A Note on Quadratic Convergence of the Homogeneous and Self-Dual Linear Programming Algorithm

Abstract

: In this note we show that Ye-Todd-Mizuno's O( p nL)-iteration homogeneous and self-dual linear programming (LP) algorithm possesses quadratic convergence of the duality gap to zero. In the case of infeasibility, this result shows that a homogenizing variable quadratically converges to zero and implies that the iterates of the (original) LP variable quadratically diverge. Thus, we have established a complete asymptotic convergence result for interior-point algorithms without any assumption on the LP problem. Key words: Linear Programming, interior point algorithms, homogeneity, self-dual, quadratic convergence. The Institute of Applied Mathematics, Academia Sinica, Beijing, China. y Department of Management Sciences, The University of Iowa, Iowa City, Iowa 52242, USA. This research was supported in part by NSF grant DDM-9207347 and the K.C. WONG Education Foundation, Hong Kong, through Academia Sinica, Beijing, China. 1 Introduction Consider linear programs in the following..