Some Aspects of Stability in Stochastic Programming

Annals of Operations Research

Volumn 100, Issue 1-4, 2000, Pages 55-84

  Schultz, Rüdiger ^a  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

Author keywords

Parametric optimization; Stability analysis; Stochastic programming

Indexed keywords

EID: 0034562594     PISSN: 02545330     EISSN: 15729338     Source Type: Journal    
DOI: 10.1023/a:1019258932012     Document Type: Article

References (114)

1. Z. Artstein, Sensitivity with respect to the underlying information in stochastic programs, Journal of Computational and Applied Mathematics 56 (1994) 127-136.
2. Z. Artstein, Probing for information in two-stage stochastic programming and the associated consistency, in: Asymptotic Statistics, eds. P. Mandl and M. Hušková (Springer, Berlin, 1994) pp. 21-33.
3. Z. Artstein and R.J.-B. Wets, Sensors and information in optimization under stochastic uncertainty, Mathematics of Operations Research 18 (1993) 523-547.
4. Z. Artstein and R.J.-B. Wets, Stability results for stochastic programs and sensors, allowing for discontinuous objective functions, SIAM Journal on Optimization 4 (1994) 537-550.
5. Z. Artstein and R.J.-B. Wets, Consistency of minimizers and the SLLN for stochastic programs, Journal of Convex Analysis 2 (1995) 1-17.
6. H. Attouch, Variational Convergence for Functions and Operators (Pitman, London, 1984).
7. H. Attouch and R.J.-B. Wets, Epigraphical processes: laws of large numbers for random lsc functions, Technical Report, Department of Mathematics, University of California, Davis (1991).
8. H. Attouch and R.J.-B. Wets, Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM Journal on Optimization 3 (1993) 359-381.
9. J.P. Aubin, Lipschitz behaviour of solutions to convex minimization problems, Mathematics of Operations Research 9 (1984) 87-111.
10. B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-Linear Parametric Optimization (Akademie-Verlag, Berlin, 1982).
11. B. Bank and R. Mandel, Parametric Integer Optimization (Akademie-Verlag, Berlin, 1988).
12. C. Berge, Topological Spaces (Macmillan, New York, 1963).
13. R.N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions (Wiley, New York, 1976).
14. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
15. J.R. Birge and F. Louveaux, Introduction to Stochastic Programming (Springer, New York, 1997).
16. J.R. Birge and L. Qi, Subdifferential convergence in stochastic programs, SIAM Journal on Optimization 5 (1995) 436-453.
17. J.R. Birge and L. Qi, Continuous approximation schemes for stochastic programs, Annals of Operations Research 56 (1995) 15-38.
18. J.R. Birge and R.J.-B. Wets, Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse, Mathematical Programming Study 27 (1986) 54-102.
19. C.E. Blair and R.J.-B. Jeroslow, The value function of a mixed integer program: I, Discrete Mathematics 19 (1977) 121-138.
20. J.F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour, SIAM Review 40 (1998) 228-264.
21. C. Borell, Convex set functions in d-space, Periodica Mathematica Hungarica 6 (1975) 111-136.
22. F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
23. J.M. Danskin, The Theory of Max-Min and its Application to Weapons Allocations Problems (Springer, New York, 1967).
24. G.B. Dantzig, J. Folkman and N. Shapiro, On the continuity of the minimum set of a continuous function, Journal of Mathematical Analysis and Applications 17 (1967) 519-548.
25. D. Dentcheva, Differentiable selections of set-valued mappings with application in stochastic programming, Journal of Mathematical Analysis and Applications 223 (1998) 371-396.
26. D. Dentcheva and W. Römisch, Differential stability of two-stage stochastic programs, SIAM Journal on Optimization 11 (2000) 87-112.
27. R.M. Dudley, Real Analysis and Probability (Wadsworth & Brooks/Cole, Pacific Grove, CA, 1989).
28. J. Dupačová, Stability in stochastic programming with recourse-estimated parameters, Mathematical Programming 28 (1984) 72-83.
29. J. Dupačová, Stability in stochastic programming - probabilistic constraints, in: Stochastic Optimization, eds. V.I. Arkin, A. Shiraev and R.J.-B. Wets (Springer, Berlin, 1986) pp. 314-325.
30. J. Dupačová, Stochastic programming with incomplete information: A survey of results on postoptimization and sensitivity analysis, Optimization 18 (1987) 507-532.
31. J. Dupačová, Stability and sensitivity analysis for stochastic programming, Annals of Operations Research 27 (1990) 115-142.
32. J. Dupačová and W. Römisch, Quantitative stability for scenario-based stochastic programs, in: Prague Stochastics '98, eds. M. Hušková et al. (JCMF, Prague, 1998) pp. 119-124.
33. J. Dupačová and R.J.-B. Wets, Asymptotic behaviour of statistical estimators and of optimal solutions of stochastic optimization problems, The Annals of Statistics 16 (1988) 1517-1549.
34. A.V. Fiacco, Sensitivity analysis for nonlinear programming using penalty functions, Mathematical Programming 10 (1976) 287-311.
35. A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983).
36. O. Fiedler and W. Römisch, Stability in multistage stochastic programming, Annals of Operations Research 56 (1995) 79-93.
37. S.J. Gartska, Distribution functions in stochastic programs with recourse: A parametric analysis, Mathematical Programming 6 (1974) 339-351.
38. C.R. Givens and R.M. Shortt, A class of Wasserstein metrics for probability distributions, Michigan Mathematical Journal 31 (1984) 231-240.
39. N. Gröwe, Estimated stochastic programs with chance constraints, European Journal of Operational Research 101 (1997) 285-305.
40. R. Henrion and W. Römisch, Metric regularity and quantitative stability in stochastic programs with probabilistic constraints, Mathematical Programming 84 (1999) 55-88.
41. R. Henrion and W. Römisch, Stability of solutions to chance constrained stochastic programs, in: Parametric Optimization and Related Topics, eds. J. Guddatt, R. Hirabaya Shi, H.Th. Jongen and F. Twilt (Peter Lang, Frankfurt a. M., 2000) pp. 95-114.
42. P. Kall, Stochastic Linear Programming (Springer, Berlin, 1976).
43. P. Kall, On approximations and stability in stochastic programming, in: Parametric Optimization and Related Topics, eds. J. Guddat, H.Th. Jongen, B. Kummer and F. Nožička (Akademie-Verlag, Berlin, 1987) pp. 387-407.
44. P. Kall, A. Ruszczyński and K. Frauendorfer, Approximation techniques in stochastic programming, in: Numerical Techniques for Stochastic Optimization, eds. Yu. Ermoliev and R.J.-B. Wets (Springer, Berlin, 1988) pp. 33-64.
45. P. Kall and S.W. Wallace, Stochastic Programming (Wiley, Chichester, 1994).
46. Yu.M. Kaniovski, A.J. King and R.J.-B. Wets, Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems, Annals of Operations Research 56 (1995) 189-208.
47. V. Kaňková, Stability in the stochastic programming, Kybernetika 14 (1978) 339-349.
48. A.J. King, Asymptotic behavior of solutions in stochastic optimization: Nonsmooth analysis and the derivation of non-normal limit distributions, Ph.D. dissertation, University of Washington, Seattle (1986).
49. A.J. King, Generalized delta theorems for multivalued mappings and measurable selections, Mathematics of Operations Research 14 (1989) 720-736.
50. A.J. King and R.T. Rockafellar, Asymptotic theory for solutions in statistical estimation and stochastic programming, Mathematics of Operations Research 18 (1993) 148-162.
51. A.J. King and R.J.-B. Wets, Epi-consistency of convex stochastic programs, Stochastics and Stochastics Reports 34 (1991) 83-92.
52. D. Klatte, On the stability of local and global optimal solutions in parametric problems of nonlinear programming, Part I and Part II, in: Seminarbericht Nr. 75, Sektion Mathematik (Humboldt-Universität zu Berlin, Berlin, 1985) pp. 1-39.
53. D. Klatte, A note on quantitative stability results in nonlinear optimization, in: Seminarbericht Nr. 90, Sektion Mathematik (Humboldt-Universität zu Berlin, Berlin, 1987) pp. 77-86.
54. D. Klatte, On quantitative stability for non-isolated minima, Control and Cybernetics 23 (1994) 183-200.
55. D. Klatte and G. Thiere, Error bounds for solutions of linear equations and inequalities, Mathematical Methods of Operations Research 41 (1995) 191-214.
56. L. Korf and R.J.-B. Wets, Ergodic limit laws for stochastic optimization problems, Technical Report, Department of Mathematics, University of California, Davis (1997).
57. J. Kuelbs and R.M. Dudley, Log log laws for empirical measures, Annals of Probability 8 (1980) 405-418.
58. K. Marti, Approximationen der Entscheidungsprobleme mit linearer Ergebnisfunktion und positiv homogener, subadditiver Verlustfunktion, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 31 (1975) 203-233.
59. B.S. Mordukhovich, Complete characterization of openness, metric regularity and Lipschitzian properties of multifunctions, Transactions of the American Mathematical Society 340 (1993) 1-35.
60. B.S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, Journal of Mathematical Analysis and Applications 183 (1994) 250-288.
61. G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988).
62. G.Ch. Pflug, Asymptotic stochastic programs, Mathematics of Operations Research 21 (1996) 769-789.
63. G.Ch. Pflug, Stochastic programs and statistical data, Annals of Operations Research 85 (1999) 59-78.
64. G.Ch. Pflug, A. Ruszczyński and R. Schultz, On the Glivenko-Cantelli problem in stochastic programming: Linear recourse and extensions, Mathematics of Operations Research 23 (1998) 204-220.
65. G.Ch. Pflug, A. Ruszczyński and R. Schultz, On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse, Mathematical Methods of Operations Reserach 47 (1998) 39-49.
66. D. Pollard, Convergence of Stochastic Processes (Springer, New York, 1984).
67. A. Prékopa, Logarithmic concave measures with applications to stochastic programming, Acta Scientiarum Mathematicarum 32 (1971) 301-316.
68. A. Prékopa, Stochastic Programming (Kluwer, Dordrecht, 1995).
69. S.T. Rachev, Probability Metrics and the Stability of Stochastic Models (Wiley, New York, 1991).
70. S.T. Rachev and W. Römisch, Quantitative stability of stochastic programs: The approach via probability metrics, Manuscript, Humboldt-Universität zu Berlin, Institut für Mathematik (1999).
71. Y. Rinott, On convexity of measures, Annals of Probability 4 (1976) 1020-1026.
72. S.M. Robinson, Perturbed, Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms, Mathematical Programming 7 (1974) 1-16.
73. S.M. Robinson, Regularity and stability for convex multivalued functions, Mathematics of Operations Research 1 (1976) 130-143.
74. S.M. Robinson, Persistence and continuity of local optimizers, IIASA Collaborative Paper CP-84-5, IIASA Laxenburg, Austria (1984).
75. S.M. Robinson, Local epi-continuity and local optimization, Mathematical Programming 37 (1987) 208-222.
76. S.M. Robinson and R.J.-B. Wets, Stability in two-stage stochastic programming, SIAM Journal on Control and Optimization 25 (1987) 1409-1416.
77. R.T. Rockafellar, Integral functionals, normal integrands and measurable selections, in: Nonlinear Operators and the Calculus of Variations, eds. G.P. Gossez et al., Lecture Notes in Mathematics, Vol. 543 (Springer, New York, 1976) pp. 157-207.
78. R.T. Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Analysis: Theory, Methods and Applications 9 (1985) 867-885.
79. R.T. Rockafellar and R.J.-B. Wets, Variational Analysis (Springer, Berlin, 1998).
80. W. Römisch and R. Schultz, Distribution sensitivity in stochastic programming, Mathematical Programming 50 (1991) 197-226.
81. W. Römisch and R. Schultz, Distribution sensitivity for certain classes of chance-constrained models with application to power dispatch, Journal of Optimization Theory and Applications 71 (1991) 569-588.
82. W. Römisch and R. Schultz, Stability analysis for stochastic programs, Annals of Operations Research 30 (1991) 241-266.
83. W. Römisch and R. Schultz, Lipschitz stability for stochastic programs with complete recourse, SIAM Journal on Optimization 6 (1996) 531-547.
84. W. Römisch and A. Wakolbinger, Obtaining convergence rates for approximations in stochastic programming, in: Parametric Optimization and Related Topics, eds. J. Guddat, H.Th. Jongen, B. Kummer and F. Nožička (Akademie-Verlag, Berlin, 1987) pp. 327-343.
85. R.Y. Rubinstein and A. Shapiro, Discrete Event Systems. Sensitivity Analysis and Stochastic Optimization by the Score Function Method (Wiley, Chichester, 1993).
86. G. Salinetti, Approximations for chance-constrained programming problems, Stochastics 10 (1983) 157-179.
87. R. Schultz, Continuity and stability in two-stage stochastic integer programming, in: Stochastic Optimization, Numerical Methods and Technical Applications, ed. K. Marti, Lecture Notes in Economics and Mathematical Systems, Vol. 379 (Springer, Berlin, 1992) pp. 81-92.
88. R. Schultz, Continuity properties of expectation functions in stochastic integer programming, Mathematics of Operations Research 18 (1993) 578-589.
89. R. Schultz, Strong convexity in stochastic programs with complete recourse, Journal of Computational and Applied Mathematics 56 (1994) 3-22.
90. R. Schultz, On structure and stability in stochastic programs with random technology matrix and complete integer recourse, Mathematical Programming 70 (1995) 73-89.
91. R. Schultz, Rates of convergence in stochastic programs with complete integer recourse, SIAM Journal on Optimization 6 (1996) 1138-1152.
92. R.J. Serfling, Approximation Theorems of Mathematical Statistics (Wiley, New York, 1980).
93. A. Shapiro, Asymptotic properties of statistical estimators in stochastic programming, The Annals of Statistics 17 (1989) 841-858.
94. A. Shapiro, On concepts of directional differentiability, Journal of Optimization Theory and Applications 66 (1990) 477-487.
95. A. Shapiro, On differential stability in stochastic programming, Mathematical Programming 47 (1990) 107-116.
96. A. Shapiro, Asymptotic analysis of stochastic programs, Annals of Operations Research 30 (1991) 169-186.
97. A. Shapiro, Perturbation analysis of optimization problems in Banach spaces, Numerical Functional Analysis and Optimization 13 (1992) 97-116.
98. A. Shapiro, Asymptotic behavior of optimal solutions in stochastic programming, Mathematics of Operations Research 18 (1993) 829-845.
99. A. Shapiro, Quantitative stability in stochastic programming, Mathematical Programming 67 (1994) 99-108.
100. G.R. Shorack and J.A. Wellner, Empirical Processes with Applications to Statistics (Wiley, New York, 1986).
101. L. Stougie, Design and analysis of algorithms for stochastic integer programming, Ph.D. Thesis, Center for Mathematics and Computer Science, Amsterdam (1985).
102. L. Stougie and M.H. Van der Vlerk, Stochastic integer programming, in: Annotated Bibliographies in Combinatorial Optimization, eds. M. Dell'Amico, F. Maffioli and S. Martello (Wiley, Chichester, 1997) pp. 127-141.
103. A.W. Van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes (Springer, New York, 1996).
104. V.N. Vapnik and A. V. Červonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and Applications 16 (1971) 264-280.
105. S. Vogel, Stability results for stochastic programming problems, Optimization 19 (1988) 269-288.
106. S. Vogel, A stochastic approach to stability in stochastic programming, Journal of Computational and Applied Mathematics 56 (1994) 65-96.
107. D.W. Walkup and R.J.-B. Wets, Lifting projections of convex polyhedra, Pacific Journal of Mathematics 28 (1969) 465-475.
108. J. Wang, Continuity of feasible solution sets of probabilistic constrained programs, Journal of Optimization Theory and Applications 63 (1989) 79-89.
109. J. Wang, Stability of multistage stochastic programming, Annals of Operations Research 56 (1995) 313-322.
110. R.J.-B. Wets, Stochastic programs with fixed recourse: the equivalent deterministic program, SIAM Review 16 (1974) 309-339.
111. R.J.-B. Wets, Stochastic programming: solution techniques and approximation schemes, in: Mathematical Programming: State-of-the-Art 1982, eds. A. Bachem, M. Grötschel and B. Korte (Springer, Berlin, 1983) pp. 560-603.
112. R.J.-B. Wets, Stochastic programming, in: Handbooks in Operations Research and Management Science, Vol.1, Optimization, eds. G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd (North-Holland, 1989) pp. 573-629.
113. R.J.-B. Wets, Constrained estimation: Consistency and asymptotics, Applied Stochastic Models and Data Analysis 7 (1991) 17-32.
114. R.J.-B. Wets, Challenges in stochastic programming, Mathematical Programming 75 (1996) 115-135.