Newton's method and its use in optimization

European Journal of Operational Research

Volumn 181, Issue 3, 2007, Pages 1086-1096

  Polyak, B T ^a  

^a TRAPEZNIKOV INSTITUTE OF CONTROL SCIENCES   (Russian Federation)

Author keywords

Convergence; Global behavior; Interior point methods; Newton's method; Nonlinear programming

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DATA MINING; GLOBAL OPTIMIZATION; NONLINEAR PROGRAMMING; OPERATIONS RESEARCH;

GLOBAL BEHAVIOR; GLOBAL CONVERGENCE; INTERIOR POINT METHODS; UNCONSTRAINED MINIMIZATION;

NEWTON-RAPHSON METHOD;

EID: 33947685523     PISSN: 03772217     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ejor.2005.06.076     Document Type: Article

References (67)

1. Arnold V.I. Small denominators and problem of stability in classical and celestial mechanics. Uspekhi Mathematicheskih Nauk 18 6 (1963) 91-192
2. Barnsley M. Fractals Everywhere (1993), Academic Press, London
3. Bennet A.A. Newton's method in general analysis. Proceedings of National Academy of Sciences, USA 2 10 (1916) 592-598
4. Ben-Tal A., and Nemirovski A. Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications (2001), SIAM, Philadelphia
5. Bertsekas D.P. Constrained Optimization and Lagrange Multiplier Method (1982), Academic Press, New York
6. Bertsekas D.P. Nonlinear Programming (1999), Athena Scientific, Belmont
7. L.T. Biegler, I.E. Grossmann, Part I: Retrospective on optimization, Part II: Future perspective on optimization, Preprints, Chemical Engineering Department, Carnegie-Mellon University, 2004.
8. Boyd S., and Vandenberghe L. Convex Optimization (2004), Cambridge University Press, Cambridge
9. Chen X., Nashed Z., and Qi L. Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM Journal on Numerical Analysis 38 (2000) 1200-1216
10. Collatz L. Functional Analysis and Numerical Mathematics (1966), Academic Press, New York
11. Conn A.B., Gould N.I.M., and Toint Ph.L. Trust Region Methods (2000), SIAM, Philadelphia
12. Curry J.H., Garnett L., and Sullivan D. On the iteration of a rational function: Computer experiments with Newton's method. Communications on Mathematical Physics 91 (1983) 267-277
13. Deulfhard P. Newton Methods for Nonlinear Problems: Affine Invariant and Adaptive Algorithms (2004), Springer, Berlin
14. R.M. Dickau, Newton's method. Available from: .
15. Electronic library ZIB. Available from: .
16. Facchinei F., and Pang J.S. Finite-Dimensional Variational Inequalities and Complementarity Problems (2003), Springer, New York
17. Fine H. On Newton's method of approximation. Proceedings of National Academy of Sciences, USA 2 9 (1916) 546-552
18. Fukushima M., and Qi L. Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (1999), Kluwer, Dordrecht
19. Goldfeld S., Quandt R., and Trotter H. Maximization by quadratic hill climbing. Econometrica 34 (1966) 541-551
20. Gill P.E., Murray W., and Saunders M.A. SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM Journal on Optimization 12 (2002) 979-1006
21. Graves L.M. Some mapping theorems. Duke Mathematical Journal 17 (1950) 111-114
22. Ioffe A.D. On the local surjection property. Nonlinear Analysis: Theory, Methods and Applications 11 5 (1987) 565-592
23. D.E. Joyce, Newton basins. Available from: .
24. Julia G. Sur l'iteration des fonctions rationelles. Journal de Mathematiques Pure et Applique 8 (1918) 47-245
25. Kantorovich L.V. On Newton's method for functional equations. Doklady AN SSSR 59 7 (1948) 1237-1240
26. Kantorovich L.V. Functional analysis and applied mathematics. Uspekhi Mathematicheskih Nauk 3 1 (1948) 89-185 Translated as N.B.S. Report 1509, Washington, DC, 1952
27. Kantorovich L.V. On Newton's method. Trudy Mathematicheskogo Instituta Imeni V.A. Steklova 28 (1949) 104-144
28. Kantorovich L.V. Principle of majorants and Newton's method. Doklady AN SSSR 76 1 (1951) 17-20
29. Kantorovich L.V. Some further applications of principle of majorants. Doklady AN SSSR 80 6 (1951) 849-852
30. Kantorovich L.V. On approximate solution of functional equations. Uspekhi Mathematicheskih Nauk 11 6 (1956) 99-116
31. Kantorovich L.V. Some further applications of Newton's method. Vestnik LGU, Seriya Matematika Mekhanika 7 (1957) 68-103
32. Kantorovich L.V., and Akilov G.P. Functional Analysis in Normed Spaces (1959), Fizmatgiz, Moscow
33. Kantorovich L.V., and Akilov G.P. Functional Analysis (1977), Nauka, Moscow
34. Klatte D., and Kummer B. Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (2002), Kluwer, Dordrecht
35. Krasnoselski M.A., Vainikko G.M., Zabreiko P.P., Rutitski Ya.B., and Stetsenko V.Ya. Approximate Solution of Operator Equations (1969), Nauka, Moscow English translation: Walter-Noordhoff, Groningen, 1972
36. Levenberg K. A method for the solution of certain nonlinear problems in least squares. Quarterly of Applied Mathematics 2 (1944) 164-168
37. Levitin E.S., and Polyak B.T. Constrained minimization methods. USSR Computational Mathematics and Mathematical Physics 6 5 (1966) 1-50
38. Ljusternik L.A. On conditional extrema of functionals. Mathematicheskij Sbornik 41 3 (1934) 390-401
39. Mandelbrot B. The Fractal Geometry of Nature (1983), W.H. Freeman, New York
40. Marquardt D. An algorithm for least squares estimation of nonlinear parameters. SIAM Journal of Applied Mathematics 11 (1963) 431-441
41. J.H. Mathews, Bibliography for Newton's method. Available from: .
42. Mifflin R. Semismooth and semiconvex functions in constrained optimization. SIAM Journal on Control and Optimization 15 (1977) 959-972
43. Mysovskikh I.P. On convergence of L.V. Kantorovich's method for functional equations and its applications. Doklady AN SSSR 70 4 (1950) 565-568
44. Nesterov Yu., and Nemirovski A. Interior-Point Polynomial Algorithms in Convex Programming (1994), SIAM, Philadelphia
45. Yu. Nesterov, B. Polyak, Cubic regularization of a Newton scheme and its global performance, CORE Discussion Papers 2003, No 41, Mathematical Programming, submitted for publication.
46. Nesterov Yu. Introductory Lectures on Convex Programming (2004), Kluwer, Boston
47. Ostrowski A.M. Solution of Equations and Systems of Equations (1960), Academic Press, Basel
48. Ortega J.M., and Rheinboldt W.C. Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press, New York-London
49. Peitgen H.O., Saupe D., and Haeseler F. Cayley's problem and Julia sets. Mathematical Intelligencer 6 2 (1984) 11-20
50. In: Peitgen H.O. (Ed). Newton's Method and Dynamical Systems (1989), Springer, Berlin
51. Polyak B.T. Gradient methods for solving equations and inequalities. USSR Computational Mathematics and Mathematical Physics 4 6 (1964) 17-32
52. Polyak B.T. Convexity of nonlinear image of a small ball with applications to optimization. Set-Valued Analysis 9 1-2 (2001) 159-168
53. Polyak B.T. The convexity principle and its applications. Bulletin of Brazilian Mathematical Society 34 1 (2003) 59-75
54. 2 bounded controls. Dynamics of Continuous, Discrete and Impulsive Systems 11 2-3 (2004) 255-268
55. Polyak B.T. Introduction to Optimization (1987), Optimization Software, New York
56. Polyak B.T. Iterative methods using Lagrange multipliers for solving extremal problems with equality-type constraints. USSR Computational Mathematics and Mathematical Physics 10 5 (1970) 42-52
57. Polyak B.T. Minimization of nonsmooth functionals. USSR Computational Mathematics and Mathematical Physics 9 3 (1969) 14-29
58. Qi L., and Sun J. A nonsmooth version of Newton's method. Mathematical Programming 58 (1993) 353-367
59. Ramm A.G. Acceleration of convergence: A continuous analog of the Newton's method. Applicable Analysis 81 4 (2002) 1001-1004
60. Rheinboldt W.C. Methods for Solving Systems of Nonlinear Equations (1998), SIAM, Philadelphia
61. Robinson S.M. Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear programming algorithms. Mathematical Programming 7 (1974) 1-16
62. Robinson S.M. Newton's method for a class of nonsmooth functions. Set-Valued Analysis 2 (1994) 291-305
63. Smale S. A convergent process of price adjustment and global Newton's methods. Journal of Mathematical Economics 3 (1976) 107-120
64. Smale S. Newton's method estimates from data at one point. In: Ewing R., Gross K., and Martin C. (Eds). The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics (1986), Springer, Berlin
65. Smale S. Complexity theory and numerical analysis. Acta Numerica 6 (1997) 523-551
66. Wright S.J. Primal-Dual Interior-Point Methods (1997), SIAM, Philadelphia
67. Ypma T.J. Historical development of the Newton-Raphson method. SIAM Review 37 4 (1995) 531-551