PUBLICATIONS

 
 
 

Books

Lectures on Stochastic Programming: Modeling and Theory, by Shapiro, A., Dentcheva, D. and Ruszczynski, A., SIAM, Philadelphia, 2009. SIAM web site of this book.

Stochastic Programming,
Handbook in Operations Research and Management Science,
edited by  Ruszczynski, A.  and Shapiro, A.,
Elsevier, 2003.

Bonnans, J.F. and Shapiro, A.,Perturbation Analysis of Optimization Problems,
Springer, New York, 2000, Chinese edition, Science Press, 2008.
Corrections

Rubinstein, R.Y. and Shapiro, A., Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method, John Wiley and Sons, New York, 1993.


Preprints

Shapiro, A. and Philpott, A, "A Tutorial on Stochastic Programming "

Lan, G., Nemirovski, A. and Shapiro, A., "Validation Analysis of Robust Stochastic Approximation Method", Published electronically in:  Optimization Online

Nemirovski, A. and Shapiro, A., "On complexity of Shmoys - Swamy class of two-stage linear stochastic programming problems",
Published electronically in:  Optimization Online

Shapiro, A. and Nemirovski, A., "Duality of linear conic problems",
Published electronically in:  Optimization Online

Shapiro, A., "On differentiability of symmetric matrix valued functions",
Published electronically in:  Optimization Online

Ahmed, S. and  Shapiro, A., "The Sample Average Approximation Method for Stochastic Programs with Integer Recourse",
Published electronically in:  Optimization Online
 

Parts of Books

1. Shapiro, A., ``Minimum Rank Factor Analysis", in: Encyclopedia of Statistical Sciences, pp. 532-534, Vol. 5, S. Kotz, N.L. Johnson and C.B. Read, Eds., New York, Wiley, 1985.

2. Shapiro, A., ``Minimum Trace Factor Analysis", in: Encyclopedia of Statistical Sciences, pp. 534-537, Vol. 5, S. Kotz, N.L. Johnson and C.B. Read, Eds., New York, Wiley, 1985.

3. Rubinstein, R.Y., Shapiro, A. and Uryas'ev, S., ``The Score Function Method", in: Encyclopedia of Operations Research and Management Science, Gass, S.I. and C.M. Harris, Editors, pp. 614-617, Kluwer Academic Publishers, 1996.

4. Shapiro, A., ``Optimality conditions and sensitivity analysis of cone-constrained and semi-definite programs", in: Topics in semidefinite and interior-point methods, pp. 1-15, Fields Inst. Commun., 18, Amer. Math. Soc., Providence, RI, 1998.

5. Shapiro, A., ``First and Second Order Optimality Conditions and Perturbation Analysis of Semi-Infinite Programming Problems", in: R. Reemtsen and J.-J. Rueckmann, Eds., Semi-Infinite Programming, pp. 103-133, Kluwer Academic Publishers, Boston-London-Dordrecht, 1998.

6. Shapiro, A., ``Semidefinite Programming: Optimality Conditions and Stability", Encyclopedia of Optimization, Kluwer Academic Publishers, to appear.

7. Shapiro, A., `` Duality, optimality conditions and perturbation analysis", in: Semidefinite Programming and Applications Handbook, R. Saigal and L. Vandenberghe and H.Wolkowicz, Eds., KluwerAcademic Publishers, Boston, pp.67-92, 2000.

8. Shapiro, A., ``Statistical inference of   stochastic optimization problems", in:  Probabilistic Constrained
Optimization: Methodology and Applications.  Uryasev, S. (Ed.), Kluwer Academic Publishers, pp. 282-304, 2000.

9. Kleywegt, A.J. and Shapiro, A., "Stochastic Optimization",  3rd Edition of the Handbook of Industrial Engineering, Gavriel Salvendy, Ed., John Wiley, New York, pp. 2625 - 2650, 2001.

10. Shapiro, A.  "On duality theory of conic linear problems",   Miguel A. Goberna and Marco A. López, Eds., Semi-Infinite Programming: Recent Advances,   Kluwer Academic Publishers, pp. 135 - 165, 2001.

11. Nemirovski, A. and Shapiro, A., "Scenario approximations of chance constraints",
Published electronically in:  Optimization Online
Published in: Probabilistic and Randomized Methods for Design under Uncertainty, pp. 3-48, G. Calafiore and F. Dabbene (Eds.), Springer, London, 2005.

12. Shapiro, A. and Nemirovski, A., "On complexity of stochastic programming problems",
Published electronically in:  Optimization Online
Published in: Continuous Optimization: Current Trends and Applications, pp. 111-144, V. Jeyakumar and A.M. Rubinov (Eds.), Springer, 2005.

13. Ruszczynski, A. and Shapiro, A., "Optimization of Risk Measures",
Published in: Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene (Eds.), Springer, London, 2005.

14. Shapiro, A., "Statistical Inference of Moment Structures", Handbook of Latent Variable and Related Models, Sik-Yum Lee (Ed.), Elsevier, Amsterdam, 2007.


Published Journal Papers (refereed)

1. Shapiro, A., ``Rank-Reducibility of a Symmetric Matrix and Sampling Theory of Minimum Trace Factor Analysis", Psychometrika, vol. 47, pp. 187-199, 1982.

2. Shapiro, A., ``Optimally Scaled Matrices, Necessary and Sufficient Conditions", Numerische Mathematik, vol. 39, pp. 239-245, 1982.

3. Shapiro, A., ``Weighted Minimum Trace Factor Analysis", Psychometrika, vol. 47, pp. 243-264, 1982.

4. Della Riccia, G. and Shapiro, A., ``Minimum Rank and Minimum Trace of Covariance Matrices", Psychometrika, vol. 47, pp. 443-448, 1982.

5. Shapiro, A., ``On Unsolvability of Inverse Eigenvalue Problems Almost Everywhere", Linear Algebra and Applications, vol. 49, pp. 27-31, 1983.

6. Della Riccia, G. and Shapiro, A., ``Fisher Discriminant Analysis and Factor Analysis", IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-5, pp. 99-104, 1983.

7. Shapiro, A., ``Asymptotic Distribution Theory in the Analysis of Covariance Structures (a unified approach)", South African Statistical Journal, vol. 17, pp. 33-81, 1983.

8. Shapiro, A. and Browne , M.W., ``On the Investigation of Local Identifiability: a Counter-Example", Psychometrika, vol. 48, pp. 303-304, 1983.

9. Shapiro, A., ``A Note on the Consistency of Estimators in the Analysis of Moment Structures", British Journal of Mathematical and Statistical Psychology, vol. 37, pp. 84-88, 1984.

10. Shapiro, A., ``On Optimality Conditions in Quasidifferentiable Optimization", SIAM J. Control and Optimization, vol. 22, pp. 610-617, 1984.

11. Shapiro, A., ``Optimal Block Diagonal l2-Scaling of Matrices", SIAM J. Numerical Analysis, vol. 22, pp. 81-94, 1985.

12. Shapiro, A., ``Asymptotic Distribution of Test Statistics in the Analysis of Moment Structures Under Inequality Constraints", Biometrika, vol. 72, pp. 133-144, 1985.

13. Shapiro, A., ``Asymptotic Equivalence of Minimum Discrepancy Function Estimators to GLS Estimators ", South African Statistical Journal, vol. 19, pp. 73-81, 1985.

14. Shapiro, A., ``Extremal Problems on the Set of Nonnegative Definite Matrices", Linear Algebra and Applications, vol. 67, pp. 7-18, 1985.

15. Shapiro, A., ``A Note on the Asymptotic Distribution of the Greatest Lower Bound to Reliability", Psychometrika, vol. 50, pp. 243-244, 1985.

16. Shapiro, A., ``Identifiability of Factor Analysis, Some Results and Open Problems", Linear Algebra and Applications, vol. 70, pp. 1-7, 1985.

17. Steiger, J.H., Shapiro, A. and Browne, M.W., ``On the Multivariate Asymptotic Distribution of Sequential Chi-Square Statistics", Psychometrika, vol. 50, pp. 253-264, 1985.

18. Shapiro, A., ``Second-Order Derivatives of Extremal-Value Functions and Optimality Conditions for Semi-Infinite Programs", Mathematics of Operations Research, vol. 10, pp. 207-219, 1985.

19. Shapiro, A., ``Second-Order Sensitivity Analysis and Asymptotic Theory of Parametrized Nonlinear Programs ", Mathematical Programming, vol. 33, pp. 280-299, 1985.

20. Shapiro, A., ``Asymptotic Theory of Overparametrized Structural Models", Journal of the American Statistical Association, vol. 81, pp. 142-149, 1986.

21. Shapiro, A., ``Quasidifferential Calculus and First-Order Optimality Conditions in Nonsmooth Optimization", Mathematical Programming Studies, vol. 29, pp. 56-68, 1986.

22. Browne, M.W. and Shapiro, A., ``The Asymptotic Covariance Matrix of Sample Correlation Coefficients under General Conditions", Linear Algebra and Applications, vol. 82, pp. 169-176, 1986.

23. Shapiro, A., ``On Differentiability of metric projections in R^n, 1: Boundary Case", Proceedings of the American Mathematical Society, vol. 99, pp. 123-128, 1987.

24. Shapiro, A., ``A Conjecture Related to Chi-Bar-Squared Distributions", The American cMathematical Monthly, vol. 94, pp. 46-48, 1987.

25. Shapiro, A., ``Robustness Properties of the MDF Analysis of Moment Structures", South African Statistical Journal, vol. 21, pp. 39-62, 1987.

26. Shapiro, A. and Browne, M.W., ``Analysis of Covariance Structures Under Elliptical Distributions," Journal of the American Statistical Association, Vol. 82, pp. 1092-1097, 1987.

27. Browne, M.W. and Shapiro, A., ``Adjustments for Kurtosis in Factor Analysis with Elliptically Distributed Errors", Journal Royal Statistical Society, Series B, vol. 49, pp. 346-352, 1987.

28. Kano, Y. and Shapiro, A., ``On Asymptotic Variance of Uniqueness Estimators in Factor Analysis", South African Statistical Journal, vol. 21, pp. 131-136, 1987.

29. Shapiro, A., ``Directional Differentiability of Metric Projections Onto Moving Sets at Boundary Points", Journal of Mathematical Analysis and Applications, vol. 131, pp. 392-403, 1988.

30. Shapiro, A., ``Towards a Unified Theory of Inequality Constrained Testing in Multivariate Analysis," International Statistical Review, vol. 56, pp. 49-62, 1988.

31. Shapiro, A., ``Sensitivity Analysis of Nonlinear Programs and Differentiability Properties of Metric Projections", SIAM J. Control and Optimization, vol. 26, pp. 628-645, 1988.

32. Browne, M.W. and Shapiro, A., ``Robustness of Normal Theory Methods in the Analysis of Linear Latent Variate Models", British Journal of Mathematical and Statistical Psychology, vol. 41, pp. 193-208, 1988.

33. Botha, J.D., Shapiro, A. and Steiger, J.H., ``Uniform Indices of Fit for Factor Analysis Models", Multivariate Behavioral Research, vol. 23, pp. 443-450, 1988.

34. Shapiro, A., ``Perturbation Theory of Nonlinear Programs When the Set of Optimal Solutions is Not a Singleton", Applied Mathematics & Optimization. An International Journal, vol. 18, pp. 215-229, 1988.

35. Shapiro, A. and Botha, J.D., ``Dual Algorithms for Orthogonal Procrustes Rotations", SIAM Journal on Matrix Analysis and Applications, vol. 9, pp. 378-383, 1988.

36. Shapiro, A. and Browne, M.W., ``On the Asymptotic Bias of Estimators Under Parameter Drift", Statistics and Probability Letters, vol. 7, pp. 221-224, 1988.

37. Shapiro, A., ``Asymptotic Properties of Statistical Estimators in Stochastic Programming," The Annals of Statistics, Vol. 17, pp. 841-858, 1989.

38. Shapiro, A., ``On Differential Stability in Stochastic Programming", Mathematical Programming, vol. 47, pp. 107-116, 1990.

39. Shapiro, A. and Browne, M.W., ``On the Treatment of Correlation Structures as Covariance Structures", Linear Algebra and Applications, vol. 127, pp. 567-587, 1990.

40. Shapiro, A., ``On Concepts of Directional Differentiability", Journal of Optimization Theory and Applications, vol. 66, pp. 447-487, 1990.

41. Shapiro, A. and Cohen, A., ``Testing and Estimation of Equal Variances for Correlated Variables", Statistics and Probability Letters, vol. 10, pp. 231-234, 1990.

42. Rubinstein, R.Y. and Shapiro, A., ``Optimization of Static Simulation Models by the Score Function Method", Mathematics and Computers in Simulation, vol. 32, pp. 373-392, 1990.

43. Shapiro, A. and Botha, J.D., ``Variogram Fitting with a General Class of Conditionally Nonnegative Definite Functions", Computational Statistics & Data Analysis, vol. 11, pp. 87-96, 1991.

44. Shapiro, A., ``Upper Bounds for Nearly Optimal Diagonal Scaling of Matrices", Linear and Multilinear Algebra, vol. 29, pp. 145-147, 1991.

45. Shapiro, A., ``Asymptotic Analysis of Stochastic Programs", Annals of Operations Research, vol. 30, pp. 169-186, 1991.

46. Browne, M.W. and Shapiro, A., ``Invariance of Covariance Structures under Groups of Transformations", Metrika, vol. 38, pp. 345-355, 1991.

47. Shapiro, A. and Delport, J.L., ``Statistical Analysis of Jointed Rock Data", International Journal of Rock Mechanics and Mining Sciences, vol. 28, pp. 375-382, 1991.

48. Shapiro, A., ``Perturbation Analysis of Optimization Problems in Banach Spaces", Numerical Functional Analysis and Optimization, vol. 13, pp. 97-116, 1992.

49. Shapiro, A. and Bonnans, J.F., ``Sensitivity Analysis of Parametrized Programs under Cone Constraints", SIAM J. Control and Optimization, vol. 30, pp. 1409-1422, 1992.

50. Shapiro, A., Botha, J.D., Pastore, A. and Lesk, A., ``A Method for Multiple Superposition of Structures", Acta Crystallographica, vol. A48, pp. 11-14, 1992.

51. Bonnans, J.F., Ioffe, A.D. and Shapiro, A., ``Developpement de Solutions Exactes et Approchees en Programmation Non Linearire", C.R. Acad. Sc. Paris, vol. 315, Serie I, pp. 119-123, 1992.

52. Shapiro, A. and Al-Khayyal, F., ``First-Order Conditions for Isolated Locally Optimal Solutions", Journal of Optimization Theory and Applications, vol. 77, pp. 189-196, 1993.

53. Shapiro, A., ``Asymptotic Behavior of Optimal Solutions in Stochastic Programming," Mathematics of Operations Research, vol. 18, pp. 829-845, 1993.

54. Shapiro, A., ``Existence and Differentiability of Metric Projections in Hilbert Spaces", SIAM Journal on Optimization, vol. 4, pp. 130-141, 1994.

55. Shapiro, A., ``Sensitivity Analysis of Parametrized Programs via Generalized Equations", SIAM J. Control and Optimization, vol. 32, pp. 553-571, 1994.

56. Shapiro, A. and Wardi, Y., ``Nondifferentiability of the Steady-State Function in Discrete Event Dynamic Systems", IEEE Transactions on Automatic Control, vol. 39, pp. 1707-1711, 1994.

57. Shapiro, A., ``On Lipschitzian Stability of Optimal Solutions of Parametrized Semi-Infinite Programs", Mathematics of Operations Research, vol. 19, pp. 743-752, 1994.

58. Shapiro, A., ``Quantitative Stability in Stochastic Programming", Mathematical Programming, Series A, vol. 67, pp. 99-108, 1994.

59. Shapiro, A., ``Directionally Nondifferentiable Metric Projection", Journal of Optimization Theory and Applications, vol. 81, pp. 203-204, 1994.

60. Shapiro, A. and Fan, M.K.H., ``On Eigenvalue Optimization", SIAM J. Optimization, 5, pp. 552-569, 1995.

61. Shapiro, A., ``Directional Differentiability of the Optimal value Function in Convex Semi-Infinite Programming", Mathematical Programming, Series A, 70, pp. 149-157, 1995.

62. Shapiro, A. and Wardi, Y., ``Convergence Analysis of Stochastic Algorithms", Mathematics of Operations Research, 21, pp. 615-628, 1996.

63. Shapiro, A. and Wardi, Y., ``Convergence Analysis of Gradient Descent Stochastic Algorithms", Journal of Optimization Theory and Applications, 91, pp. 439-454, 1996.

64. Rehman, S.U., and Shapiro, A., ``An integral transform approach to cross-variograms modeling", Computational Statistics and Data Analysis, 22, pp. 213-233, 1996.

65. Shapiro, A., ``Simulation-based optimization - convergence analysis and statistical inference", Commun. Statist.-Stochastic Models, 12, pp. 425-454, 1996.

66. Shapiro, A., ``On Uniqueness of Lagrange Multipliers in Optimization Problems Subject to Cone Constraints", SIAM Journal on Optimization, vol. 7, pp. 508-518, 1997.

67. Shapiro, A. ``First and second order analysis of nonlinear semidefinite programs", Mathematical Programming, Series B, vol. 77, pp. 301-320, 1997.

68. Shapiro, A. and Homem-de-Mello, T., ``A Simulation-Based Approach to Stochastic Programming with Recourse", Mathematical Programming, vol. 81, pp. 301-325, 1998.

69. Bonnans, J.F. and Shapiro, A., ``Optimization problems with perturbations, A guided tour", SIAM Review, vol. 40, pp. 228-264, 1998.

70. Bonnans, J.F. and Shapiro, A., ``Nondegeneracy and quantitative stability of parameterized optimization problems with multiple solutions", SIAM J. Optimization, vol. 8, pp. 940-946, 1998.

71. Bonnans, J.F., Cominetti, R. and Shapiro, A., ``Sensitivity analysis of optimization problems under second order regular constraints", Mathematics of Operations Research, vol. 23, pp. 806-831, 1998.

72. Ten Berge, J.M.F., Krijnen, W.P., Wansbeck, T. and Shapiro, A., ``Some New Results on Correlation Preserving Factor Scores Prediction Methods", Linear Algebra and Its Applications, vol. 289, pp. 311-318, 1999.

73. Bonnans, J.F., Cominetti, R. and Shapiro, A., ``Second order optimality conditions based on parabolic second order tangent sets", SIAM J. Optimization, vol. 9, pp. 466-492, 1999.

74. Homem-de-Mello, T., Shapiro, A. and Spearman, M.L., ``Finding optimal material release times using simulation based optimization", Management Science, vol. 45, pp. 86-102, 1999.

75. Ruckmann, J.J. and Shapiro, A., ``On first order optimality conditions in generalized semi-infinite programming", Journal Optimization Theory and Applications, vol. 101, pp. 677-691, 1999.

76. Shapiro, A. and Ten Berge, J.M.F.,  ``The asymptotic bias of Minimum Trace Factor Analysis, with applications to the greatest lower bound to reliability",   Psychometrika, vol. 65,  pp. 413-425, 2000.

77. Shapiro, A., "On the asymptotics of constrained local M-estimators", The Annals of Statistics, vol. 28,  pp. 948-960, 2000.

78. Shapiro, A. and   Homem-de-Mello, T., "On rate of convergence of Monte Carlo approximations of stochastic programs",
SIAM J. Optimization, vol. 11, pp. 70-86, 2000.

79. Ruckmann, J.J. and Shapiro, A., "Second order optimality conditions in generalized semi-infinite programming",  Set-Valued Analysis, vol. 9, pp. 169-186, 2001.

80. Kleywegt, A.J.,  Shapiro, A. and Homem-de-Mello, T.,  "The sample average approximation method for stochastic discrete optimization", SIAM J. Optimization, vol. 12, pp. 479-502, 2001.

81. Shapiro, A. and Ten Berge, J.M.F.,  "Statistical Inference of Minimum Rank  Factor Analysis", Psychometrika, vol. 67, pp. 79-94, 2002.

82. Doveh, E., Shapiro, A. and Feigin, P.D., "Testing of monotonicity in regression models", Journal of Statistical Planning and Inference,  vol. 107, pp. 289-306, 2002.

83. Shapiro, A. and Kleywegt, A.J.,  "Minimax analysis of stochastic problems",  Optimization Methods and Software, vol. 17, pp. 523-542, 2002.

84. Shapiro, A.,  Homem-de-Mello,T. and  Kim, J.C., "Conditioning of convex piecewise linear stochastic programs", Mathematical Programming, Ser. A, vol. 94, pp. 1-19, 2002.

85. Verweij, B.,  Ahmed, S.,  Kleywegt, A.J., Nemhauser, G. and Shapiro, A.,  "The sample average approximation method applied to stochastic routing problems: a computational study",  Computational Optimization and Applications, vol. 24, pp. 289-333, 2003.

86. Shapiro, A.,  "Sensitivity analysis of generalized equations", Journal of Mathematical Sciences, vol. 115, pp. 2554-2565, 2003.

87. Shapiro, A.,  "Inference of statistical bounds for multistage stochastic programming problems",Mathematical Methods of Operations Research, vol. 58, pp. 57-68, 2003.

88. Shapiro, A.,  "On a class of nonsmooth composite functions", Mathematics of Operations Research, vol. 28, pp. 677-692, 2003.

89. Shapiro, A.,  "Scheffe's method for constructing simultaneous confidence intervals subject to cone constraints", Statistics and Probability Letters, vol. 64, pp.403-406, 2003.

90. Shapiro, A. and Ahmed, S.,  "On a class of minimax stochastic programs", SIAM J. Optimization, vol. 14, pp. 1237-1249, 2004.

91. Shapiro, A. and Sun, Jie., "Some properties of the augmented Lagrangian in cone constrained optimization", Mathematics of Operations Research, vol. 29, pp. 479-491, 2004.

92. Santoso , T., Ahmed, S., Goetschalckx, M. and Shapiro, A.,  "A stochastic programming approach for supply chain network design under uncertainty", European Journal of Operational Research, vol. 167, pp. 96 -115, 2005.

93. Shapiro, A., "Sensitivity analysis of parameterized variational inequalities", Mathematics of Operations Research, vol. 30, pp. 109 - 126, 2005.

94. Qi., L., Shapiro, A. and  Ling, C., "Differentiability and semismoothness properties of integral functions and their applications", Mathematical Programming, Ser. A,   vol. 102, pp. 223 -248, 2005.

95. Shapiro, A, "On duality theory of convex semi-infinite programming", Optimization,  vol. 54, pp. 535 - 543, 2005.

96. Shapiro, A, "On complexity of multistage stochastic programs", Operations Research Letters,   vol. 34, pp. 1 - 8, 2006.

97. Linderoth, J., Shapiro, A., Wright, S.,  "The empirical behavior of sampling methods for stochastic programming",  Annals of Operations Research, vol. 142, pp. 215-241, 2006.

98. Shapiro, A., "Worst-case distribution analysis of stochastic programs", Mathematical Programming, Ser. B, vol. 107, pp. 91-96, 2006.

99. Shapiro, A., "Stochastic programming with equilibrium constraints", Journal Optimization Theory and Applications, vol. 128, pp. 223-243, 2006.

100. Ruszczynski, A. and Shapiro, A., "Optimization of convex risk functions",  Mathematics of Operations Research, vol. 31, pp. 433-452, 2006.

101. Ruszczynski, A. and Shapiro, A., "Conditional Risk Mappings",   Mathematics of Operations Research, vol. 31, pp. 544-561, 2006.

102. Qian, Z. and  Shapiro, A., "Simulation-based approach to estimation of latent variable models", Computational Statistics and Data Analysis,  vol. 51, pp. 1243-1259, 2006.

103. Blomvall, J. and Shapiro, A., "Solving Multistage Asset Investment Problems by Monte Carlo Based Optimization", Mathematical Programming, Series B, vol. 108, pp. 571-595, 2006.

104. Nemirovski, A. and Shapiro, A., "Convex approximations of chance constrained programs",
SIAM J. Optimization, vol. 17, pp. 969-996, 2006.

105. Shapiro, A. and Huifu Xu, "Uniform Laws of Large Numbers for Set-Valued Mappings and Subdifferentials of Random Functions",  Journal of Mathematical Analysis and Applications, vol. 325 , pp. 1390-1399, 2007.

106. Ahmed, S., Cakmak, U. and Shapiro, A.,"Coherent Risk Measures in Inventory Problems",
European Journal of Operational Research, vol. 182, pp. 226-238, 2007.

107. Trindade, A., Uryasev, S., Shapiro, A. and Zrazhevsky, G., "Financial prediction with constrained tail risk ",
Journal of Banking and Finance, vol. 31, pp.3524-3538, 2007.

108. Shapiro, A., "Stochastic Programming Approach to Optimization under Uncertainty",
Mathematical Programming, Series B, vol. 112, pp. 183-220, 2008.

109. Shapiro, A., "Asymptotics of minimax stochastic programs", Statistics and Probability Letters, vol. 78, pp. 150-157, 2008.

110. Shapiro, A. and Huifu Xu, "Stochastic Mathematical Programs with Equilibrium Constraints, Modeling and Sample Average Approximation",  Optimization, vol. 57, pp. 395-418, 2008.

111. Ruckmann, J.J. and  Shapiro, A., "Augmented Lagrangians in semi-infinite programming", Mathematical Programming, Series B,  vol. 116, pp. 499-512, 2009.

112. Shapiro, A., "Asymptotic normality of test statistics under alternative hypotheses", Journal of Multivariate Analysis, vol. 100, pp. 936-945, 2009.

113. Nemirovski, A., Juditsky, A., Lan, G. and  Shapiro, A., "Robust stochastic approximation approach to stochastic programming", SIAM J. Optimization, vol 19, pp. 1574 - 1609, 2009.

114. Shapiro, A., "Semi-infinite programming, duality, discretization and optimality conditions", Optimization, vol. 58, pp. 133-161, 2009..

115. Pagnoncelli, B., Ahmed, S. and Shapiro, A., "Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications ", Journal Optimization Theory and Applications, vol. 142, pp. 399-416, 2009.

116.Shapiro, A. " On a time consistency concept in risk averse multi-stage stochastic programming", Operations Research Letters, vol. 37, pp. 143-147, 2009.