Global Convergence of the Affine Scaling Method for Convex Quadratic Programming

Renato D. C. Monteiro and Takashi Tsuchiya

In this paper we give a global convergence proof of the second order affine scaling algorithm for convex quadratic programming problems, where the next iterate is the point which minimizes the objective function over the intersection of the feasible region with the ellipsoid centered at the current point and whose radius is a fixed fraction $\beta \in (0,1]$ of the radius of the largest ``scaled'' ellipsoid inscribed in the nonnegative orthant. The analysis is based on the local Karmarkar potential function introduced by Tsuchiya. For any $\beta \in (0, 2/3]$ and without assuming any nondegeneracy assumption on the problem, it is shown that the sequences of primal iterates and dual estimates converge to optimal solutions of the quadratic program and its dual, respectively.

Preprint, Georgia Institute of Technology, March, 1995.