Semidefinite programming for uncertain linear equations in static analysis of structures

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 1, 2008, Pages 102-115

  Kanno, Yoshihiro ^a   Takewaki, Izuru ^b  

^a UNIVERSITY OF TOKYO   (Japan)

^b KYOTO UNIVERSITY   (Japan)

Author keywords

Confidential bound; Data uncertainty; Interior point method; Interval analysis; Semidefinite program; Uncertain linear equations

Indexed keywords

DEMODULATION; ELASTIC MODULI; LINEAR PROGRAMMING; NONLINEAR PROGRAMMING; NUMERICAL ANALYSIS; SOLUTIONS; UNCERTAINTY ANALYSIS;

CONFIDENTIAL BOUND; DATA UNCERTAINTY; INTERIOR-POINT METHOD; INTERVAL ANALYSIS; SEMIDEFINITE PROGRAM; UNCERTAIN LINEAR EQUATIONS;

LINEAR EQUATIONS;

EID: 54849425777     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.04.003     Document Type: Article

References (40)

1. Alefeld G., Kreinovich V., and Mayer G. On the shape of the symmetric, persymmetric, and skew-symmetric solution set. SIAM J. Matrix Anal. Appl. 18 (1997) 693-705
2. Alefeld G., and Mayer G. Interval analysis: theory and applications. J. Comput. Appl. Math. 121 (2000) 421-464
3. Ben-Haim Y. Information-gap Decision Theory: Decisions under Severe Uncertainty. second ed. (2006), Academic Press, London, UK
4. Ben-Haim Y., and Elishakoff I. Convex Models of Uncertainty in Applied Mechanics (1990), Elsevier, New York, NY
5. Ben-Tal A., El Ghaoui L., and Nemirovski A. Robustness. In: Wolkowicz H., Saigal R., and Vandenberghe L. (Eds). Handbook of Semidefinite Programming - Theory, Algorithms, and Applications (2000), Kluwer, Dordrecht, The Netherlands 139-162
6. Ben-Tal A., and Nemirovski A. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications (2001), SIAM, Philadelphia, PA
7. Ben-Tal A., and Nemirovski A. Robust optimization - methodology and applications. Math. Prog. B92 (2002) 453-480
8. Bernstein D.S. Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear System Theory (2005), Princeton University Press, Princeton, NJ
9. Borchers B. CSDP 2.3 user's guide, a C library for semidefinite programming. Optim. Methods Softw. 11/12 (1999) 597-611
10. Boyd S., El Ghaoui L., Feron E., and Balakrishnan V. Linear Matrix Inequalities in System and Control Theory (1994), SIAM, Philadelphia, PA
11. Calafiore G., and El Ghaoui L. Ellipsoidal bounds for uncertain linear equations and dynamical systems. Automatica 40 (2004) 773-787
12. Chen S., Lian H., and Yang X. Interval static displacement analysis for structures with interval parameters. Int. J. Numer. Methods Engrg. 53 (2002) 393-407
13. Chen S., Lian H., and Yang X. Interval eigenvalue analysis for structures with interval parameters. Finite Elem. Anal. Des. 39 (2003) 419-431
14. De Gersem H., Moens D., Desmet W., and Vandepitte D. Interval and fuzzy dynamic analysis of finite element models with superelements. Comput. Struct. 85 (2007) 304-319
15. El Ghaoui L., Oustry F., and Lebret H. Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9 (1998) 33-52
16. Hansen E., and Sengupta S. Bounding solutions of systems of equations using interval analysis. BIT Numer. Math. 21 (1981) 203-211
17. Helmberg C. Semidefinite programming. Eur. J. Oper. Res. 137 (2002) 461-482
18. Kanno Y., Ohsaki M., and Katoh N. Sequential semidefinite programming for optimization of framed structures under multimodal buckling constraints. Int. J. Struct. Stab. Dyn. 1 (2001) 585-602
19. Kanno Y., and Takewaki I. Confidence ellipsoids for static response of trusses with load and structural uncertainties. Comput. Methods Appl. Mech. Engrg. 196 (2006) 393-403
20. Kanno Y., and Takewaki I. Robustness analysis of trusses with separable load and structural uncertainties. Int. J. Solids Struct. 43 (2006) 2646-2669
21. Kojima M., Shindoh S., and Hara S. Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim. 7 (1997) 86-125
22. Köylüoǧlu H.U., Çakmak A.S., and Nielsen S.R.K. Interval algebra to deal with pattern loading and structural uncertainties. J. Engrg. Mech. (ASCE) 121 (1995) 1149-1157
23. McWilliam S. Anti-optimization of uncertain structures using interval analysis. Comput. Struct. 79 (2001) 421-430
24. Moens D., and Vandepitte D. A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1257-1555
25. Muhanna R.L., and Mullen R.L. Uncertainty in mechanics problems - interval-based approach. J. Engrg. Mech. (ASCE) 127 (2001) 557-566
26. Neumaier A. Interval Methods for Systems of Equations (1990), Cambridge University Press, Cambridge, UK
27. Neumaier A., and Pownuk A. Linear systems with large uncertainties, with applications to truss structures. Reliab. Comput. 13 (2007) 149-172
28. Ohsaki M., Fujisawa K., Katoh N., and Kanno Y. Semi-definite programming for topology optimization of truss under multiple eigenvalue constraints. Comput. Methods Appl. Mech. Engrg. 180 (1999) 203-217
29. Pantelides C.P., and Ganzerli S. Design of trusses under uncertain loads using convex models. J. Struct. Engrg. (ASCE) 124 (1998) 318-329
30. Qiu Z. Comparison of static response of structures using convex models and interval analysis method. Int. J. Numer. Methods Engrg. 56 (2003) 1735-1753
31. Qiu Z., and Elishakoff I. Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput. Methods Appl. Mech. Engrg. 152 (1998) 361-372
32. Rohn J. Overestimations in bounding solutions of perturbed linear equations. Linear Algebra Appl. 262 (1997) 55-65
33. Sharaya I.A. On maximal inner estimation of the solution sets of linear systems with interval parameters. Reliab. Comput. 7 (2001) 409-424
34. Shary S.P. Solving the linear interval tolerance problem. Math. Comput. Simulat. 39 (1995) 53-85
35. Strang G. Linear Algebra and its Applications (1976), Academic Press, New York, NY
36. Sturm J.F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12 (1999) 625-653
37. Takewaki I., and Ben-Haim Y. Info-gap robust design with load and model uncertainties. J. Sound Vibr. 288 (2005) 551-570
38. Toh K.C., Todd M.J., and Tütüncü R.H. SDPT3 - a MATLAB software package for semidefinite programming, version 1.3. Optim. Methods Softw. 11/12 (1999) 545-581
39. Yamashita M., Fujisawa K., and Kojima M. Implementation and evaluation of SDPA6.0 (SemiDefinite Programming Algorithm 6.0). Optim. Methods Softw. 18 (2003) 491-505
40. Using MATLAB, The MathWorks, Inc., Natick, MA, 2002.