In this paper we present a convex optimization problem for solving the rational covariance extension problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou＇s seminal existence result and his conjecture, a stronger version of which we have resolved in [Byrnes et al., IEEE Trans. Automat. Control, AC-40 (1995), pp. 1841-1857].