Rigorous error bounds for the optimal value in semidefinite programming

SIAM Journal on Numerical Analysis

Volumn 46, Issue 1, 2007, Pages 180-200

  Jansson, Christian ^a   Chaykin, Denis ^a   Keil, Christian ^a  

^a HAMBURG UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Interval arithmetic; Linear programming; NETLIB LP library; Rigorous error bounds; SDPLIB; Semidefinite programming; Sensitivity analysis

Indexed keywords

COMBINATORIAL MATHEMATICS; COMBINATORIAL OPTIMIZATION; CONTROL SYSTEMS; CONTROL THEORY; DEMODULATION; DIGITAL ARITHMETIC; GLOBAL OPTIMIZATION; LINEARIZATION; OPTIMAL SYSTEMS; OPTIMIZATION; PROGRAMMING THEORY; SENSITIVITY ANALYSIS;

INTERVAL ARITHMETIC; NETLIB LP LIBRARY; RIGOROUS ERROR BOUNDS; SDPLIB; SEMIDEFINITE PROGRAMMING;

LINEAR PROGRAMMING;

EID: 55549133270     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050622870     Document Type: Article

References (49)

1. G. ALEFELD AND J. HERZBERGER, Introduction to Interval Computations, Academic Press, New York, 1983.
2. D. L. APPLEGATE, W. COOK, S. DASH, AND D. G. ESPINOZA, QSopt ex, http://www.dii.uchile. cl/~daespino/QSoptExact doc/main.html.
3. V. BALAKRISHNAN AND E. FERON, eds., Linear matrix inequalities in control theory and applications, Internat. J. Robust Nonlinear Control, 6 (1996), pp. 895-1100.
4. M. BERZ et al., COSY Infinity, http://www.bt.pa.msu.edu/ index files/cosy.htm.
5. B. BORCHERS, SDPLIB 1.2, a library of semidefinite programming test problems, Optim. Methods Softw., 11 (1999), pp. 683-690.
6. D. CHAYKIN, A C++ implementation of rigorous error bounds in semidefinite programming, in progress.
7. C. A. FLOUDAS, Deterministic Global Optimization - Theory, Methods and Applications, Nonconvex Optim. Appl. 37, Kluwer Academic Publishers, Dordrecht, The Netherlands, Boston, London, 2000.
8. R. FOURER AND D. M. GAY, Experience with a primal presolve algorithm, in Large Scale Optimization: State of the Art, W. W. Hager, D. W. Hearn, and P. M. Pardalos, eds., Kluwer Academic Publishers, Norwell, MA, Dordrecht, The Netherlands, 1994, pp. 135-154.
9. R. M. FREUND, F. ORDÓ̃NEZ, AND K. C. TOH, Behavioral measures and their correlation with IPM iteration on semi-definite programming problems, Math. Program., 109 (2007), pp. 445-475.
10. K. FUJISAWA, Y. FUTAKATA, M. KOJIMA, K. NAKATA, AND M. YAMASHITA, SDPA-M (SemiDefinite Programming Algorithm in MATLAB) User's Manual - Version 2.00, Research report B-359, Tokyo Institute of Technology, Tokyo, Japan, 2000.
11. L. GRANVILLIERS AND F. BENHAMOU, Algorithm 852: RealPaver: An interval solver using constraint satisfaction techniques, ACM Trans. Math. Software, 32 (2006), pp. 138-156.
12. M. GRÖTSCHEL, L. LOVÁSZ, AND A. SCHRIJVER, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica, 1 (1981), pp. 169-197.
13. E. R. HANSEN, Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
14. C. JANSSON, VSDP: Verified SemiDefinite Programming, User's Guide, Beta Version 0.1, to appear.
15. C. JANSSON, A self-validating method for solving linear programming problems with interval input data, in Scientific Computation with Automatic Result Verification, Comput. Suppl. 6, Springer-Verlag, Vienna, 1988, pp. 33-45.
16. C. JANSSON, A rigorous lower bound for the optimal value of convex optimization problems, J. Global Optim., 28 (2004), pp. 121-137.
17. C. JANSSON, Rigorous lower and upper bounds in linear programming, SIAM J. Optim., 14 (2004), pp. 914-935.
18. R. B. KEARFOTT, GlobSol, http://interval.louisiana.edu.
19. R. B. KEARFOTT, Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
20. R. B. KEARFOTT, On proving existence of feasible points in equality constrained optimization problems, Math. Program., 83 (1998), pp. 89-100.
21. C. KEIL, Lurupa - Rigorous Error Bounds in Linear Programming, in Algebraic and Numerical Algorithms and Computer-Assisted Proofs, B. Buchberger, S. Oishi, M. Plum, and S. Rump, eds., no. 05391 in Dagstuhl Seminar Proceedings, Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany, 2006, http://drops.dagstuhl. de/opus/volltexte/2006/445.
22. C. KEIL, Verified linear programming - a comparison, Reliable Computing, submitted.
23. C. KEIL AND C. JANSSON, Computational experience with rigorous error bounds for the Netlib linear programming library, Reliab. Comput., 12 (2006), pp. 303-321.
24. M. KIYOMI, exlp, http://members.jcom.home.ne.jp/ masashi777/exlp.html.
25. O. KNÜPPEL, PROFIL/BIAS and extensions, Version 2.0, Technical report, Inst. f. Informatik III, Technische Universität Hamburg-Harburg, 1998.
26. T. KOCH, perPlex, urlhttp://www.zib.de/koch/perplex.
27. R. KRAWCZYK, Fehlerabschätzung bei linearer Optimierung, in Interval Mathematics, K. Nickel, ed., Lecture Notes in Comput. Sci. 29, Springer-Verlag, Berlin, 1975, pp. 215-222.
28. Y. LEBBAH, ICOS (Interval COnstraints Solver), http://www.essi.fr/~lebbah/icos/index.html.
29. MATLAB USER's GUIDE, VERSION 6, THE MATHWORKS INC., NATICK, MA, 2000.
30. G. MAYER, Result verification for eigenvectors and eigenvalues, in Topics in Validated Computations, Proceedings of the IMACS-GAMM International Workshop, Oldenburg, Germany, Stud. Comput. Math. 5, J. Herzberger, ed., Elsevier, Amsterdam, 1994, pp. 209-276.
31. R. E. MOORE, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.
32. A. NEMIROVSKI, Lectures on Modern Convex Optimization, http://www2.isye.gatech.edu/~nemirovs/Lect ModConvOpt.pdf, 2003.
33. NETLIB linear programming library, http://www.netlib.org/lp.
34. A. NEUMAIER, Interval methods for systems of equations, Encyclopedia Math. Appl., Cambridge University Press, Cambridge, UK, 1990.
35. A. NEUMAIER, Introduction to Numerical Analysis, Cambridge University Press, Cambridge, UK, 2001.
36. A. NEUMAIER, Complete search in continuous global optimization and constraint satisfaction, Acta Numer., 13 (2004), pp. 271-369.
37. A. NEUMAIER AND O. SHCHERBINA, Safe bounds in linear and mixed-integer programming, Math. Program., 99 (2004), pp. 283-296.
38. F. ORDÓ̃NEZ AND R. M. FREUND, Computational experience and the explanatory value of condition measures for linear optimization, SIAM J. Optim., 14 (2003), pp. 307-333.
39. M. V. RAMANA, L. TUNÇEL, AND H. WOLKOWICZ, Strong duality for semidefinite programming, SIAM J. Optim., 7 (1997), pp. 641-662.
40. J. RENEGAR, Linear programming, complexity theory and elementary functional analysis, Math. Program., 70 (1995), pp. 279-351.
41. J. B. ROSEN AND S. SUZUKI, Construction of nonlinear programming test problems, Communication of ACM, 8 (1965), p. 113.
42. S. M. RUMP, Validated solution of large linear systems, in Validation Numerics: Theory and Applications, Comput. Suppl. 9, R. Albrecht, G. Alefeld, and H. Stetter, eds., Springer-Verlag, Berlin, 1993, pp. 191-212.
43. S. M. RUMP, Verification methods for dense and sparse systems of equations, in Topics in Validated Computations - Studies in Computational Mathematics, J. Herzberger, ed., Elsevier, Amsterdam, 1994, pp. 63-136.
44. S. M. RUMP, INTLAB - Interval Laboratory, a MATLAB Toolbox for Verified Computations, Version 5.1, 2005.
45. R. E. SKELTON AND T. IWASAKI, Increased roles of linear algebra in control education, IEEE Control Syst. Mag., 15 (1995), pp. 76-89.
46. M. J. TODD, Detecting infeasibility in infeasible-interior- point methods for optimization, in Foundations of Computational Mathematics: Minneapolis, 2002, London Math. Soc. Lecture Note Ser. 312, F. Cucker, R. DeVore, P. Olver, and E. Süli, eds., Cambridge University Press, Cambridge, UK, 2004, pp. 157-192.
47. R. H. TÜTÜNCÜ, K. C. TOH, AND M. J. TODD, Solving semidefinite-quadratic-linear programs using SDPT3, Math. Program., 95B (2003), pp. 189-217.
48. P. VAN HENTENRYCK, P. MICHEL, AND Y. DEVILLE, Numerica: A Modelling Language for Global Optimization, MIT Press, Cambridge, MA, 1997.
49. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.