Chordal Sparsity, Decomposing SDPs and the Lyapunov Equation

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1 Chordal Sarsity, Decoosing SDPs and the Lyaunov Equation Richard P. Mason and Antonis Paachristodoulou Abstract Analysis questions in control theory are often forulated as Linear Matrix Inequalities and solved using convex otiisation algoriths. For large LMIs it is iortant to exloit structure and sarsity within the roble in order to solve the associated Seidefinite Progras efficiently. In this aer we decoose SDPs by taking advantage of chordal sarsity, and aly our ethod to the roble of constructing Lyaunov functions for linear systes. By choosing Lyaunov functions with a chordal grahical structure we convert the seidefinite constraint in the roble into an equivalent set of saller seidefinite constraints, thereby facilitating the solution of the roble. The aroach has the otential to be alied to other robles such as stabilising controller synthesis, odel reduction and the KYP lea. I. INTRODUCTION Many robles in control theory can be forulated as Linear Matrix Inequalities LMIs) and solved using convex otiisation algoriths [1], [2]. In this aer we will focus on the Lyaunov LMI: given A R n n, find P 0 such that Q = A T P + PA 0, 1) where A B denotes that A B is ositive definite. It is well known that this roble can be solved using linear algebra by icking a Q 0 and solving for P [3]. Our otivation for studying the Lyaunov LMI is that it aears as a block within any key LMIs in control theory. For exale, it aears within the KYP-lea and the siultaneous stabilisation roble. For soe of these robles there is no analytical solution and so we turn to iterative algoriths such as interior-oint ethods to solve the associated SDPs. The roble is that these interior-oint ethods do not scale well when P is a dense atrix. One aroach to itigate this roble is to restrict P to be a sarse atrix as this reduces the nuber of free variables in the LMI [4]. In general this is conservative, but it is known that for certain classes of systes a sarse P is sufficient to find a feasible solution to the Lyaunov LMI [5], [6]. In this aer we use two theores fro linear algebra that connect ositive seidefinite atrices and chordal grahs and aly the to solve 1). These results first aeared in aers [7], [8] and have since been alied by several researchers in otiisation to decoose large SDPs, see [9] [12]. R. P. M. and A. P. are with the Deartent of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK. R. P. M. was suorted by the Life Sciences Interface Doctoral Training Centre at the University of Oxford. A. P. was suorted in art by the Engineering and Physical Sciences Research Council rojects EP/J012041/1, EP/I031944/1 and EP/J010537/1. richard.ason,antonis}@eng.ox.ac.uk We aly these theores to the Lyaunov LMI by choosing the sarsity attern of P so that P and Q have chordal sarsity atterns. This enables us to decoose the constraint in the Lyaunov LMI into ultile constraints on subatrices of P and Q. When the resulting LMI is sarse this decoosition ethod allows us to solve the roble significantly faster than by using the standard dense ethod. A. Solving the Lyaunov LMI by Interior-Point Methods We wish to reforulate 1) as a standard SDP to see how the density of P affects the coutational difficulty of the roble. First we write the Lyaunov LMI in the for [ P 0 0 A T P + PA ] 0. 2) Note that since P is an n n syetric atrix it has u to = nn + 1)/2 free variables. Let W 1,W 2,...,W be the standard basis atrices for S n and define the atrices A 1,A 2,...,A S 2n by [ ] Wi 0 A i = 0 A T, i = 1,2,...,. 3) W i +W i A We ay then forulate the LMI feasibility roble as the constraint of an SDP axiise b T y i=1 Z 0 y i A i + Z = A 0 where y R, Z S 2n, b = 0 and A 0 = εi, ε > 0. The dual SDP to 4) is A 0 X A i X = b i, i = 1,..., X 0 where X S 2n and A B = TrA T B) = i, j A i j B i j. The SDP above can then be solved using a rial-dual interioroint ethod via a self-dual ebedding) [13] [16]. Prialdual interior-oint ethods generate a sequence of oints X k,y k,z k ) in the interior of the feasible set using iterations of the for X k+1,y k+1,z k+1 ) = X k,y k,z k ) + α k X, y, Z) where α k is a ste length and X, y, Z) is a Newton ste direction that ust be couted at each iteration. Calculating the ste direction involves finding the solution to a square syste of linear equations called the Schur coleent syste. When P is dense the size of the Schur 4) 5)

2 Let JC) = i, j) C C 1 i j n} for every C V. In a connected grah the axial cliques will overla, i.e., C i C j /0 for soe C i,c j C. Given a clique tree that satisfies the clique intersection roerty we denote the inial set of overlaing eleents by Λ, where Λ = i, j,k,l) i, j) JC k C l ), C k,c l ) E }. Fig. 1: An exale of a chordal grah. coleent syste is = nn+1)/2 and hence when is n is large the tie required to solve the LMI can be rohibitive. This otivates us to exlore sarse P atrices, and in articular we will consider the case where P and Q have a chordal sarsity attern. II. PRELIMINARIES Let G = V,E) be an undirected grah, with set of nodes V = 1,2,...,N} and set of edges E V V. For a subset V V, the induced subgrah is the grah G = V,E ) with vertex set V and edge set E = E V V ). A clique is a subset C V such that for any i, j C, i, j) E. A axial clique is a clique that is not a subset of another clique. A cycle is a sequence of airwise distinct vertices γ = ν 1,ν 2,...,ν s ) having the roerty that ν 1,ν 2 ),ν 2,ν 3 ),...,ν s 1,ν s ),ν s,ν 1 ) E and s is the length of the cycle. A chord of the cycle γ is an edge ν i,ν j ) E where 1 i < j s, i, j) 1,s), and i j 2. A grah is chordal if every cycle of length 4 has a chord, see Figure 1. A vertex ν is called silicial if all its neighbours are adjacent to each other. A bijection α : V 1,...,N} is called an ordering of G. For silicity we write an ordering as α = ν 1,ν 2,...,ν N where ν i = α 1 i). An ordering α of G is a erfect eliination ordering if, for each i 1,...,N}, ν i is a silicial vertex of the subgrah induced by ν i,...,ν N }. Theore 1: [17] G is chordal if and only if G has a erfect eliination ordering. Let G be a connected grah with set of axial cliques C = C 1,C 2,...,C }. A clique tree, T = C,E ) with E C C, is a connected grah with no cycles. A clique tree is said to satisfy the clique intersection roerty if for every air of distinct C i,c j C, the set C i C j is contained in every clique on the unique ath connecting C i and C j in the clique tree. Theore 2: [18] A connected grah G is chordal if and only if there exists a clique tree T = C,E ) for which the clique intersection roerty holds. III. CHORDAL DECOMPOSITIONS OF POSITIVE SEMIDEFINITE MATRICES Let G = V,E) be an undirected grah and assue that i,i) E for all i V, i.e., each node has a self-loo. A artial syetric atrix, X, is a syetric N N atrix where eleent X i j is secified if and only if i, j) E. A coletion of X is an N N atrix M which satisfies M i j = X i j for all i, j) E. We say that M is a ositive coletion of X if and only if M is a coletion of X and M is ositive seidefinite. We will use the following notation: S N E,?) = the set of N N artial syetric atrices with eleents defined on E S N +E,?) = X S N E,?) M 0,M i j = X i j i, j) E} S N E,0) = X S N X i j = 0 if i, j) E} S N +E,0) = X S N E,0) X 0} S C = X S N X i j = 0 if i, j) C C} for every C V S C + = X S C X 0} for every C V XC) = X S C such that X i j = X i j i, j) C C) for every X S N and every C V. U i j = the N N syetric atrix with 1 in the i, j)th and j,i)th eleents and 0 elsewhere. Theore 3: [7] Let G = V,E) be a chordal grah with set of axial cliques C = C 1,...,C }. Suose that X S N E,?). Then X S N +E,?) if and only if XC k ) 0 for k = 1,2,...,. Dual to the cone S N +E,?) is the cone S + E,0) and the following can be derived fro Theore 3 using this dual relationshi [19]. Theore 4: [8] Let G = V,E) be a chordal grah with set of axial cliques C = C 1,...,C }. Suose that A S N E,0). Then A S N +E,0) if and only if there exists a set of atrices A 1,A 2...,A } such that A = A k, A k S C k +, k = 1,...,. For our uroses we restate Theore 4 using clique trees. Theore 5: [12] Let G = V,E) be a chordal grah with set of axial cliques C = C 1,...,C }. Suose that A S N E,0). Then A S N +E,0) if and only if the syste of LMIs A k L k θ) S C k +, k = 1,2,...,, is feasible, where θ is a vector of variables of the for θ = θ i jkl i, j,k,l) Λ) and L k θ) = i, j,l) i, j,k,l) Λ U i j θ i jkl i, j,h) i, j,h,k) Λ for every θ = θ i jkl i, j,k,l) Λ), k 1,2,..., }. θ i jhk U i j

3 IV. CHORDAL DECOMPOSITION OF THE LYAPUNOV LMI In this section we return to the SDP forulation of the Lyaunov LMI 4)-5) and exlain how it can be decoosed using Theores 3 and 4. Suose that A S N E,0), where G = V, E) is not necessarily chordal. A chordal extension of G, denoted by G ch = V,F), is a chordal grah where the edge set satisfies E F. Since i, j) E A i j = 0, for an arbitrary X S N A X = i, j) E A i j X i j = A i j X i j. 6) A given grah G ay have any chordal extensions and the roble of finding the chordal extension that adds the iniu nuber of edges is NP hard, but fortunately there are effective heuristics for finding inial chordal extensions. In this aer we find a chordal extension for G by eruting the adjacency atrix using a iniu degree ordering and then erforing a sybolic Cholesky factorisation. Now consider 5). Suose that A 0,A 1,...,A S N E,0) and relace the inner roducts using 6). Then we can write the rial for of the SDP in an equivalent for [A 0 ] i j X i j [A q ] i j X i j = b q, q = 1,2,..., X S N +F,?), where we have used the fact that the values of the objective and constraint functions are not affected by coleting X. Fro Theore 3 the seidefinite constraint will be satisfied if and only if the subatrices of X corresonding to the axial cliques of G = V,F) are ositive seidefinite. Let C = C 1,C 2,...,C } be the axial cliques of G = V,F), then 7) can be written as: [A 0 ] i j X i j [A q ] i j X i j = b q, q = 1,2,..., X k S C k +, k = 1,2,...,. We ay decoose any A q SE,0) into a suation of atrices of the for A k q S C k since E F, i.e., A q = 7) 8) A k q, A k q S C k, k = 1,2,...,. 9) Using 9) we can write the inner roduct of A q with an arbitrary atrix X S N as A q X = [A q ] i j X i j = [A k q] i j X i j ). Since A k q S C k A q X = we ay relace this equality with ) A k q XC k ) for every X S N. 10) By using 10) we have ) A k 0 XC k) ) A k q XC k ) = b q, q = 1,2,..., XC k ) S C k +, k = 1,2,...,. 11) Unfortunately this is not a standard SDP as the subatrices in the seidefinite constraints XC k ) 0 share eleents and so are not indeendent. To convert this into a standard SDP Fukuda et. al. introduce new indeendent variables X 1,X 2,...,X where X k S C k + and new constraints that ensure equality between the overlaing eleents [9]. The SDP roble then becoes A k 0 X k) A k q X k) = b q, q = 1,2,..., U i j X k U i j X l = 0, i, j,k,l Λ) X k S C k +, k = 1,2,..., 12) where the constraint U i j X k U i j X l = 0 enforces that the variable X i j shared by axial cliques k and l ust be equal 1. Note that the single large seidefinite constraint in 5) has been relaced by ultile seidefinite constraints on saller atrix variables and that the SDP is now in a block diagonal for. This is iortant because SDP solvers such as SeDuMi can exloit this block diagonal for to calculate the next iterate of the Newton ste ore efficiently. A. Dual SDP We now decoose the dual SDP using Theore 4. Consider 4). Under our assution on the aggregate sarsity attern of the LMI the feasibility constraints of the dual SDP are q y q A q + Z = A 0, Z S N +E,0). 13) i=1 Let G ch = V,F) be a chordal extension of G = V,E), with set of axial cliques C = C 1,C 2,...,C }. Using this chordal extension we can write 13) in the equivalent for y q A q + Z = A 0, Z S N +F,0). 14) q=1 Note that the constraint Z i j = 0 if i, j) F \ E is ilicit in these constraints, since Ay) = A 0 q=1 y qa q S N E,0) for all y R. Let A k q for k = 1,2,..., and q = 0,1,2,..., be atrices that satisfy ) A k 0 q=1y q A k q = A 0 q i=1 y q A q. 1 Technically, Λ consists of two disjoint trees since the Lyaunov SDP is block diagonal with two blocks.

4 The following roosition uses Theore 4 to decoose the constraints of the dual SDP. Proosition 1: The set of constraints 14) is feasible if and only if there exist atrices Z k S C k and L k θ) S C k for k = 1,2,..., that satisfy Z k = A k 0 q=1y q A k q L k θ), k = 1,2,..., Z k S C k +, k = 1,2,..., where L k θ) are as defined in Theore 5. Proof: Let Z = Z k and note that Lk θ) = 0 to give the first condition. The second condition then follows fro Theore 4, Z S N +F,0) Z = Z k, Z k S C k +. With these equivalent constraints the decoosition of the dual SDP is axiize b T y y q A k q + L k θ) + Z k = A k 0, k = 1,2,..., q=1 Z k S C k +, k = 1,2,...,. 15) Note that the SDPs 12) and 15) are Lagrange duals of one another derivation given in the Aendix). For each equality constraint U i j X k U i j X l = 0 in the rial roble there is an associated dual variable θ i jkl. We note that this decoosition has been exloited to decoose robles in distributed robust stability analysis using IQCs [20]. For further inforation on the conversion rocess we refer the reader to aers [9], [10], [12]. Solvers that exloit chordal sarsity in general SDPs are available, in articular SMCP [11] and SDPA-C [21]. Also available is SarseCoLO which autoates the reforulation of SDPs with chordal sarsity to facilitate solution using standard solvers [12]. V. STRUCTURED SYSTEMS Throughout this section let ẋt) = Axt) be the syste that we wish to construct a Lyaunov function for. We consider five classes of A atrices that allow us to choose P so that P and Q have a chordal sarsity attern. By finding chordal sarsity atterns we are then able to efficiently test for Lyaunov functions of this for using the chordal decoosition of the SDP. A. Banded Matrices We say that A R n n is a banded atrix of bandwidth d if there exists an integer 1 < d < n such that A i j 0 if and only if i j d. This for of A atrix corresonds to the case where the syste consists of a chain of overlaing subsystes. Note that banded atrices are a subset of chordal grahs. In this case we can choose P to also be a banded atrix of order d = 1,2,...,n 1 and this gives a sequence of banded Q atrices of increasing order u to a colete grah. B. Cyclical Matrices Another class of A atrices for which we can find P and Q with chordal sarsity atterns is the class of cyclical atrices. In a cyclical atrix the nodes are connected in a ring or a cycle toology which can be reresented by the following sarsity attern A i j = 1 if i j 1 1 if i, j) = 1,n) or n,1) Let the sarsity attern of P be defined by 1 if i = j 1 if i + j = n + 1 P i j = 1 if i + j = n ) 17) Theore 6: Given A R n n with a cyclic sarsity attern, the sarsity attern of P defined in 17) is chordal and results in a Q = A T P + PA with a chordal sarsity attern. Proof given in the aendix). C. Tree Matrices Let G = V,E) be a tree, i.e, a grah that is connected and has no cycles. We ick one node to be the root of the tree and label it r. We say that node u is the arent of node v and v is the child of u if u is the unique node adjacent to v on the unique ath fro r to v. We define the siblings of a node u to be the set of all nodes with the sae arent as u including u). Given node i V, we denote its arent by Pari), its children by Chi) and its siblings by Sibi). We nuber the nodes in a toological ordering, eaning that each arent has a higher nuber than its children. Let Mi) = i} Pari) Chi). Given a tree T, define the sarsity attern of A by 1 if j Mi) A i j = 18) Let the sarsity attern of P to be defined by 1 if j Sibi) P i j = 19) Theore 7: Given A R n n with a sarsity attern defined by the tree T, the sarsity attern of P defined in 19) is chordal and results in a Q = A T P + PA with a chordal sarsity attern. Proof given in the aendix). D. Metzler Matrices and Triangular Matrices A atrix A R n n is said to be Metzler if all its off diagonal eleents are nonnegative i.e, a i j 0 i j. Proosition 2: [6] Let A R n n be a Metzler atrix. Then the following stateents are equivalent: i) the atrix A is Hurwitz ii) there exists a ositive definite, diagonal atrix P 0 such that Q = A T P + PA 0. If we further assue that A is sarse, then by choosing P to be diagonal we ensure that the structure of A is reserved in Q. Let G = V,E) be the grahical structure of Q, then

5 we can find a chordal extension G = V,F) where E F. Therefore, given a sarse Metzler atrix we can aly the chordal decoosition to the SDP so that we can efficiently test whether the syste is stable. Reark 1: It is known that stable triangular atrices also adit a diagonal Lyaunov function [22]. Therefore when a atrix is triangular and sarse we can aly exactly the sae decoosition as with the Metzler atrices. E. Sarse Non-chordal Matrices Chordal extensions allow us to generalise the chordal decoosition aroach to non-chordal sarse systes. Given a sarse atrix A R n n, choose a sarsity attern for P, i.e., GP) = V,R). This ilies a sarsity attern for Q i.e., GQ) = V,S) where Q = A T P+PA. The sarsity atterns of P and Q will not be chordal in general, but we can chordal extend the to G ch P) = V,T ) and G ch Q) = V,U) and then aly the chordal decoosition as described in the revious section. This suggests a ethod to iteratively search for sarse Lyaunov functions: Algorith: Given A R n n. Initialise = 1 while: n end P = banded atrix with bandwidth. Coute GQ) where Q = A T P + PA Chordal extend GP) and GQ) to G ch P) and G ch Q) Solve the chordal SDP 12) or 15). if: SDP is feasible else: end A Lyaunov function has been found = + 1 VI. NUMERICAL RESULTS In this section we give nuerical results that deonstrate the iroveents in efficiency ossible when chordal sarsity in an LMI is taken advantage of. We ake use of SarseCoLO which detects chordal sarsity in an LMI and rerocesses the data so that the atrix variables are block diagonal. The conversion is erfored using the clique tree conversion ethod described in Section IV. After this rerocessing ste, SarseCoLO then calls either SeDuMi, SDPA or SDPT3 to solve the roble [12]. To test the chordal decoosition ethod we generated rando atrices with negative eigenvalues and iosed on the the roerties described in SectionV to create sarse Metzler, sarse triangular, banded, cyclical and tree structured A atrices. Then we coared the tie taken for SeDuMi and SarseCoLO+SeDuMi to solve the Lyaunov LMI. For all of our exerients the SDP was considered to be solved when the rial-dual ga had been reduced to less than ε = The exerients were run on a MacBook Pro with a 2.9GHz rocessor and 8GB of RAM. For the case of banded atrices we chose A and P to be banded with order d = 5. To generate the trees we sily started with the trivial tree ν} and then added edges to a node not in the tree, reeating until all of the nodes were connected to the tree. For the Metzler and triangular cases we generated sarse grahs and then found a chordal extension to deterine how to decoose the roble. To generate sarse non-chordal atrices we first generated sarse chordal grahs and then added in rando eleents using the MAT- LAB coand srandsyn, density) where density = 0.1/n. Table I shows the CPU tie in seconds required to solve the SDPs for the five different sarsity atterns. The variable axc corresonds to the largest axial clique in the sarsity attern of the LMI. MaxC was ket constant as the nuber of nodes was increased. The nubers in the brackets e.g., 230,17) are the size and nuber of blocks the Schur coleent atrix resectively. We see that the size of the Schur coleent atrix is larger when we aly the chordal decoosition ethod. This is due to the extra variables that are introduced the dual variables of the equality constraints θ)). However, the roble is decoosed into a nuber of blocks of saller size which SeDuMi can exloit to solve the roble ore efficiently. Hence the chordal decoosition ethod is significantly ore efficient in these sarse cases with sall axial cliques. As the nuber of nonzero eleents in the A atrix increases, this tends to increase the density of the atrices in the LMI. This leads to larger axial cliques, with ore overlaing eleents, which in the chordal decoosition requires us to introduce ore dual variables. This increases the size of the Schur coleent atrix. Table II shows the effect of increasing axial clique sizes on the tie taken to solve the Lyaunov LMI for sarse Metzler atrices with n = 400. We see that for sall axc the chordal decoosition is ore efficient than the standard dense ethod, but as axc is ade larger the increased size of the Schur coleent atrix outweighs the advantages of decoosing the roble into blocks. VII. CONCLUSION We have shown that chordal sarsity in the Lyaunov LMI can be exloited to irove the efficiency with which the associated SDPs are solved. We alied the ethod to construct Lyaunov functions for randoly generated linear systes fro five different classes. It was shown that when the size of the axial cliques is sall the chordal decoosition ethod is significantly quicker than the standard ethod. This akes it a roising aroach when alied to large, sarse systes which are often found in alications. In the future we will aly this ethodology to other LMIs arising in systes and control theory, such as in

6 Banded Matrices axc = 11) n SeDuMi SarseCoLO+SeDuMi ,2) ,27) ,2) ,51) ,2) ,76) ,2) ,101) Cyclical Matrices axc = 4) n SeDuMi SarseCoLO+SeDuMi ,2) ,33) ,2) ,65) ,2) ,95) ,2) ,127) Tree Matrices axc = 15) n SeDuMi SarseCoLO+SeDuMi ,2) ,73) ,2) ,146) ,2) ,217) ,2) ,289) Sarse Metzler axc = 10) n SeDuMi SarseCoLO+SeDuMi ,2) ,17) ,2) ,37) ,2) ,53) ,2) ,71) Sarse Triangular axc = 10) n SeDuMi SarseCoLO+SeDuMi ,2) ,24) ,2) ,41) ,2) ,57) ,2) ,70) Sarse Non-chordal axc = 12) n SeDuMi SarseCoLO+SeDuMi ,2) ,31) ,2) ,73) ,2) ,97) ,2) ,70) TABLE I: SeDuMi CPU tie vs SarseCoLO+SeDuMi CPU tie in seconds to solve the Lyaunov LMI for different sarsity atterns, size of Schur coleent atrix, no. of blocks). Sarse Metzler n= 400) axc nnza) SeDuMi SarseCoLO+SeDuMi ,2) ,35) ,2) ,36) ,2) ,44) ,2) ,192) ,2) ,172) TABLE II: SeDuMi CPU tie vs SarseCoLO+SeDuMi CPU tie in seconds to solve the Lyaunov LMI for Metzler atrices with n = 400 nodes whilst varying the size of the axial cliques, size of Schur coleent atrix, no. of blocks). the KYP lea. This will allow us to bound the inutouut roerties of large, sarse systes. Another roising direction is to aly this aroach to the stability analysis of systes with olynoial vector fields by using Su of Squares. We will also investigate how to use inforation about the syste, in ters of dynaics or sectral roerties, in order to infor the choice of sarsity attern for the Lyaunov function VIII. APPENDIX A. Derivation of the dual SDP fro the rial SDP The Lagrangian function for 12) is LX,y,θ,Z) = i, j,k,l Λ A k 0 X k) q=1y q θ i jkl U i j X k U i j X l) A k q X k) b q ) Z k X k) where Z k S C k + for k = 1,2,...,. The dual function is defined by gy,θ,z) = inf LX,y,θ,Z). Define X Lk θ) to be L k θ) = i, j,l):i, j,k,l) Λ U i j θ i jkl i, j,h):i, j,h,k) Λ U i j θ i jhk. Taking the artial derivative of the Lagrangian function with resect to X k we have L X k = Ak 0 y q A k q L k θ) Z k = 0, 20) q=1 and fro this we see that the dual function is given by b gy,θ,z) = T y if 20) holds and Z k S C k + k, otherwise. Maxiising this dual function gives 15). B. Proof of Theore 6 Let A and P have sarsity atterns as defined in 16) and 17). Note that P can be rearranged to be a banded atrix and hence have a chordal sarsity attern. The sarsity attern of Q = A T P + PA is given by Q i j = 1 if i j 1 1 if n i + j n ) The roof that Q as defined above is chordal is by induction. We use the notation that Q n is the Q atrix for an n node cycle. For n = 1,2,3 there cannot be a cycle of length greater or equal to 4 so these grahs ust be chordal. We first consider the even case and take Q 4 as our base case. Since it is a colete grah it is chordal. Next consider Q n with n even and assue that it is chordal. Construct Q n+2 by relabelling the nodes of Q n by increenting ν i := ν i+1, for i = 1,2,...,n and then adding two new nodes ν 1,ν n+2 } to the erihery of the network and connecting the according to 21), see Figure 2. Let the erfect eliination ordering for Q n be denoted by α. If we can find a erfect eliination ordering for Q n+2 of the for ν 1,ν n+2,α then we have shown that Q n+2 is chordal. Consider the first row of Q n+2 Q n+2 1,k = 1, for k 1,2,n + 1,n + 2}, In order for node 1 to be silicial, nodes 2,n+1 and n+2 ust for a clique. Fro 21) we see that Qi n+2 j = 1 for n + 2 i + j n + 5 and hence these nodes do indeed for

7 a) Q 6 b) Q 8 Fig. 2: Constructing Q 8 fro Q 6 Drawn without self-loos) a clique. Siilarly for node n + 2, after eliinating node 1 the n + 2th row of Q n+2 is Qn+2,k n+2 = 1, for k 2,3,n + 1,n + 2}, Again using the definition of Q n+2 we see that the nodes 2,3 and n+2 for a clique and hence n+2 is a silicial vertex. Hence the ordering ν 1,ν n+2,α is a erfect eliination ordering, and by induction Q n is chordal for all even n. The sae arguent can be alied to the odd n case, which coletes the roof. C. Proof of Theore 7 Let A and P have sarsity atterns as defined in 18) and 19). Note that P can be rearranged to be block diagonal and hence have a chordal sarsity attern. The eleent-wise equation for the sarsity attern of Q is Q i j 0 n A ik P k j + n P ik A k j 0 where we have used the fact that the sarsity attern of A is syetric. Fro the definitions of the sarsity attern of A and P we conclude that the sarsity attern of Q is given by 1 if Mi) Sib j) /0) or Sibi) M j) /0) Q i j = 22) Let the toological ordering α = 1,2,...,n} be a candidate for a erfect eliination ordering. The subgrah induced by ν i,...,ν n } cannot include the children of ν i or the children of its siblings, since these nodes have a lower nuber in the toological ordering. The reaining nodes that ν i is connected to are: its siblings with a higher nuber in the ordering, its arent and the siblings of its arent. It reains to be shown that all of the siblings of i are adjacent to all of the siblings of the arent of node i. Let j Sibi) and k SibPari)), using 22) we have M j) Sibk) = Pari) /0 Q jk = 1,Q k j = 1. Hence the siblings of the arent of node i are adjacent to all of the siblings of i. Therefore ν i is a silicial vertex and α is a erfect eliination ordering. We conclude that Q is a chordal grah. REFERENCES [1] S. Boyd, L. E. Ghaoui, E. 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