A Simple Path-following Algorithm for the Feasibility Problem in Semidefinite Programming and for Matrix Scaling over the Semidefinite Cone

Let E be the Hilbert space of symmetric matrices of the form diag(A,M), where A is n× n, and M is an l× l diagonal matrix, and the inner product 〈x, y〉 ≡ Trace(xy). Given x ∈ E, we write x ≥ 0 (x > 0) if it is positive semidefinite (positive definite). Let Q : E → E be a symmetric positive semidefinite linear operator, and μ = min{φ(x) = 0.5Trace(xQx) : ‖x‖ = 1, x ≥ 0}. The feasibility problem in SDP can be formulated as the problem of testing if μ = 0 for some Q. Let ǫ ∈ (0,1) be a given accuracy, u = Qe− e, e the identity matrix in E, and N = n+ l. We describe a simple path-following algorithm that in case μ = 0, in O( √ N ln[N‖u‖/ǫ]) Newton iterations produces x ≥ 0, ‖x‖ = 1 such that Trace(xQx) ≤ ǫ. If μ > 0, in O( √ N ln[N‖u‖/μǫ]) Newton iterations the algorithm produces d > 0 such that ‖DQDe − e‖ ≤ ǫ, where D is the operator that maps w ∈ E to d1/2wd1/2. Moreover, we use the algorithm to prove: μ > 0, if and only if there exists d > 0 such that DQDe = e, if and only if there exists d > 0 such that Qd > 0. Thus the matrix scaling problem is a natural dual to the feasibility problem in SDP. Although the above complexities can be deduced from a path-following algorithm for general self-concordant homogeneous programming and for matrix scaling obtained by Kalantari [8], for the problems considered here the present analysis is quite elementary and short, while complete. This simplicity is mainly due to a new inequality derived in this paper that relates norm of scaled quantities at two successive Newton iterations and implies quadratic rate of convergence. The present algorithm is not only much simpler than the existing path-following algorithms for solving the feasibility problem in SDP, but is also capable of solving the matrix scaling problem. When n = 0, the algorithm reduces to the diagonal matrix scaling/linear programming algorithm of Khachiyan and Kalantari [11]. As in the case of LP the algorithm of this paper can be used to solve the general SDP problem.