A General Framework for Establishing Polynomial Convergence of Long-Step Methods for Semidefinite Programming

This paper considers feasible long-step primal-dual path-following methods for semidefinite programming based on Newton directions associated with central path equations of the form Φ(PXP T , P−T SP−1) − νI = 0, where the map Φ and the nonsingular matrix P satisfy several key properties. An iteration-complexity bound for the long-step method is derived in terms of an upper bound on a certain scaled norm of the second derivative of Φ. As a consequence of our general framework, we derive polynomial iteration-complexity bounds for long-step algorithms based on the following four maps: Φ(X, S) = (XS + SX)/2, Φ(X, S) = Lx SLx, Φ(X, S) = X 1/2SX1/2, and Φ(X, S) = W 1/2XSW−1/2, where Lx is the lower Cholesky factor of X and W is the unique symmetric matrix satisfying S = WXW .