Interval mathematics, algebraic equations and optimization

Journal of Computational and Applied Mathematics

Volumn 124, Issue 1-2, 2000, Pages 263-280

  Wolfe, M A ^a  

^a UNIVERSITY OF ST ANDREWS   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; LINEAR ALGEBRA; OPTIMIZATION;

GAUSSIAN ALGORITHMS; INTERVAL MATHEMATICS;

COMPUTATION THEORY;

EID: 0034541549     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00421-0     Document Type: Article

References (67)

1. C.S. Adjiman I.P. Androulakis C.A. Floudas A global optimization method, α BB, for general twice-differentiable constrained NLPs – II, Implementation and computational results Comput. Chem. Eng. 22 1998 1159 1179
2. C.S. Adjiman S. Dallwig C.A. Floudas A. Neumaier A global optimization method, α BB, for general twice-differentiable constrained NLPs – I, Theoretical advances Comput. Chem. Eng. 22 1998 1137 1158
3. G. Alefeld Inclusion methods for systems of nonlinear equations – the interval Newton method and modifications J. Herzberger Topics in Validated Computations 1994 Elsevier Dordrecht 7 26
4. G. Alefeld D. Claudio The basic properties of interval arithmetic, its software realizations and some applications Comput. Struct. 67 1998 3 8
5. G. Alefeld A. Gienger F. Potra Efficient numerical validation of solutions of nonlinear systems SIAM J. Numer. Anal. 31 1994 252 260
6. G. Alefeld J. Herzberger Introduction to Interval Computations 1983 Academic Press New York
7. G. Alefeld V. Kreinovich G. Mayer The shape of the symmetric solution set R.B. Kearfott V. Kreinovich Applications of Interval Computations 1996 Kluwer Academic Publishers Dordrecht 61 79
8. M.S. Bazarraa and C.M. Shetty Foundations of Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 122 1976 Springer Berlin
9. A. Ben-Israel T.N.E. Greville Generalized Inverses: Theory and Applications 1974 Wiley New York
10. C.A. Floudos P.M. Pardolos A Collection of Test Problems for Constrained Optimization, Lecture Notes in Computer Science, Vol. 455 1990 Springer Berlin
11. A. Frommer G. Mayer Efficient methods for enclosing solutions of systems of nonlinear equations Computing 44 1990 221 235
12. A. Frommer G. Mayer On the R-order of Newton-like methods for enclosing solutions of nonlinear equations SIAM J. Numer. Anal. 27 1990 105 116
13. A. Frommer, G. Mayer, Two-stage interval iterative methods, preprint, 1993.
14. R. Hammer M. Hocks U. Kulisch D. Ratz Numerical Toolbox for Verified Computing I 1993 Springer Berlin
15. E.R. Hansen Bounding the solution of interval linear equations SIAM J. Numer. Anal. 29 1992 1493 1503
16. E. Hansen Global Optimization using Interval Analysis 1992 Marcel Dekker New York
17. E.R. Hansen G.W. Walster Bounds for Lagrange multipliers and optimal points Comput. Math. Appl. 25 1993 59 69
18. E.R. Hansen G.W. Walster Nonlinear equations and optimization Comput. Math. Appl. 25 1993 125 145
19. C. Jansson O. Knüppel A branch and bound algorithm for bound constrained optimization problems without derivatives J. Global Optim. 7 1995 297 331
20. R.B. Kearfott Decomposition of arithmetic expressions to improve the behaviour of interval iteration for nonlinear systems Computing 47 1991 169 191
21. R.B. Kearfott Rigorous Global Search: Continuous Problems 1996 Kluwer Academic Publishers Dordrecht
22. R.B. Kearfott V. Kreinovich (Eds.) Applications of Interval Computations 1996 Kluwer Academic Publishers Dordrecht
23. L. Kolev A new method for global solution of systems of non-linear equations Reliab. Comput. 4 1998 125 146
24. V. Kreinovich A. Lakeyev Linear interval equations: computing enclosures with bounded relative or absolute overestimation is NP-hard Reliab. Comput. 2 1996 341 350
25. V. Kreinovich A. Lakeyev S. Noskov Optimal solution of interval linear systems is intractible (NP-hard) Interval Comput. 1 1993 6 14
26. G. Mayer Some remarks on two interval-arithmetic modifications of the Newton method Computing 48 1992 125 128
27. G. Mayer Success in epsilon-inflation G. Alefeld A. Frommer Scientific Computing and Validated Numerics 1996 Akademie Verlag Berlin 98 104
28. G. Mayer Epsilon inflation in verification algorithms J. Comput. Appl. Math. 60 1995 147 169
29. G. Mayer J. Rohn On the applicability of the interval Gaussian algorithm Reliab. Comput. 4 1998 205 222
30. R.E. Moore, Interval arithmetic and automatic error analysis in digital computing, Ph.D. Thesis, Stanford University, 1962.
31. R.E. Moore J.B. Kioustelidis A simple test for accuracy of approximate solutions to nonlinear (or linear) systems SIAM J. Numer. Anal. 17 1980 521 529
32. A. Neumaier Interval Methods for Systems of Equations 1990 Cambridge University Press Cambridge
33. S. Ning R.B. Kearfott A comparison of some methods for solving linear interval equations SIAM J. Numer. Anal. 34 1997 1289 1305
34. J.M. Ortega W.C. Rheinboldt Iterative Solution of Nonlinear Equations in Several Variables 1970 Academic Press New York
35. M.C. Pandian A convergence test and componentwise error estimates for Newton methods SIAM J. Numer. Anal. 22 1985 779 791
36. L. Qi A note on the Moore test for nonlinear systems SIAM J. Numer. Anal. 19 1982 851 857
37. L.B. Rall Computational Solution of Nonlinear Operator Equations 1969 Wiley New York
38. H. Ratschek J. Rokne New Computer Methods for Global Optimization 1988 Ellis Horwood Chichester
39. H. Ratschek R.L. Voller Global optimization over unbounded domains SIAM J. Control Optim. 28 1990 528 539
40. H. Ratschek J. Rokne Interval methods R. Horst P.M. Pardolos Handbook of Global Optimization 1995 Kluwer Academic Publishers Dordrecht 751 828
41. J. Rohn On nonconvexity of the solution set of a system of linear interval equations BIT 30 1989 161 165
42. J. Rohn Cheap and tight bounds: the recent result by E. Hansen can be made more efficient Interval Comput. 4 1993 13 21
43. J. Rohn Enclosing solutions of overdetermined systems of linear interval equations Reliab. Comput. 2 1996 167 171
44. J. Rohn Linear interval equations: computing enclosures with bounded relative overestimation is NP-hard R.B. Kearfott V. Kreinovich Applications of Interval Computations 1996 Kluwer Academic Publishers Dordrecht 81 89
45. J. Rohn On overestimation produced by the interval Gaussian algorithm Reliab. Comput. 3 1997 363 368
46. S.M. Rump Verification methods for dense and sparse systems of equations J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 63 135
47. S.M. Rump Improved iteration schemes for validation algorithms for dense and sparse nonlinear systems Computing 57 1996 77 84
48. S.M. Rump A note on epsilon inflation Reliab. Comput. 4 1998 371 375
49. S.P. Shary Algebraic approach in the “outer problem” for interval linear equations Reliab. Comput. 3 1997 103 135
50. Z. Shen Z. Huang M.A. Wolfe An interval maximum entropy method for a discrete minimax problem Appl. Math. Comput. 87 1997 49 68
51. Z. Shen A. Neumaier A note on Moore's interval test for zeros of nonlinear systems Computing 40 1988 85 90
52. Z. Shen A. Neumaier M.C. Eiermann Solving minimax problems by interval methods BIT 30 1990 742 751
53. Z. Shen M.A. Wolfe On certain computable tests and componentwise error bounds Computing 50 1993 353 368
54. M.A. Wolfe, Interval methods for linear systems, in: E. Spedicato, M. Vespucci (Eds.), NATO Advanced Study Institute Computer Algorithms for Solving Linear Algebraic Equations: the State of the Art, Universita’ Degli Studi di Bergamo, 1990, pp. 128–140.
55. M.A. Wolfe An interval algorithm for constrained global optimization J. Comput. Appl. Math. 50 1994 605 612
56. M.A. Wolfe On global optimization in R using interval arithmetic Optim. Methods Software 3 1994 60 76
57. M.A. Wolfe An interval algorithm for bound-constrained global optimization Optim. Methods Software 6 1995 145 159
58. M.A. Wolfe Interval methods for global optimization Appl. Math. Comput. 75 1996 179 206
59. M.A. Wolfe On a second derivative test due to Qi Reliab. Comput. 4 1998 223 234
60. M.A. Wolfe, A note on a uniqueness theorem for the second-derivative test of Qi, to appear.
61. 1 using interval arithmetic Reliab. Comput. 5 1999 371 383
62. M.A. Wolfe, On bounding solutions of underdetermined systems, to appear.
63. M.A. Wolfe L.S. Zhang An interval algorithm for nondifferentiable global optimization Appl. Math. Comput. 63 1994 101 122
64. T. Yamamoto Error bounds for Newton's iterates derived from the Kantorovich theorem Numer. Math. 48 1986 91 98
65. K. Yamamura An algorithm for representing functions of many variables by superpositions of functions of one variable and addition IEEE Trans. Circuits Systems 1 43 1996 338 340
66. K. Yamamura H. Kawata A. Tokue Interval solution of nonlinear equations using linear programming BIT 38 1 1998 186 199
67. Zhang Danqing, Li Weiguo, Shen Zuhe, Solving underdetermined systems with interval methods, Reliab. Comput. 5 (1999) 23–33.