Chordal Sparsity, Decomposing SDPs and the Lyapunov Equation Richard P. Mason and Antonis Papachristodoulou Abstract— Analysis questions in control theory are often formulated as Linear Matrix Inequalities and solved using convex optimisation algorithms. For large LMIs it is important to exploit structure and sparsity within the problem in order to solve the associated Semidefinite Programs efficiently. In this paper we decompose SDPs by taking advantage of chordal sparsity, and apply our method to the problem of constructing Lyapunov functions for linear systems. By choosing Lyapunov functions with a chordal graphical structure we convert the semidefinite constraint in the problem into an equivalent set of smaller semidefinite constraints, thereby facilitating the solution of the problem. The approach has the potential to be applied to other problems such as stabilising controller synthesis, model reduction and the KYP lemma. I. I NTRODUCTION Many problems in control theory can be formulated as Linear Matrix Inequalities (LMIs) and solved using convex optimisation algorithms [1], [2]. In this paper we will focus on the Lyapunov 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 positive definite. It is well known that this problem can be solved using linear algebra by picking a Q ≺ 0 and solving for P [3]. Our motivation for studying the Lyapunov LMI is that it appears as a block within many key LMIs in control theory. For example, it appears within the KYP-lemma and the simultaneous stabilisation problem. For some of these problems there is no analytical solution and so we turn to iterative algorithms such as interior-point methods to solve the associated SDPs. The problem is that these interior-point methods do not scale well when P is a dense matrix. One approach to mitigate this problem is to restrict P to be a sparse matrix as this reduces the number of free variables in the LMI [4]. In general this is conservative, but it is known that for certain classes of systems a sparse P is sufficient to find a feasible solution to the Lyapunov LMI [5], [6]. In this paper we use two theorems from linear algebra that connect positive semidefinite matrices and chordal graphs and apply them to solve (1). These results first appeared in papers [7], [8] and have since been applied by several researchers in optimisation to decompose large SDPs, see [9]–[12]. R. P. M. and A. P. are with the Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK. R. P. M. was sup- ported by the Life Sciences Interface Doctoral Training Centre at the Uni- versity of Oxford. A. P. was supported in part by the Engineering and Phys- ical Sciences Research Council projects EP/J012041/1, EP/I031944/1 and EP/J010537/1. {richard.mason,antonis}@eng.ox.ac.uk We apply these theorems to the Lyapunov LMI by choos- ing the sparsity pattern of P so that P and Q have chordal sparsity patterns. This enables us to decompose the con- straint in the Lyapunov LMI into multiple constraints on submatrices of P and Q. When the resulting LMI is sparse this decomposition method allows us to solve the problem significantly faster than by using the standard dense method. A. Solving the Lyapunov LMI by Interior-Point Methods We wish to reformulate (1) as a standard SDP to see how the density of P affects the computational difficulty of the problem. First we write the Lyapunov LMI in the form  −P 0 0 A T P + PA  ≺ 0. (2) Note that since P is an n × n symmetric matrix it has up to m = n(n + 1)/2 free variables. Let W 1 , W 2 ,..., W m be the standard basis matrices for S n and define the matrices A 1 , A 2 ,..., A m ∈ S 2n by A i =  −W i 0 0 A T W i + W i A  , i = 1, 2,..., m. (3) We may then formulate the LMI feasibility problem as the constraint of an SDP maximise b T y subject to m ∑ i=1 y i A i + Z = A 0 Z  0 (4) where y ∈ R m , Z ∈ S 2n , b = 0 and A 0 = −ε I , ε > 0. The dual SDP to (4) is minimize A 0 • X subject to A i • X = b i , i = 1,..., m X  0 (5) where X ∈ S 2n and A • B = Tr(A T B)= ∑ i, j A ij B ij . The SDP above can then be solved using a primal-dual interior- point method (via a self-dual embedding) [13]–[16]. Primal- dual interior-point methods generate a sequence of points (X k , y k , Z k ) in the interior of the feasible set using iterations of the form (X k+1 , y k+1 , Z k+1 )=(X k , y k , Z k )+ α k (ΔX , Δy , ΔZ) where α k is a step length and (ΔX , Δy , ΔZ) is a Newton step direction that must be computed at each iteration. Calculating the step direction involves finding the solution to a square system of linear equations called the Schur complement system. When P is dense the size of the Schur