New Trends in Mathematical Programming pp 137-157 | Cite as

# On Primal—Dual Path—Following Algorithms for Semidefinite Programming

## Abstract

Interior point methods for semidefinite programming have recently been studied intensively, due to their polynomial complexity and practical efficiency. Most of these methods are extensions of linear programming algorithms. The primal-dual central path following method for linear programming by Jansen et al. [6] has recently been extended to semidefinite programming by Jiang [7], utilizing the Nesterov-Todd direction and introducing a new distance measure. In this note we refine and extend this analysis: A weaker condition for a feasible full Newton step is established, and quadratic convergence to target points on the central path is shown. Moreover, we show how to compute large dynamic target updates which still allow full Newton steps.

## Keywords

Interior Point Method Central Path Semidefinite Program Quadratic Convergence Superlinear Convergence## Preview

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## References

- 1.F. Alizadeh and J.-P.A. Haeberley and M.L. Overton,
*Primal-dual methods for semidefinite programming: convergence rates, stability and numerical results*, 721, NYU Computer Science Dept, New York University, New York, NY (1996).Google Scholar - 2.K. M. Anstreicher and M. Fampa,
*A long-step path following algorithm for semidefinite programming problems*, Working Paper, Department of Management Sciences, University of Iowa, USA, (1996).Google Scholar - 3.L. Faybusovich,
*Semi-definite programming: a path-following algorithm for a linear-quadratic functional*, SIAM Journal on Optimization, 6(1996), pp. 10071024.Google Scholar - 4.D. Goldfarb and K. Scheinberg,
*Interior point trajectories in semidefinite programming*, Dept. of IEOR, Columbia University, New York, NY, (1996).Google Scholar - 5.
*B. He and E. de Klerk and C. Roos and T. Terlaky*Method of approximate centers for semi-definite programming*96–27, 1996, Faculty of Technical Mathematics and Computer Science, Delft University of Technology, (To appear in*Optimization Methods and Software.)Google Scholar - 6.B. Jansen and C. Roos and T. Terlaky and J.-Ph. Vial,
*Primal-dual algorithms for linear programming based on the logarithmic barrier method*, Journal of Optimization Theory and Applications, 83 (1994), pp. 1–26.MathSciNetCrossRefzbMATHGoogle Scholar - 7.J. Jiang,
*A long step primal-dual path following method for semidefinite programming*, Dept of Applied Mathematics, Tsinghua University, Beijing, China (1996).Google Scholar - 8.E. de Klerk and C. Roos and T. Terlaky,
*Initialization in semidefinite programming via a self-dual, skew-symmetric embedding*, 96–10, Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands, (1996). ( To appear in OR Letters. )Google Scholar - 9.E. de Klerk and C. Roos and T. Terlaky,
*Polynomial primal-dual affine scaling algorithms in semidefinite programming*, 96–42, Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands, (1996). ( To appear J. Comb. Opt. )Google Scholar - 10.M. Kojima and M. Shida and S. Shindoh,
*A Note on the Nesterov-Todd and the Kojima-Shindoh-Hara search directions in Semidefinite Programming*, B313, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan (1996).Google Scholar - 11.M. Kojima and S. Shindoh and S. Hara, Interior point methods for the monotone semidefinite linear complementarity problems
*No. 282, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan (1994). (To appear in*SIAM Journal on Optimization)Google Scholar - 12.Z.-Q. Luo and J. F. Sturm and S. Zhang,
*Superlinear convergence oa a Symmetric primal-dual path following algorithm for semidefinite programming*, 9607/A, Tinbergen Institute, Erasmus University Rotterdam (1996).Google Scholar - 13.I. J. Lustig and R. E. Marsten and D. F. Shanno,
*Interior point methods: Computational state of the art*, ORSA Journal on Computing, 6 (1994), pp. 1–15.MathSciNetCrossRefzbMATHGoogle Scholar - 14.S. Mizuno and M. J. Todd and Y. Ye, On
*adaptive step primal-dual interior-point algorithms for linear programming*, Mathematics of Operations Research, 18 (1993) pp. 964–981.MathSciNetCrossRefzbMATHGoogle Scholar - 15.Y. Nesterov and M. J. Todd, Self
*-scaled barriers and interior-point methods for convex programming*, 1091, School of OR and IE, Cornell University, Ithaca, New York, USA, (1994). ( To appear in Mathematics of Operations Research )Google Scholar - 16.C. Roos and T. Terlaky and J.-Ph. Vial,
*Theory and Algorithms for Linear Optimization: An interior point approach*, To appear December 1996, John Wiley & Sons, New York.Google Scholar - 17.J. F. Sturm and S. Zhang,
*Symmetric primal-dual path following algorithms for semidefinite programming*, 9554/A, Tinbergen Institute, Erasmus University Rotterdam, (1995).Google Scholar - 18.M. J. Todd and K. C. Toh and R. H. Tütüncü,
*On the Nesterov-Todd direction in semidefinite programming*, School of OR and Industrial Engineering, Cornell University, Ithaca, New York 14853–3801, (1996).Google Scholar