# Asymptotic behavior of the central path for a special class of degenerate SDP problems

## Abstract.

This paper studies the asymptotic behavior of the central path (*X*(*ν*),*S*(*ν*),*y*(*ν*)) as *ν*↓0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose “degenerate diagonal blocks” Open image in new window of the central path are assumed to satisfy Open image in new window We establish the convergence of the central path towards a primal-dual optimal solution, which is characterized as being the unique optimal solution of a certain log-barrier problem. A characterization of the class of SDP problems which satisfy our assumptions are also provided. It is shown that the re-parametrization *t*>0→(*X*(*t*^{4}),*S*(*t*^{4}),*y*(*t*^{4})) of the central path is analytic at *t*=0. The limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is Open image in new window Finally, we apply our results to the convex quadratically constrained convex programming (CQCCP) problem and characterize the class of CQCCP problems which can be formulated as SDPs satisfying the assumptions of this paper. In particular, we show that CQCCP problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class.

## Keywords

Limiting behavior Central path Semidefinite programming Convex quadratic programming Convex quadratically constrained programming## Preview

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## References

- 1.Adler, I., Monteiro, R.D.C.: Limiting behavior of the affine scaling continuous trajectories for linear programming problems. Mathematical Programming
**50**, 29–51 (1991)CrossRefGoogle Scholar - 2.Asic, M.D., Kovacevic-Vujcic, V.V., Radosavljevic-Nikolic, M.D.: A note on limiting behavior of the projective and the affine rescaling algorithms. In: J.C. Lagarias, M.J. Todd, (eds.), Mathematical Developments Arising from Linear Programming: Proceedings of a Joint Summer Research Conference held at Bowdoin College, Brunswick, Maine, USA, June/July 1988, volume 114 of Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, USA, 1990, pp. 151–157Google Scholar
- 3.Bayer, D.A., Lagarias, J.C.: The nonlinear geometry of linear programming, Part I: Affine and projective scaling trajectories. Transactions of the American Mathematical Society
**314**(2), 499–526 (1989)Google Scholar - 4.de Klerk, E.: Aspects of semidefinite programming: interior point algorithms and selected applications. Applied Optimization Series 65. Kluwer Academic Press, Dordrecht, The Netherlands, 2002Google Scholar
- 5.de Klerk, E., Roos, C., Terlaky, T.: Initialization in semidefinite programming via a self-dual,skew- symmetric embedding. Operations Research Letters
**20**, 213–221 (1997)CrossRefGoogle Scholar - 6.de Klerk, E., Roos, C., Terlaky, T.: Infeasible-start semidefinite programming algorithms via self-dual embeddings. Fields Institute Communications
**18**, 215–236 (1998)Google Scholar - 7.Goldfarb, D., Scheinberg, K.:Interior point trajectories in semidefinite programming. SIAM Journal on Optimization
**8**, 871–886 (1998)CrossRefGoogle Scholar - 8.Graña Drummond, L.M., Peterzil, H.Y.: The central path in smooth convex semidefinite programs. Optimization
**51**, 207–233 (2002)Google Scholar - 9.Güler, O.: Limiting behavior of the weighted central paths in linear programming. Mathematical Programming
**65**, 347–363 (1994)CrossRefGoogle Scholar - 10.Halická, M.: Analytical properties of the central path at the boundary point in linear programming. Mathematical Programming
**84**, 335–355 (1999)CrossRefGoogle Scholar - 11.Halická, M.: Two simple proofs of analyticity of the central path in linear programming. Operations Research Letters
**28**, 9–19 (2001)CrossRefGoogle Scholar - 12.Halická, M.: Analyticity of the central path at the boundary point in semidefinite programming. European Journal of Operational Research
**143**, 311–324 (2002)CrossRefGoogle Scholar - 13.Halická, M., de Klerk, E., Roos, C.: Limiting behavior of the central path in semidefinite optimization. Preprint, Faculty of Technical Mathematics and Informatics, TU Delft, NL–2628 CD Delft, The Netherlands, June 2002Google Scholar
- 14.Halická, M., de Klerk, E., Roos, C.: On the convergence of the central path in semidefinite optimization. SIAM Journal on Optimization
**12**, 1090–1099 (2002)CrossRefGoogle Scholar - 15.Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization algorithms I. Volume 305 of Comprehensive Study in Mathematics. Springer-Verlag, New York, 1993Google Scholar
- 16.Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach to interior point algorithms for linear complementarity problems. Volume 538 of Lecture Notes in Computer Science. Springer Verlag, Berlin, Germany, 1991Google Scholar
- 17.Kojima, M., Mizuno, S., Noma, T.: Limiting behavior of trajectories by a continuation method for monotone complementarity problems. Mathematics of Operations Research
**15**(4), 662–675 (1990)Google Scholar - 18.Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM Journal on Optimization
**7**, 86–125 (1997)CrossRefGoogle Scholar - 19.Lu, Z., Monteiro, R.D.C.: Error bounds and limiting behavior of weighted paths associated with the SDP map
*X*^{1/2}*SX*^{1/2}. Manuscript, School of ISyE, Georgia Tech, Atlanta, GA, 30332, USA, June 2003Google Scholar - 20.Lu, Z., Monteiro, R.D.C.: Limiting behavior of the Alizadeh-Haeberly-Overton weighted paths in semidefinite programming. Manuscript, School of ISyE, Georgia Tech, Atlanta, GA, 30332, USA, July 2003Google Scholar
- 21.Luo, Z-Q., Sturm, J. F., Zhang, S.: Superlinear convergence of a symmetric primal-dual path-following algorithm for semidefinite programming. SIAM Journal on Optimization
**8**, 59–81 (1998)CrossRefGoogle Scholar - 22.Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Operations Research Letters
**7**, 21–26 (1988)CrossRefGoogle Scholar - 23.McLinden, L.: An analogue of Moreau’s proximation theorem, with application to the nonlinear complementarity problem. Pacific Journal of Mathematics
**88**, 101–161 (1980)Google Scholar - 24.McLinden, L.: The complementarity problem for maximal monotone multifunctions. In: R.W. Cottle, F. Giannessi, J.-L. Lions, (eds.), Variational Inequalities and Complementarity Problems, Wiley, New York, 1980, pp. 251–270Google Scholar
- 25.Megiddo, N.: Pathways to the optimal set in linear programming. In: N. Megiddo, (ed.), Progress in Mathematical Programming: Interior point and Related Methods, Springer Verlag, New York, 1989, pp. 131–158; Identical version In: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, 1986, pp. 1–35Google Scholar
- 26.Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud., Princeton University Press, 1968Google Scholar
- 27.Monteiro, R.D.C.: Convergence and boundary behavior of the projective scaling trajectories for linear programming. Mathematics of Operations Research
**16**(4), 842–858 (1991)Google Scholar - 28.Monteiro, R.D.C., Pang, J.-S.: Properties of an interior-point mapping for mixed complementarity problems. Mathematics of Operations Research
**21**, 629–654 (1996)Google Scholar - 29.Monteiro, R.D.C., Pang, J.-S.: On two interior-point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research
**23**, 39–60 (1998)Google Scholar - 30.Monteiro, R.D.C., Todd, M.J.: Path-following methods for semidefinite programming. In: R. Saigal, L. Vandenberghe, H. Wolkowicz, (eds.), Handbook of Semidefinite Programming. Kluwer Academic Publishers, Boston-Dordrecht-London, 2000Google Scholar
- 31.Monteiro, R.D.C., Tsuchiya, T.: Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem. Mathematics of Operations Research
**21**, 793–814 (1996)Google Scholar - 32.Monteiro, R.D.C., Zanjácomo, P.R.: General interior-point maps and existence of weighted paths for nonlinear semidefinite complementarity problems. Mathematics of Operations Research
**25**, 381–399 (2000)CrossRefGoogle Scholar - 33.Monteiro, R.D.C., Zhou, F.: On the existence and convergence of the central path for convex programming and some duality results. Computational Optimization and Applications
**10**, 51–77 (1998)CrossRefGoogle Scholar - 34.Preiss, M., Stoer, J.: Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems. Mathematical Programming
**99**, 499–520 (2004)CrossRefGoogle Scholar - 35.Sporre, G., Forsgren, A.: Characterization of the limit point of the central path in semidefinite programming. Technical Report TRITA-MAT-2002-OS12, Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, June 2002Google Scholar
- 36.Stoer, J., Wechs, M.: Infeasible-interior-point paths for sufficient linear complementarity problems. Mathematical Programming
**83**, 403–423 (1998)CrossRefGoogle Scholar - 37.Stoer, J., Wechs, M.: On the analyticity properties of infeasible-interior point paths for monotone linear complementarity problems. Numerische Mathematik
**81**, 631–645 (1999)CrossRefGoogle Scholar - 38.Wechs, M.: The analyticity of interior-point-paths at strictly complementary solutions of linear programs. Optimization, Methods and Software
**9**, 209–243 (1998)Google Scholar - 39.Witzgall, C., Boggs, P.T., Domich, P.D.: On the convergence behavior of trajectories for linear programming. In: J.C. Lagarias, M.J. Todd, (eds.), Mathematical Developments Arising from Linear Programming: Proceedings of a Joint Summer Research Conference held at Bowdoin College, Brunswick, Maine, USA, June/July 1988, volume 114 of Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, USA, 1990, pp. 161–187Google Scholar