Sample Average Approximation Method for Chance Constrained ...

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15. Markowitz, H.: Portfolio selection. J. Financ. 7:77–97 (1952)16. Wang, Y., Chen, Z., Zhang, K.: A chance-constrained portfolio selection problemunder t-distrbution. Asia Pac. J. Oper. Res. 24(4):535–556 (2007)17. Campi, M.C., Garatti, S.: The exact feasibility of randomized solutions of robustconvex programs. Optimization online (www.optimization-online.org) (2007)18. Calafiore, G., Campi, M.C.: The scenario approach to robust control design. IEEET. Automat. Contr. 51:742–753 (2006)19. Klein Haneveld, W.K., van der Vlerk, M.H.: Stochastic Programming (lecturenotes), (2007).20. Law, A., Kelton, W.D.: Simulation Modeling and Analysis, Industrial Engineeringand Management Science Series. McGraw-Hill Science/Engineering/Math (1999).34
1.001.000.950.950.900.900.850.850.800.998 1.004 1.0100.801.001.011.02Figure 1: Normal returns <strong>for</strong> γ = 0.Figure 2: Lognormal returns, γ = 0.1.001.000.950.950.900.900.850.850.800.998 1.004 1.0100.801.010 1.0201.030Figure 3: Normal returns, γ = 0.05.Figure 4: Lognormal returns, γ = 0.05.1.001.000.950.950.900.900.850.850.800.9951.000 1.005 1.0100.801.005 1.010 1.015 1.020 1.025 1.030Figure 5: Normal returns, γ = 0.10.Figure 6: Lognormal returns, γ = 0.10.35
Page 1 and 2: Sample Average Approximation Method
Page 4 and 5: distribution in chance constraint b
Page 6 and 7: Carathéodory function, i.e., G(x,
Page 8 and 9: (i) for any sequence x k converging
Page 10 and 11: y a version of the uniform law of l
Page 12 and 13: f(¯x) ≥ ϑ ∗ . Also f(ˆx N )
Page 14 and 15: us choose two positive integers M a
Page 16 and 17: Note that, of course, E [ r T x ] =
Page 18 and 19: problem (5), with γ = 0, to be inf
Page 20 and 21: estimation of the parameters was ta
Page 22 and 23: original problem, we observe in Fig
Page 24 and 25: 3.3 Upper BoundsA method to compute
Page 26 and 27: priate choice of the parameters sin
Page 28 and 29: It is possible to write the SAA of
Page 30 and 31: 5 ConclusionsWe have discussed chan
Page 32 and 33: References1. Charnes, A., Cooper, W
Page 36: 1.011.0051.00.99530 80 130 1800.10.

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