# Scenario Approximation of Robust and Chance-Constrained Programs

## Abstract

We consider scenario approximation of problems given by the optimization of a function over a constraint that is too difficult to be handled but can be efficiently approximated by a finite collection of constraints corresponding to alternative scenarios. The covered programs include min-max games, and semi-infinite, robust and chance-constrained programming problems. We prove convergence of the solutions of the approximated programs to the given ones, using mainly epigraphical convergence, a kind of variational convergence that has demonstrated to be a valuable tool in optimization problems.

## Keywords

Mathematical programming Epigraphical convergence Scenario approximation Sampling## Notes

### Acknowledgements

We are grateful to Christian Hess and Enrico Miglierina for useful comments and discussions, and to the anonymous referees, the Associate Editor Masao Fukushima and the Editor Franco Giannessi for valuable suggestions that helped improve the article substantially.

## References

- 1.Still, G.: Discretization in semi-infinite programming: The rate of convergence. Math. Program., Ser. A
**91**(1), 53–69 (2001) MathSciNetzbMATHGoogle Scholar - 2.Calafiore, G., Campi, M.C.: Robust convex programs: Randomized solutions and applications in control. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, December 2003 pp. 2423–2428 (2003) Google Scholar
- 3.Campi, M.C., Calafiore, G.: Decision making in an uncertain environment: The scenario-based optimization approach. In: Kárný, M., Kracík, J., Andrýsek, J. (eds.) Multiple participant decision making. International Series on Advanced Intelligence, vol. 9, pp. 99–111. Advanced Knowledge International (2004) Google Scholar
- 4.Calafiore, G., Campi, M.C.: Uncertain convex programs: Randomized solutions and confidence levels. Math. Program.
**102**, 25–46 (2005) MathSciNetzbMATHCrossRefGoogle Scholar - 5.Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty, pp. 3–48. Springer, Berlin (2006) CrossRefGoogle Scholar
- 6.Pucci de Farias, D., Van Roy, B.: On constraint sampling in the linear programming approach to approximate dynamic programming. Math. Oper. Res.
**29**(3), 462–478 (2004) MathSciNetzbMATHCrossRefGoogle Scholar - 7.Reemtsen, R.: Semi-infinite programming: Discretization methods. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 3417–3424. Springer, New York (2009) CrossRefGoogle Scholar
- 8.Reemtsen, R.: Discretization methods for the solution of semi-infinite programming problems. J. Optim. Theory Appl.
**71**(1), 85–103 (1991) MathSciNetzbMATHCrossRefGoogle Scholar - 9.Reemtsen, R.: Some outer approximation methods for semi-infinite optimization problems. J. Comput. Appl. Math.
**53**(1), 87–108 (1994) MathSciNetzbMATHCrossRefGoogle Scholar - 10.Combettes, P.: Strong convergence of block-iterative outer approximation methods for convex optimization. SIAM J. Control Optim.
**38**(2), 538–565 (2000) (electronic) MathSciNetzbMATHCrossRefGoogle Scholar - 11.Shapiro, A.: Monte Carlo sampling methods. In: Stochastic programming. Handbooks Oper. Res. Management Sci., vol. 10, pp. 353–425. Elsevier, Amsterdam (2003) CrossRefGoogle Scholar
- 12.Huber, P.J.: The 1972 Wald lecture. Robust statistics: A review. Ann. Math. Stat.
**43**, 1041–1067 (1972) zbMATHCrossRefGoogle Scholar - 13.Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992) zbMATHCrossRefGoogle Scholar
- 14.Polak, E.: On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Rev.
**29**(1), 21–89 (1987) MathSciNetCrossRefGoogle Scholar - 15.Hettich, R., Kortanek, K.O.: Semi-infinite programming: Theory, methods, and applications. SIAM Rev.
**35**(3), 380–429 (1993) MathSciNetzbMATHCrossRefGoogle Scholar - 16.Žaković, S., Rustem, B.: Semi-infinite programming and applications to minimax problems. Ann. Oper. Res.
**124**, 81–110 (2003) MathSciNetzbMATHCrossRefGoogle Scholar - 17.Hess, C., Seri, R., Choirat, C.: Approximations results for robust optimization. Working paper (2010) Google Scholar
- 18.Hiriart-Urruty, J.B.: Contributions à la programmation mathématique: cas déterministe et stochastique. Ph.D. thesis, Université de Clermont-Ferrand II, Clermont (1977) Google Scholar
- 19.Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Manag. Sci.
**4**, 183–195 (1958) CrossRefGoogle Scholar - 20.Charnes, A., Cooper, W.W.: Chance-constrained programming. Manag. Sci.
**6**, 73–79 (1959/1960) MathSciNetCrossRefGoogle Scholar - 21.Charnes, A., Cooper, W.W.: Chance constraints and normal deviates. J. Am. Stat. Assoc.
**57**, 134–148 (1962) MathSciNetzbMATHCrossRefGoogle Scholar - 22.Sengupta, J.K.: Stochastic linear programming with chance constraints. Int. Econ. Rev.
**11**, 101–116 (1970) zbMATHCrossRefGoogle Scholar - 23.Still, G.: Generalized semi-infinite programming: Theory, methods. Eur. J. Oper. Res.
**119**, 301–313 (1999) zbMATHCrossRefGoogle Scholar - 24.Still, G.: Generalized semi-infinite programming: Numerical aspects. Optimization
**49**(3), 223–242 (2001) MathSciNetzbMATHCrossRefGoogle Scholar - 25.Bai, D., Carpenter, T.J., Mulvey, J.M.: Making a case for robust models. Manag. Sci.
**43**, 895–907 (1997) zbMATHCrossRefGoogle Scholar - 26.Redaelli, G.: Convergence problems in stochastic programming models with probabilistic constraints. Riv. Mat. Sci. Econ. Soc.
**21**(1–2), 147–164 (1998) MathSciNetGoogle Scholar - 27.Pennanen, T., Koivu, M.: Epi-convergent discretizations of stochastic programs via integration quadratures. Numer. Math.
**100**, 141–163 (2005) MathSciNetzbMATHCrossRefGoogle Scholar - 28.Choirat, C., Hess, C., Seri, R.: A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Ann. Probab.
**31**(1), 63–92 (2003) MathSciNetzbMATHCrossRefGoogle Scholar - 29.Choirat, C., Hess, C., Seri, R.: Approximation of stochastic programming problems. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 45–60. Springer, Berlin (2006) CrossRefGoogle Scholar
- 30.Dal Maso, G.: An Introduction to
*Γ*-Convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser, Boston (1993) CrossRefGoogle Scholar - 31.Jagannathan, R.: Chance-constrained programming with joint constraints. Oper. Res.
**22**(2), 358–372 (1974) MathSciNetzbMATHCrossRefGoogle Scholar - 32.López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res.
**180**(2), 491–518 (2007) zbMATHCrossRefGoogle Scholar - 33.Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998) zbMATHCrossRefGoogle Scholar
- 34.Allen, F.M., Braswell, R.N., Rao, P.V.: Distribution-free approximations for chance constraints. Oper. Res.
**22**(3), 610–621 (1974) MathSciNetzbMATHCrossRefGoogle Scholar - 35.Gray, R.M., Kieffer, J.C.: Asymptotically mean stationary measures. Ann. Probab.
**8**(5), 962–973 (1980) MathSciNetzbMATHCrossRefGoogle Scholar - 36.Wets, R.J.B.: Stochastic programs with chance constraints: Generalized convexity and approximation issues. In: Generalized Convexity, Generalized Monotonicity: Recent Results, Luminy, 1996. Nonconvex Optim. Appl., vol. 27, pp. 61–74. Kluwer Academic, Dordrecht (1998) CrossRefGoogle Scholar
- 37.Pagnoncelli, B.K., Ahmed, S., Shapiro, A.: Sample average approximation method for chance constrained programming: Theory and applications. J. Optim. Theory Appl.
**142**(2), 399–416 (2009) MathSciNetzbMATHCrossRefGoogle Scholar - 38.Elker, J., Pollard, D., Stute, W.: Glivenko–Cantelli theorems for classes of convex sets. Adv. Appl. Probab.
**11**(4), 820–833 (1979) MathSciNetzbMATHCrossRefGoogle Scholar - 39.Steele, J.M.: Empirical discrepancies and subadditive processes. Ann. Probab.
**6**(1), 118–127 (1978) MathSciNetzbMATHCrossRefGoogle Scholar - 40.Shorack, G.R., Wellner, J.A.: Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986) zbMATHGoogle Scholar
- 41.Nobel, A.: A counterexample concerning uniform ergodic theorems for a class of functions. Stat. Probab. Lett.
**24**(2), 165–168 (1995) MathSciNetzbMATHCrossRefGoogle Scholar - 42.Henrion, R., Römisch, W.: Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Math. Program., Ser. A
**84**(1), 55–88 (1999) zbMATHGoogle Scholar - 43.Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim.
**19**(2), 674–699 (2008) MathSciNetzbMATHCrossRefGoogle Scholar - 44.Cheney, E.W.: Introduction to Approximation Theory. AMS Chelsea, Providence (1998). Reprint of the second (1982) edition zbMATHGoogle Scholar
- 45.Barrodale, I., Phillips, C.: Algorithm 495: Solution of an overdetermined system of linear equations in the Chebyshev norm. ACM Trans. Math. Softw.
**1**(3), 264–270 (1975) zbMATHCrossRefGoogle Scholar - 46.Devroye, L.: Laws of the iterated logarithm for order statistics of uniform spacings. Ann. Probab.
**9**(5), 860–867 (1981) MathSciNetzbMATHCrossRefGoogle Scholar - 47.Aliprantis, C.D., Border, K.C.: Infinite-Dimensional Analysis. Springer, Berlin (1999) zbMATHCrossRefGoogle Scholar
- 48.Beer, G., Rockafellar, R.T., Wets, R.J.B.: A characterization of epi-convergence in terms of convergence of level sets. Proc. Am. Math. Soc.
**116**(3), 753–761 (1992) MathSciNetzbMATHCrossRefGoogle Scholar - 49.Molchanov, I.S.: A limit theorem for solutions of inequalities. Scand. J. Stat.
**25**(1), 235–242 (1998) MathSciNetzbMATHCrossRefGoogle Scholar - 50.Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman, Boston (1984) zbMATHGoogle Scholar
- 51.Breiman, L.: Probability. Classics in Applied Mathematics, vol. 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992) zbMATHCrossRefGoogle Scholar