On Long Step Path Following and SUMT for Linear and Quadratic Programming - Anstreicher, Kurt M. (SIAM journal on optimization, 1996)
We consider a long step barrier algorithm for the minimization of a convex quadratic objective subject to linear inequality constraints. The algorithm is a dual version of a method developed by Anstreicher et al. [Algorithmica, 10 (1993), pp. 365-382], and requires $O ( nL )$ or $O( \sqrt{n} L )$ iterations to solve a problem with n constraints, depending on how the barrier parameter is reduced. As a corollary of our analysis we demonstrate that the classical SUMT algorithm, exactly as implemented in 1968, solves linear and quadratic programs in $O( \sqrt{n} L\log L )$ iterations, with proper initialization and choice of parameters.