Mathematical Programming

, Volume 102, Issue 1, pp 25–46 | Cite as

Uncertain convex programs: randomized solutions and confidence levels

  • Giuseppe CalafioreEmail author
  • M.C. Campi


Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance’ parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations.

In this paper, we consider an alternative ‘randomized’ or ‘scenario’ approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.


Confidence Level Problem Formulation Engineering Problem Main Approach Uncertain Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apkarian, P., Tuan, H.D.: Parameterized LMIs in control theory. SIAM J. Control Optim 38 (4), 1241–1264 (2000)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Barmish, B.R., Sherbakov, P.: On avoiding vertexization of robustness problems: the approximate feasibility concept. IEEE Trans. Aut. Control 47, 819–824 (2002)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim 7 (4), 991–1016 (1997)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23 (4), 769–805 (1998)MathSciNetGoogle Scholar
  5. 5.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25 (1), 1–13 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ben-Tal, A., Nemirovski, A.: On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM J. Optim. 12 (3), 811–833 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Calafiore, G., Campi, M.C., El Ghaoui, L.: Identification of reliable predictor models for unknown systems: a data-consistency approach based on learning theory. In: Proceedings of IFAC 2002 World Congress, Barcelona, Spain, 2002Google Scholar
  8. 8.
    Calafiore, G., Dabbene, F.: A probabilistic framework for problems with real structured uncertainty in systems and control. Automatica 38 (8), 1265–1276 (2002)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Calafiore, G., Polyak, B.: Stochastic algorithms for exact and approximate feasibility of robust LMIs. IEEE Trans. Aut. Control 46 (11), 1755–1759, November 2001CrossRefMathSciNetGoogle Scholar
  10. 10.
    Charnes, A., Cooper, W.W.: Chance constrained programming. Manag. Sci. 6, 73–79 (1959)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dantzig, G.B.: Linear programming under uncertainty. Manage. Sci. 1, 197–206 (1955)zbMATHMathSciNetGoogle Scholar
  12. 12.
    de Farias, D.P., Van Roy, B.: On constraint sampling in the linear programming approach to approximate dynamic programming. Technical Report Dept. Management Sci. Stanford University, 2001Google Scholar
  13. 13.
    El Ghaoui, L., Calafiore, G.: Robust filtering for discrete-time systems with bounded noise and parametric uncertainty IEEE Trans. Aut. Control 46 (7), 1084–1089, July 2001CrossRefGoogle Scholar
  14. 14.
    El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18 (4), 1035–1064 (1997)CrossRefGoogle Scholar
  15. 15.
    El Ghaoui, L., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9 (1), 33–52 (1998)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Goberna, M.A., Lopez, M.A.: Linear Semi-Infinite Optimization. Wiley, 1998Google Scholar
  17. 17.
    Goemans, M.X., Williamson, D.P.: .878-approximation for MAX CUT and MAX 2SAT. Proc. 26th ACM Sym. Theor. Computing, 1994, pp. 422–431Google Scholar
  18. 18.
    Guestrin, C., Koller, D., Parr, R.: Efficient solution algorithms for factored MDPs. To appear in J. Artificial Intelligence Research, submitted 2002Google Scholar
  19. 19.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Statistical Association 58, 13–30 (1963)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Kelley, J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8 (4), 703–712 (1961)MathSciNetGoogle Scholar
  21. 21.
    Krishnan, K.: Linear programming (LP) approaches to semidefinite programming (SDP) problems. Rensselaer Polytechnic Institute, Ph.D. Thesis, Troy, New York, 2001Google Scholar
  22. 22.
    Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra and its Applications 284, 193–228 November 1998Google Scholar
  23. 23.
    Luebbecke, M.E., Desrosiers, J.: Selected topics in column generation. Les Cahiers de GERAD G-2002-64, 2002, pp. 193–228Google Scholar
  24. 24.
    Morrison, J.R., Kumar, P.R.: New linear program performance bound for queuing networks J. Optim. Theor. Appl. 100 (3), 575–597 (1999)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge, 1995Google Scholar
  26. 26.
    Mulvey, J.M., Vanderbei, R.J., Zenios, S.A.: Robust optimization of large-scale systems. Oper. Res. 43, 264–281 (1995)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Prékopa, A.: Stochastic Programming. Kluwer, 1995Google Scholar
  28. 28.
    Schuurmans, D., Patrascu, R.: Direct value-approximation for factored MDPs. Adv. Neural Infor. Processing (NIPS), 2001Google Scholar
  29. 29.
    Shapiro, A., Homem-de-Mello, T.: On rate of convergence of Monte Carlo approximations of stochastic programs. SIAM J. Optim. 11, 70–86 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Still, G.: Discretization in semi-infinite programming: Rate of convergence. Math. Program. Ser. A 91, 53–69 (2001)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Todd, M.J.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38 (1), 49–95, March 1996MathSciNetGoogle Scholar
  33. 33.
    Vapnik, V.N., Ya.Chervonenkis, A.: On the uniform convergence of relative frequencies to their probabilities. Theor. Probab. Appl. 16 (2), 264–280 (1971)MathSciNetGoogle Scholar
  34. 34.
    Vidyasagar, M.: A theory of learning and generalization. Springer-Verlag, London, 1997Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Dipartimento di Automatica e InformaticaPolitecnico di TorinoTorinoItaly
  2. 2.Dipartimento di Automatica per l’AutomazioneUniversità di BresciaBresciaItaly

Personalised recommendations