Abstract
This paper presents a theoretical result on convergence of a primal affine-scaling method for convex quadratic programs. It is shown that, as long as the stepsize is less than a threshold value which depends on the input data only, Ye and Tse's interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions. In addition, its local convergence rate is at least linear and the dual iterates have an ergodically convergent property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E.R. Barnes, “A variation on Karmarkar's algorithm for solving linear programming problems,”Mathematical Programming 36 (1989) 174–182.
I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Soviet Mathematics Doklady 8 (1967) 674–675.
I.I. Dikin, “On the speed of an iterative process,”Upravlyaemye Sistemi 12 (1974) 54–60.
R.C. Monteiro, I. Adler and M.G.C. Resende, “A polynomial-time primal—dual affine scaling algorithm for linear and convex quadratic programming and its power series extension,”Mathematics of Operations Research 15 (1990) 191–214.
P. Tseng and Z.-Q. Luo, “On the convergence of the affine-scaling algorithm,” Preprints CICS-P-169, M.I.T. (Cambridge, MA, 1989).
T. Tsuchiya, “Global convergence of the affine scaling methods for degenerate linear programming problems,” Research Memorandum No. 373, The Institute of Statistical Mathematics (Tokyo, Japan, 1990).
T. Tsuchiya, “Global convergence property of the affine scaling methods for primal degenerate linear programming problems,” Research Memorandum No. 367, The Institute of Statistical Mathematics (Tokyo, Japan, 1989).
R.J. Vanderbei and J.C. Lagarias, “Dikin's convergence result for the affine-scaling algorithm,” in: J. Lagarias and M. Todd,Contemporary Mathematics No. 114 (AMS, Providence, RI, 1990) pp. 109–119.
R.J. Vanderbei, M.S. Meketon and B.A. Freeman, “A modification of Karmarkar's linear programming algorithm,”Arithmica 1 (1986) 395–407.
Y. Ye, “An extension of Karmarkar's projective algorithm and the trust region method for quadratic programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming — Interior Point and Related Methods (Springer, Berlin, 1989).
Y. Ye and E. Tse, “An extension of Karmarkar's projective algorithm for convex quadratic programming,”Mathematical Programming 44 (1989) 157–180.
Author information
Authors and Affiliations
Additional information
Research supported in part by the NSF under grant DDM-8721709.
Rights and permissions
About this article
Cite this article
Sun, J. A convergence proof for an affine-scaling algorithm for convex quadratic programming without nondegeneracy assumptions. Mathematical Programming 60, 69–79 (1993). https://doi.org/10.1007/BF01580601
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580601