Semidefinite programs: New search directions, smoothing-type methods, and numerical results

SIAM Journal on Optimization

Volumn 13, Issue 1, 2003, Pages 1-23

  Kanzow, Christian ^a   Nagel, Christian ^a  

^a UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

Global convergence; Newton's method; Semidefinite programs; Smoothing type methods; Superlinear convergence

Indexed keywords

EID: 0037289395     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623401390525     Document Type: Article

References (33)

1. F. ALLIZADEH, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
2. F. ALIZABEH, J.-P. A. HAEBERLY, AND M. L. OVERTON, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM J. Optim., 8 (1998), pp. 746-768.
3. B. BORCHERS, SDPLIB 1.2, A library of semidefinite programming test problems, Optim. Methods Softw., 11 (1999), pp. 597-611.
4. J. V. BURKE AND S. XU, A non-interior predictor-corrector path-following method for LCP, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 45-63.
5. J. V. BURKE AND S. XU, A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem, Math. Program., 87 (2000), pp. 113-130.
6. B. CHEN AND P. T. HARKER, A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 1168-1190.
7. X. CHEN, H. QI, AND P. TSENG, Analysis of Nonsmooth Symmetric-Matrix Functions with Applications to Semidefinite Complementarity Problems, Technical report, Department of Mathematics, University of Washington, Seattle, WA, 2000.
8. X. CHEN AND P. TSENG, Non-Interior Continuation Methods for Solving Semidefinite Complementarity Problems, Technical report, Department of Mathematics, University of Washington, Seattle, WA, 1999.
9. S. ENGELKE AND C. KANZOW, Improved smoothing-type methods for the solution of linear programs, Numer. Math., 90 (2002), pp. 487-507.
10. S. ENGELKE AND C. KANZOW, On the solution of linear programs by Jacobian smoothing methods, Ann. Oper. Res., 103 (2001), pp. 49-70.
11. S. ENGELKE AND C. KANZOW, Predictor-Corrector Smoothing Methods for the Solution of Linear Programs, Preprint 153, Department of Mathematics, University of Hamburg, Hamburg, Germany, 2000.
12. S. ENGELKE AND C. KANZOW, Predictor-Corrector Smoothing Methods for Linear Programs with a More Flexible Update of the Smoothing Parameter, Preprint 162, Department of Mathematics, University of Hamburg, Hamburg, Germany, 2001.
13. A. FISCHER, A special Newton-type optimization method, Optimization, 24 (1992), pp. 269-284.
14. M. FUKUSHIMA, Z.-Q. LUO, AND P. TSENG, Smoothing functions for second-order-cone complementarity problems, SIAM J. Optim., 12 (2001), pp. 436-460.
15. C. HELMBERG, F. RENDL, R. J. VANDERBEL, AND H. WOLKOWICZ, An interior-point method for semidefinite programming, SIAM J. Optim., 6 (1996), pp. 342-361.
16. R. A. HORN AND C. R. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, U.K., 1985.
17. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, U.K., 1991.
18. H. JIANG, Smoothed Fischer-Burmeister Equation Methods for the Complementarity Problem, Technical report, Department of Mathematics, University of Melbourne, Melbourne, Australia, 1997.
19. C. KANZOW, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 851-868.
20. C. KANZOW AND C. NAGEL, Semidefinite Programs: New Search Directions, Smoothing-Type Methods, and Numerical Results, Preprint 163, Department of Mathematics, University of Hamburg, Hamburg, Germany, 2001.
21. C. KANZOW AND C. NAGEL, Some Practical Aspects of a Newton-Type Method for Semidefinite Programs, Preprint 243, Institute of Applied Mathematics and Statistics, University of Würzburg, Würzburg, Germany, 2001.
22. M. KOJIMA, M. SHIDA, AND S. SHINDOH, Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Programming, 80 (1998), pp. 129-160.
23. M. KOJIMA, M. SHIDA, AND S. SHINDOH, A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, SIAM J. Optim., 9 (1999), pp. 444-465.
24. Y. NESTEROV AND A. NEMIROVSKII, Interior-Point Polynomial Algorithms in Convex Programming, Studies in Appl. Math. 13, SIAM, Philadelphia, 1994.
25. S. SMALE, Algorithms for solving equations, in Proceedings of the International Congress of Mathematicians, Providence, 1987, AMS, Providence, RI, pp. 172-195.
26. D. SUN AND J. SUN, Semismooth matrix valued functions, Math. Oper. Res., 27 (2002), pp. 150-169.
27. J. SUN, D. SUN, AND L. QI, From Strong Semismoothness of the Squared Smoothing Matrix Function to Semidefinite Complementarity Problems, Applied Mathematics Report AMR 00/20, School of Mathematics, University of New South Wales, Sydney, Australia, 2000.
28. M. J. TODD, K. C. TOH, AND R. H. TÜTÜNCÜ, On the Nesterov-Todd direction in semidefinite programming, SIAM J. Optim., 8 (1998), pp. 769-796.
29. K. C. TOH, M. J. TODD, AND R. H. TÜTÜNCÜ, SDPT3-A Matlab software package for semidefinite programming, Version 2.1, Optim. Methods Softw., 11 (1999), pp. 545-581.
30. P. TSENG, Merit functions for semi-definite complementarity problems, Math. Programming, 83 (1998), pp. 159-185.
31. P. TSENG, Error bounds and superlinear convergence analysis of some Newton-type methods in optimization, in Nonlinear Optimization and Related Topics, G. D. Pillo and F. Giannessi, eds., Kluwer Academic Publishers, Norwell, MA, 2000, pp. 445-462.
32. N. YAMASHITA AND M. FUKUSHIMA, A new merit function and a descent method for semidefinite complementarity problems, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 405-420.
33. Y. ZHANG, On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming, SIAM J. Optim., 8 (1998), pp. 365-386.