t > 0). If for some W G S 3 ^ Xp, (v ec H 0 (v e c W )' = E ;= i(v e c Z ,)(vec Zt -)'; (Z; G 5£ft^X p), unless each Z,- is a constant m ultiple of W , we can find A '^ 0 , such th a t (v ec IF ) (vec W)r» X = 0 and (vec Z,-)(vec Z,)'« X ^ O , for some index i. B ut this is a contradiction to our original assum ption. Moreover, since every m atrix in B can be w ritten e ls a finite sum of m atrices of the form (vec Z )(vec Z ), Z G 5'3fJ^Xp, these forms are in fact the extrem e forms of the cone B. C H T he brief discussion of the m atrix cones B and C which we have provided above, is sufficient for the reform ulation of the BM I as an EFP, the subject which we shall exam ine in the following subsection. 25 4 -3 .3 E F P F o r m u la tio n o f t h e B M I We now use the results of the previous section to reformulate the BMI as the EFPy(iW), where B is the cone of PSD-compIetely positive matrices, and M is defined by (4.3.8). For a given M , vve say th at the E FP^(M ) is feasible if there exists an X that satis fies (4.2.4) and (4.2.5). The next result is of importance in regards to the feasibility issue. Moreover, it is the first result that connects the EFP to the Cone-LCP. P ro p o s itio n 4.3.5 For a given M , the problem EFP/c(M) is feasible if and only if fo r every Q , the Cone-LCPfc(Q, M ) is feasible. Proof: Suppose that the EFP/c(M) is feasible. Then there exists X E X , such that M{X) E int X*. Then for every Q one can choose A > 0 large enough such that Q + M(XX) = Q -j- AM(X) E X m . On the other hand, if the Cone-LCPjc (Q, M) is feasible for every Q , let Q E int (— /C*); therefore, the E F P c(M ) is clearly feasible. □ One can establish certain results in the context of the E FPk(M ), for various classes of cones X, and linear maps M , in the spirit of the complementarity problems. We will not pursue this line of investigation in the present work. Nevertheless, it should be noted that the EFP is a non-trivial problem when the cardinality of the extreme forms is infinite, finite but very large, or when checking whether an element is an extrem e form of the cone X is computationally difficult. The importance of the E F P ^(M ) in the context of the BMI is established through the following result. In fact, as the next proposition states, the BMI is a special instance of the EFP. P ro p o s itio n 4.3.6 Let B be the class o f PSD-completely positive matrices, and the linear map M be defined as (4.3.8). Then the B M I has a solution if and only if the EFPb(M ) has a solution. Moreover, the solution o f one yields the solution of the other. 26 Proof: If the BMI has a solution A', then there exists Y = diag(jy), y < = ? R P, X = (vec K )(vec V)', such that M(X) 6 int C, and hence, the EFP/j(A/) has a solution. Conversely, suppose that the EFPy(A/) has a solution X. Then there exists V € SW+ *P , such that X = (vecK )(vec V )’ and M(X) € int C, i.e., for all 2 € S’ KP > < P , (vec Z)' { X //.(v ec V){y&cV)'H[} (vec Z) > 0 i Let V = T 'Y T be such that Y is diagonal, T is nonsingular, and T ' = T -1 . Then vec Y = (T < g > T )vec V. We observe that, (vec Z ) '{ £ f f ;( v e c y )(v e c y )'/f;} (vec Z) 1 = (vec Y )' {XI //f(v ec Z )(vec Z)'H[} (vec T) t = (vec V)'{T ® T )' ( X # i(v e c Z )(vec Z )'tff} (T ® T )(vec P ) 1 = ((T ® T )(vec Z ))' ( X tf«(vec V )(vec V)'/?;} (T ® T )(vec Z) > (tt.3.9) t The last inequality follows from the fact th at if vec W = (T ® T )vec Z, 2 6 55R ^.X p, and T is nonsingular, then W € < S '5 f tpXp- It now follows directly from the above proof that Y is a solution of the BMI. O As we have mentioned in the Introduction, one can try to modify the interior point m ethods for the computational solution of the EFP. In the case where an efficient algorithm is devised for the EFP directly, the BMI can be solved by first solving an E FP and then an LMI (See Remark 4.3.1). At the present time, it is not clear how to solve the EFP effici- ntly for arbitrary convex cones using the interior point methods. Consequently, we have taken another step forward and have transformed the EFPb’(Af) to a Cone-LCP. 27 4 .3 .4 C o n e - L C P F o r m u la t io n o f t h e E F P ^ ( M ) ( B M I ) Since the BMI is equivalent to the EFPy(M ), with B being the cone of PSD- completely positive matrices and M being defined by (4.3.8), in what follows, we shall use the “BMI” and the “EFPp(A f)” synonymously. Let us denote by p = p(p + l)/2 the dimension of the space of symm etric p x p matrices. The problem E F P b(M ) (BMI) ca.n be put into the form of a Cone-LCP by em ploying the following lemma. L e m m a 4.3.7 Let Y be an extreme form of the cone B. Then there exists a sym m etric W € V S V such that W = 0 and rank (W ) = p — 1. Proof: Let Y = (vec K )(vec V )1 . Then there exists Ui € S$R.pxp(i = 1, • • •, p — 1), such th at Ui* V = 0. Let Z{ = (vec £/,-)(vec Ui)'. Then Z c Y = 0, i = 1 ,... ,p — 1. Let W = ICf”, 1 Z i• Then W is symmetric, rank (W ) = p — 1, and W G VST>. □ Consider the Cone-LCPy(Q, M ): Find X G X p (if it exists) such that: X € B Q + M{X) eB* =C X . (Q + M(X)) = 0 Our m ain result now follows. T h e o re m 4 .3 .8 The B M I has a solution if and only if there exists a sym m etric Q E int (— C), such that the Cone-LCPe{Q, M ) has a solution X *, Q + M(X*) € 'P S D , and rank (Q -f M{X*)) = p — 1. Proof: (Necessity) Suppose th a t the BMI has a solution X *. By Lemma 4.3.7, there exists symm etric W € VST), rank W = p — 1, such th at W *X * = 0. W ithout loss of generality assume ||W || = 1. (4.3.10) (4.3.11) (4.3.12) 28 Let Qa = a W — M{XtL ). Note th at since M{X) (X G B) is sym m etric , Ql} is also symmetric. It suffices to show that there exists an a > 0, such that Qa 6 int (— C). Since M { X *) G int C and B is closed, there exists /? > 0, such th a t inft/gs;||t/||=i U»M{X’) > /? > 0. Hence, for all U G B, ||t/|| = 1, V Q a = U* {a W - M (A *)) = a (t/* IV) - M(X') < a U * W - (3 < a - (3 Therefore choosing a < /?, we obtain th a t for all i/ G |jf/|| = !,£ /• Qo < 0. Hence Qa E int (— C). Moreover, 0 = & {XU W ) = X U {Qa + M ( X ’)) and by construction, rank (Q& + M(Ar*)) = p — 1 and {Qa + M{X*)) G VST). (Sufficiency) Suppose th at the X m G SOL%one~LCP(Q, M ), with Q € in t(— C), Q + M(X*) G V ST), and rank of Q + M{X~) is p — 1. It suffices to show th at rank X~ = 1. Assume th at rank X “ = q, q > 2. Then, X * = i ,(vec Z i)(vec Z,)'; Z; G 5 ' ^ xp i ' = i W ithout loss of generality assume th at Z j = 0 (i ^ j). Let, p - i Q + M{X*) = £ ( v e c IV;)(v e c W {)'; W { G Sdlp*p j'= i Then since (vec Z,)(vec Zi)r G B (1 < i < q), and (vec W i)(vec W i)' G C (1 < j < p — 1), it is easy to see th at one has to have Zi*W j — 0 for all 1 < z < q and 1 < j < p — 1. But this implies th at Z{ s are linearly dependent, which is a contradiction. 29 Therefore rank A'" = 1 and there exists Z* € S’ IftpXp, such that, A" = (vec Z ')(v e c Z")' which implies th at X m is an extreme form of B. . Moreover, since Q & int (— C) and Q *f M (X “) € C, one has the following: Vj4 £ B : A* M(X*) = A* (Q + M(X*) - Q) - A* (Q + M {XM )) - A* Q > 0 The last inequality follows from the fact th at, for all A E B, A • Q < 0. Hence M { X M ) E int C. ’ ' □ The above theorem reduces the E F P y(M ), and hence the BMI feasibility problem, to the problem of determ ining the existence of a m atrix Q int (-C) th at yields a rank one solution for the corresponding Cone-LCPn(Q, M). If such a Q cannot be found, then the BMI does not have a solution. C o ro lla ry 4.3.9 The B M I does not have a solution if the Cone-LCPu{Q, M ) is not solvable fo r any Q £ int (— C). It is noteworthy that the linear m ap M in the Cone-LCP formulation, which arises in the context of the BMI, is itself copositive with respect to the m atrix cone B. P ro p o s itio n 4.3.10 The linear map M defined by (4-3.8) is B-copositive. Conse quently if roe define M*(X) = £ ” = 1 H-XH{, and the implication: X € B, X • M{X) = 0 = ► M{X) + M*{X) = 0 (4.3.13) holds, fo r all Q, the Cone-LCPe^Q, M ) is solvable i f it is feasible. Proof: Note th at M{X) 6 VST>, for all X G B. Since V S V C C and B* = C, X • M{X) > 0, for all X € B. If the im plication (4.3.13) holds, M(X) is indeed a 30 copositive plus map (which we have not defined in this paper), with respect to the cone B. Then solvability of the feasible copositive plus Cone-LCP follows from the generalization of the Lemke’s result which is discussed for example by Isac [10]. Q If one assumes that the copositive Cone-LCPs can be solved efficiently, as was men tioned in the Introduction, then the above proposition implies th at the Cone-LCPs which arise from the BMIs can be solved efficiently for each Q. Moreover, if for a particular M , implication (4.3.13) holds, then the corresponding Cone-LCP is al ways solvable, if the BMI has a solution (using Proposition 4.3.5). If the BMI has a solution, then knowledge of the direction of Q in the corresponding Cone-LCPb(Q,M) is sufficient for finding the solution of the BMI. In other words, if this direction is known, a param etric Cone-LCP approach can be used to examine for rank one matrices in SOLg°ne~L0P(Q, M ). In general, finding this direction is not trivial. In the case where an additional assum ption on the solution set of the EFPtf(M ) (BMI) can be made, the direction can be chosen apriori. C o ro lla ry 4.3.11 Suppose that fo r all X* £ SOL§FF(M), X “. M {X *) < ¥ • M { X m ) (4.3.14) fo r all extreme form s Y o f B (Y ^ X ). Then Q in Theorem 4-3.8 can be taken to be some positive multiple o f —I . Proof: W ithout loss of generality, assume th a t |[X*|| = I. Since M{X'f) G int C, there exists a > 0, such th at X~» (—a / + M ( X “)) = 0. To guarantee th at ~ a l + M(X) 6 C, we require th at for all Z € B, Z • (—a l + M(X)) > 0. But this is implied by the assum ption of the corollary, since if (4.3.14) holds, for all extreme forms Y = £ X, Y» ( - a l + M ( X W )) > X** ( - a + M(X*)) = 0. □ Corollary 4.3.11 reduces the BMI to a param etric Cone-LCP with a B-copositive linear map. In fact, the Q m atrices th at arise in this context have a very special form. 31 Another method for solving the EFPaf/W ), besides the param etric Cone-LCP ap proach, is to incorporate the problem of finding the Q , in setting up the correspond ing Cone-LCP. For this purpose it is convenient to associate to the m atrix cones B Let H = ]Cr=i Hi ® Hi E For the linear map M defined by (4.3.8) and using the property of the Kronecker products, vec M ( X ) = H vec X. Combining the above ideas with the result of Theorem 4.3.S, one readily obtains the following corollary. C o ro lla ry 4.3.12 Let, 2 2 — — and C (living in 55FP xp ), the cones B and C, which are obtained by applying the v ec operator to these m atrix cones, i.e., B — {x E : x = vec A , A E B} and, C = {x € 9 ftp 4 : x = v ec A , A g C} It is easy to verify th at B * = C, C*=B, and th at B and C are closed convex cones in 9 R p\ M = 0 0 and Then the B M I has a solution if and onlyif the Cone-LCPsy.B(0, M ) has a solution o f the form , z = ( vec X ) \ vec X ) where —Q E int C, and Q + M (X) is a sym m etric V ST ) m atrix with rank p — 1. The above corollary reduces the BMI feasibility problem to the problem of examining the solution set of a certain Cone-LCP. This can be a “tractable” problem if the 32 solution set is finite, or if the linear map M enjoys certain “additional” properties. Since there are many results in the com plem entarity theory which pertain to the cardinality of the solution set of a Cone-LCP [10], classification of efficiently solvable instances of the BMI can be based on those results as well. 33 Chapter 5 Conclusion In this thesis, we have reduced the Bilinear M atrix Inequality (BMI) to two cone problems defined over a generalization of the cone of completely positive matrices. The first problem, which we have referred to as the Extreme Form Problem (EFP), might be solvable by certain modifications of the interior point methods for Semi- Definite Programming, although at the present tim e, it is not clear exactly what modifications are required. The second problem, referred to as the Cone-LCP, is a generalization of the well studied Linear Complementarity Problem (LCP). Since certain classes of Cone-LCPs are amenable to an interior point approach, we have chosen to transform the EFP which arises from the BMI, to a Cone-LCP. In this case, the solution of the BMI (if it exists) is found, by solving either a parametric Cone-LCP, or by examining the solution set of an augmented Cone-LCP. Having found the solution of the EFP, the solution of the BMI is found by solving a Linear M atrix Inequalitj', which can be done efficiently by using the interior point methods. This work can be continued in several directions. One such direction is to study in greater detail the properties of the various m atrix cones introduced in this pa per, and the ways of devising interior point methods for the cone problems defined on them. The Extrem e Form Problem (EFP) seems to be of independent interest, which warrants research efforts for understanding of its properties and constructing computational procedures for its solution. 34 Reference List [1] F. Alizadeh. O ptim ization over the positive semi-definite cone: Interior-point m ethods and com binatorial application. In P. M. Pardalos, editor, Advances in Optimization and Parallel Computing. Elsevier Science, 1992. [2] A. Berman. Cones, Matrices, and Mathematical Programming. Springer- Verlag, 1973. [3] S. P. Boyd, L. EL Ghaoui, E. Feron, and V. Balakrishnan. Linear M atrix Inequalities in System s and Control Theory. SIAM, Philadelphia, 1994. [4] R. W . Cottle. J. S. Pang, and R. E. Stone. The Linear Complementarity Prob lem. Academic Press, 1992. [5] K. C. Goh, M. G. Safonov, and J. H. Ly. Robust synthesis via bilinear m atrix inequalities. International Journal o f Robust and Nonlinear Control, 1995. [6] C. Gonzaga. Search directions for interior point linear programming methods. Technical report, U C B /ER L M87/44, Electronics Research Laboratory, Uni versity of California, Berkeley, March 1987. [7] M. Hall, JR. Combinatorial Theory. Wiley, 1987. [8] R. A. Horn and C. R. Johnson. M atrix Analysis. Cambridge, 1985. [9] R. A. Horn and C. R. Johnson. Topics in M atrix Analysis. Cambridge, 1991. [10] G. Isac. Complementarity Problems. Springer-Verlag, 1992. [11] M. Kojima, N. Megiddo, and Y. Ye. An interior point potential reduction algorithm for the linear com plem entarity problem. Mathematical Programming, 54:267-279, 1992. 35 [12] M. Kojima, S. Shindoh, and S. Ilara. Interior-point methods for the monotone linear com plem entarity problem in sym m etric matrices. Technical report, De partm ent of Information Science, Tokyo Institute of Technology, Japan, 1994. [13] M. Mesbahi and G. P. Papavassilopoulos. On the P-m atrix linear complemen tarity problem. Working paper, D epartm ent of Electrical Engineering-Systems, University of Southern California, 1995. [14] Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Con vex Programming. SIAM, Philadelphia, 1994. [15] Y. E. Nesterov and M. J. Todd. Self-scaled cones and interior-point methods in nonlinear programming. Technical report, D epartm ent of Industrial Engineer ing, Cornell University, 1994. [16] J. Renegar. A polynomial-time algorithm based on Newton’s m ethod for linear programming. Mathematical Programming, 40:59-94, 19SS. [17] R. T. Rockafellar. Convex Analysis. Princeton, 1970. [18] M. G. Safonov, K. C. Goh, and J. H. Ly. Control system synthesis via bilinear m atrix inequalities. In Proceedings o f the 1994 American Control Conference, Baltimore, M.D., July 1994. [19] M. G. Safonov and G. P. Papavassilopoulos. The diam eter of an intersection of ellipsoids and BMI robust synthesis. In IFAC Symposium on Robust Control. Rio de Janeiro, Brazil, Septem ber 1994. [20] J. Stoer and C. W itzgall. Convexity and Optimization in Finite Dimension. Springer-Verlag, 1970. [21] O. Toker and H. Ozbay. On th e NP-hardness of solving bilinear m atrix inequal ities and sim ultaneous stabilization w ith static output feedback. In Proceedings of the 1995 American Control Conference, 1995. to appear. [22] Y. Ye and P. M. Pardalos. A class of linear com plem entarity problems solvable in polynomial time. Linear Algebra and its Applications, 152:3-17, 1991. 36 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. 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Creator Mesbahi, Mehran (author)

Core Title Complementarity problems over matrix cones in systems and control theory

School Graduate School

Degree Master of Science

Degree Program Applied Mathematics

Degree Conferral Date 1995-05

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,mathematics,OAI-PMH Harvest,operations research

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rozovskii, Boris (committee chair), Merendino, Salvatore (committee member), Rosen, Gary (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-3045

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