A polynomial-time inexact primal-dual infeasible path-following algorithm for convex quadratic SDP

Convex quadratic semidefinite programming (QSDP) has been widely applied in solving engineering and scientific problems such as nearest correlation problems and nearest Euclidean distance matrix problems. In this paper, we study an inexact primal-dual infeasible path-following algorithm for QSDP problems of the form: minX{12X • Q(X) + C •X : A(X) = b, X 0}, where Q is a self-adjoint positive semidefinite linear operator on Sn, b ∈ Rm, and A is a linear map from Sn to Rm. This algorithm is designed for the purpose of using an iterative solver to compute an approximate search direction at each iteration. It does not require feasibility to be maintained even if some iterates happened to be feasible. By imposing mild conditions on the inexactness of the computed directions, we show that the algorithm can find an -solution in O(n2 ln(1/ )) iterations. keywords: semidefinite programming, semidefinite least squares, infeasible interior point method, inexact search direction, polynomial complexity