Improved lipschitz bounds with the first norm for function values over multidimensional simplex

Mathematical Modelling and Analysis

Volumn 13, Issue 4, 2008, Pages 553-563

  Paulavicius R ^a   Zilinskas J ^a  

^a VILNIUS UNIVERSITY   (Lithuania)

Author keywords

Branch and bound algorithms; Global optimization; Lipschitz optimization; The first norm

Indexed keywords

GLOBAL OPTIMIZATION;

BOUNDING FUNCTIONS; BRANCH-AND-BOUND ALGORITHMS; FUNCTION VALUES; GLOBAL OPTIMIZATION ALGORITHM; LIPSCHITZ; LIPSCHITZ CONDITIONS; SYSTEM OF LINEAR EQUATIONS; THE FIRST NORM;

BRANCH AND BOUND METHOD;

EID: 57849089687     PISSN: 13926292     EISSN: None     Source Type: Journal    
DOI: 10.3846/1392-6292.2008.13.553-563     Document Type: Article

References (29)

1. M. Baravykaite and R. Čiegis. An implementation of a parallel generalized branch and bound template. Math. Model, and Anal, 12(3):277-289, 2007.
2. M. Baravykaite, R. Čiegis and J. Žilinskas. Template realization of generalized branch and bound algorithm. Math. Model, and Anal, 10(3):217-236, 2005.
3. R. Čiegis and M. Baravykaite. Implementation of a black-box global optimization algorithm with a parallel branch and bound template. Lecture Notes in Computer Science. Applied Parallel Computing. State of the Art in Scientific Computing: 8th International Conference, PARA 2006, Umea, Sweden, June 20-23, 2006. Revised Selected Papers, 4699:1115-1125, 2007.
4. E. A. Galperin. The cubic algorithm. Journal of Mathematical Analysis and Applications, 112:635-640, 1985.
5. P. Hansen and B. Jaumard. Lipshitz optimization. In R. Horst and P. Parda-los(Eds.), Handbook of Global Optimization, volume 59, pp. 404-493. Kluwer Academic Publishers, 1995.
6. R. Horst. Optimization with partition sets whose feasibility is not known. Application to concave minimization, reverse convex constraints, D. C.-programming and Lipschitzian optimization. Journal of Optimization Theory and Application, 58:11-37, 1988.
7. R. Horst and H. Tuy. On the convergence of global methods in multiextremal optimization. Journal of Optimization Theory and Applications, 54:253-271, 1987.
8. R. Horst and H. Tuy. Global Optimization: Deterministics Approaches. Springer, 2 edition, 1993.
9. C. Jansson and O. Knüppel. A Global Minimization Method: The Multi-Dimensional Case. Technical report, TU Hamburg-Harburg, 1992.
10. K. Madsen and J. Žilinskas. Testing Branch-and-Bound Methods for Global Optimization. Technical Report IMM-REP-2000-05, Technical University of Denmark, 2000.
11. D. Q. Mayne and E. Polak. Outer approximation algorithm for nondifferentiable optimization problems. Journal of Optimization Theory and Applications, 42(l):19-30, 1984.
12. C. C. Meewella and D. Q. Mayne. An algorithm for global optimization of Lipshitz continuous functions. Journal of Optimization Theory and Applications, 57:307-323, 1988.
13. R. H. Mladineo. An algorithm for finding the global maximum of a multimodal, multivariate function. Mathematical Programming, 34:188-200, 1986.
14. R. H. Mladineo. Convergence rates of a global optimization algorithms. Mathematical Programming, 54:223-232, 1992.
15. R. Paulavičius and J. Žilinskas. Analysis of different norms and corresponding Lipschitz constants for global optimization. Technological and Economic Development of Economy, 12(4):301-306, 2006.
16. R. Paulavičius and J. Žilinskas. Analysis of different norms and corresponding Lipschitz constants for global optimization in multidimensional case. Information Technology and Control, 36(4):383-387, 2007.
17. J. Pinter. Extended univariate algorithms for n-dimensional global optimization. Computing, 36:91-103, 1986.
18. J. Pinter. Branch-and-bound algorithms for solving global optimization problems with Lipschitzian structure. Optimization, 19:101-110, 1988.
19. S. A. Piyavskii. An algorithm for finding the absolute extremum of a function. Zh. Vychisl. Mat. mat. Fiz, 12(4):888-896, 1972.
20. H. Tuy and R. Horst. Convergence and restart in branch and bound algorithms for global optimization. Application to concave minimization and DC-optimization problems. Mathematical Programming, 41:161-183, 1988.
21. G. R. Wood. Multidimensional bisection applied to global optimization. Computers and Mathematics with Applications, 21:161-172, 1991.
22. A. Žilinskas and J. Žilinskas. Global optimization based on a statistical model and simplicial partitioning. Computers & Mathematics with Applications, 44(7):957-967, 2002.
23. A. Žilinskas and J. Žilinskas. On efficiency of tightening bounds in interval global optimization. Lecture Notes in Computer Science, 3732:197-205, 2006.
24. J. Žilinskas. Comparison of packages for interval arithmetic. Informatica, 16(1):145-154, 2005.
25. J. Žilinskas. Estimation of functional ranges using standard and inner interval arithmetic. Informatica, 17(1):125-136, 2006.
26. J. Žilinskas. Reducing of search space of multidimensional scaling problems with data exposing symmetries. Information Technology and Control, 36(4):377-382, 2007.
27. J. Žilinskas. Branch and bound with simplicial partitions for global optimization. Math. Model, and Anal, 13(1):145-159, 2008.
28. J. Žilinskas and I. D. L. Bogle. Evaluation ranges of functions using balanced random interval arithmetic. Informatica, 14(3):403-416, 2003.
29. J. Žilinskas and I. D. L. Bogle. Balanced random interval arithmetic. Computers & Chemical Engineering, 28(5):839-851, 2004.