The stochastic root-finding problem: Overview, solutions, and open questions

ACM Transactions on Modeling and Computer Simulation

Volumn 21, Issue 3, 2011, Pages

  Pasupathy, Raghu ^a   Kim, Sujin ^b  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Sample average approximation; Stochastic approximation; Stochastic root finding

Indexed keywords

COMPUTER CODES; EUCLIDEAN SPACES; FIXED-POINT PROBLEM; IMPLEMENTATION ASPECTS; SAMPLE-AVERAGE APPROXIMATION; STOCHASTIC APPROXIMATION; STOCHASTIC ROOT FINDING; STOCHASTIC ROOT-FINDING PROBLEM; STOCHASTIC SIMULATIONS; SYSTEM OF EQUATIONS; THEORETICAL RESULT; VECTOR FUNCTIONS;

APPROXIMATION THEORY; NONLINEAR EQUATIONS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS;

PROBLEM SOLVING;

EID: 79952563147     PISSN: 10493301     EISSN: 15581195     Source Type: Journal    
DOI: 10.1145/1921598.1921603     Document Type: Article

References (74)

1. Abbasbandy, S. 2003. Improving the Newton-Raphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 145, 887-893.
2. Atlason, J., Epelman, M.A., and Henderson, S. G. 2004. Call center staffing with simulation and cutting plane methods. Ann. Oper. Res. 127, 333-358. (Pubitemid 39056536)
3. Atlason, J., Epelman, M.A., and Henderson, S. G. 2005. Optimizing call center staffing using simulation and analytic center cutting plane methods. Manage. Sci. To appear.
4. Bartle, R. G. 1976. The Elements of Real Analysis. Wiley, New York.
5. Barton, R.R. and Meckesheimer, M. 2006. Metamodel-based simulation optimization. In Simulation, S. G. Henderson and B.L. Nelson Eds. Elsevier, Amsterdam, 535-574.
6. Bayraksan, G. and Morton, D. P. 2009. A sequential sampling procedure for stochastic programming. Oper. Res. To appear.
7. Ben-Akiva, M. and Lerman, S. R. 1985. Discrete Choice Analysis: Theory and Application to Travel Demand. The MIT Press, Cambridge, MA.
8. Bhatnagar, S. 2005. Adaptive three-timescale stochastic approximation. ACM Trans. Model. Comput. Simul. 15, 1, 74-107.
9. Blum, J. 1954a. Approximation methods which converge with probability one. Ann. Math. Stat. 25, 2, 382-386.
10. Blum, J. 1954b. Multidimensional stochastic approximation. Ann. Math. Stat. 25, 4, 737-744.
11. Bratley, P., Fox, B.L., and Schrage, L. E. 1987. A Guide to Simulation. Springer, Berlin.
12. Broadie, M., Cicek, D.M., and Zeevi, A. 2009. An adaptive multidimensional version of the Kiefer-Wolfowitz stochastic approximation algorithm. In Proceedings of the Winter Simulation Conference.M.D.Rossetti et al. Eds. IEEE, Los Alamitos, CA, 601-612.
13. Chen, H. and Schmeiser, B. W. 2001. Stochastic root finding via retrospective approximation. IIE Trans. 33, 259-275. (Pubitemid 32271552)
14. Chow, Y.S. and Robbins, H. E. 1965. On the asymptotic theory of fixed-width confidence intervals for the mean. Ann. Math. Stat. 36, 457-462.
15. Chung, K.L. 1954. On a stochastic approximation method. Ann. Math. Stat. 25, 463-483.
16. Dunkel, J. and Weber, S. 2009. Stochastic root finding and efficient estimation of convex risk measures. Oper. Res. To appear.
17. Fabian, V. 1968. On asymptotic normality in stochastic approximation. Ann. Math. Stat. 39, 1327-1332.
18. Frees, E.W. and Ruppert, D. 1990. Estimation following a Robbins-Monro designed experiment. J. Amer. Stat. Asoc. 85, 1123-1129.
19. Fu, M. C. 1994. Optimization via simulation: A review. Ann. Oper. Res. 53, 199-247.
20. Giesecke, K., Schmidt, T., and Weber, S. 2008. Measuring the risk of large losses. J. Investment Manage. 6, 4, 1-15.
21. Glasserman, P. 1991. Gradient Estimation Via Perturbation Analysis. Kluwer, Amsterdam.
22. Healy, K. and Schruben, L. W. 1991. Retrospective simulation response optimization. In Proceedings of the Winter Simulation Conference. B. L. Nelson et al. Eds. IEEE, Los Alamitos, CA, 954-957.
23. Henderson, S. G. and Nelson, B. L. (eds.). 2006. Handbooks in Operations Research and Management Science: Simulation. Vol. 13. Elsevier, Amsterdam.
24. Herer, Y.T., Tzur, M., and Yucesan, E. 2006. The multilocation transshipment problem. IIE Trans. 38, 185-200. (Pubitemid 43203320)
25. Higle, J. and Sen, S. 1991. Stochastic decomposition: An algorithm for two-stage stochastic linear programs with recourse. Math. Oper. Res.16, 650-669.
26. Homem-de-Mello, T. 2003. Variable-sample methods for stochastic optimization. ACM Trans. Model. Comput. Simul. 13, 108-133.
27. Householder, A. S. 1970. The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill, New York.
28. Hsieh, M. and Glynn, P. W. 2002. Confidence regions for stochastic approximation algorithms. In Proceedings of the Winter Simulation Conference. E. Yucesan et al. Eds. IEEE, Los Alamitos, CA, 370-375.
29. Joseph, V.R., Tian, Y., and Wu, C. F. J. 2007. Adaptive designs for stochastic root-finding. Statistica Sinica 17, 1549-1565.
30. Kelly, C. T. 1995. Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA.
31. Kelly, C. T. 2006. Solving Nonlinear Equations with Newton's Method. SIAM, Philadelphia, PA.
32. Kim, S. and Henderson, S. G. 2008. The mathematics of continuous-variable simulation optimization. In Proceedings of the Winter Simulation Conference. S. J. Mason et al. Eds. IEEE, Los Alamitos, CA, 159-167.
33. Kleywegt, A.J., Shapiro, A., and Homem-de-Mello, T. 2001. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12, 479-502.
34. Kushner, H.J. and Yin, G. G. 2003. Stochastic Approximation and Recursive Algorithms and Applications. Springer, Berlin.
35. Lai, T. L. 2003. Invited paper: Stochastic approximation. Ann. Stat. 31, 2, 391-406.
36. Law, A. M. 2007. Simulation Modeling and Analysis. McGraw-Hill, New York.
37. Lawphongpanich, S., Hearn, D.W., and Smith, M. J. (eds.). 2006. Applied Optimization: Mathematical and Computational Models for Congestion Charging, Vol. 101. Springer, Berlin.
38. Ljung, L. 1977a. Analysis of recursive stochastic algorithms. IEEE Trans. Autom. Control AC-22, 551-575.
39. Ljung, L. 1977b. On positive real transfer functions and the convergence of some recursive schemes. IEEE Trans. Autom. Control AC-22, 539-551.
40. Luedtke, J. and Ahmed, S. 2008. A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. To appear.
41. Nemirovski, A., Juditsky, A., Lan, G., and Shapiro, A. 2009. Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 4.
42. Nevel'son, M.B. and Khas'minskii, R. Z. 1973. Stochastic Approximation and Recursive Estimation. American Mathematical Society, Providence, RI.
43. Noor, K.I. and Noor, M. A. 2007. Iterative methods with fourth-order convergence for nonlinear equations. Appl. Math. Computation 189, 1, 221-227. (Pubitemid 46734233)
44. Ortega, J.M. and Rheinboldt, W. C. 1970. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.
45. Pasupathy, R. 2010. On choosing parameters in retrospective-approximation algorithms for stochastic root finding and simulation optimization. Oper. Res. 58, 4, 889-901.
46. Pasupathy, R. and Schmeiser, B. W. 2009. Retrospective-approximation algorithms for multidimensional stochastic root-finding problems. ACM Trans. Model. Comput. Simul. 19, 2, 5:1-5:36.
47. Pflug, G. C. 1996. Optimization of Stochastic Models: The Interface Between Simulation and Optimization. Kluwer, Amsterdam.
48. Plambeck, E.L., Fu, B.R., Robinson, S.M., and Suri, R. 1996. Sample-path optimization of convex stochastic performance functions. Math. Program. 75, 137-176. (Pubitemid 126393009)
49. Polyak, B. T. and Juditsky, A. B. 1992. Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30, 4, 838-855.
50. Rheinboldt, W. C. 1987. Methods for Solving Systems of Nonlinear Equations in Several Variables. Society for industrial Mathematics.
51. Robbins, H. and Monro, S. 1951. A stochastic approximation method. Ann. Math. Stat. 22, 400-407.
52. Rubinstein, R.Y. and Shapiro, A. 1993. Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. Wiley, New York.
53. Ruppert, D. 1985. A Newton-Raphson version of the multivariate Robbins-Monro procedure. Ann. Stat. 13, 236-245.
54. Ruppert, D. 1991. Stochastic approximation. In Handbook in Sequential Analysis. Dekker, New York, 503-529.
55. Sacks, J. 1958. Asymptotic distribution of stochastic approximation procedure. Ann. Math. Stat. 29, 373-405.
56. Shapiro, A. 1991. Asymptotic analysis of stochastic programs. Ann. Oper. Res. 30, 169-186.
57. Shapiro, A. 2004. Monte Carlo sampling methods. In Stochastic Programming, A. Ruszczynski and Shapiro Eds. Elsevier, Amsterdam, 353-426.
58. Sheffi, Y. 1985. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice Hall, New York.
59. Spall, J. C. 1998. Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Trans. Aerospace Electron. Syst. 34, 817-823. (Pubitemid 128737755)
60. Spall, J. C. 2000. Adaptive stochastic approximation by the simultaneous perturbation method. IEEE Trans. Autom Control 45, 1839-1853. (Pubitemid 32067073)
61. Spall, J. C. 2003. Introduction to Stochastic Search and Optimization.Wiley, New York.
62. Stewart, G. W. 1973. Introduction to Matrix Computations. Academic Press, New York.
63. Todd, M. J. 1976. The Computation of Fixed Points and Applications. Springer, Berlin.
64. Traub, J. F. 1964. Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ.
65. Venter, J. H. 1967. An extension of the Robbins-Monro procedure. Ann. Stat. 38, 181-190.
66. Verweij, B., Ahmed, S., Kleywegt, A., Nemhauser, G., and Shapiro, A. 2003. The sample average approximation method applied to stochastic vehicle routing problems: a computational study. Comput. Appl. Optim. 24, 289-333.
67. Wasan, M. T. 1969. Stochastic Approximation. Cambridge University Press, Cambridge, UK.
68. Weerakoon, S. and Fernando, T. G. I. 2000. A variant of Newton's method with accelerated third-order convergence. Appl. Math. Lett. 13, 87-93.
69. Wei, C. Z. 1987. Multivariate adaptive stochastic approximation. Ann. Stat. 15, 1115-1130.
70. Wu, C. F. J. 1985. Efficient sequential designs with binary data. Ann. Stat. 13, 1498-1508.
71. Wu, C. F. J. 1986. Maximum likelihood recursion and stochastic approximation in sequential designs. In Statistical Procedures and Related Topics, J.V. Ryzin Ed. IMS, Hayward, CA, 298-313.
72. Yakowitz, S., L'Ecuyer, P., and Vazquez-Abad, F. 2000. Global stochastic optimization with low-dispersion point sets. Oper. Res. 48, 939-950. (Pubitemid 32189512)
73. Ying, Z. and Wu, C. F. J. 1997. An asymptotic theory of sequential designs based on maximum likelihood recursions. Stat. Sinica 7, 75-918.
74. Young, D. M. 1971. Iterative Solution of Large Linear Systems. Academic Press, New York.