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A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality

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Abstract

In this paper, we study the link between a Chance-Constrained optimization Problem (CCP) and its sample counterpart (SP). SP has a finite number, say N, of sampled constraints. Further, some of these sampled constraints, say k, are discarded, and the final solution is indicated by \(x^{\ast}_{N,k}\). Extending previous results on the feasibility of sample convex optimization programs, we establish the feasibility of \(x^{\ast}_{N,k}\) for the initial CCP problem.

Constraints removal allows one to improve the cost function at the price of a decreased feasibility. The cost improvement can be inspected directly from the optimization result, while the theory here developed permits to keep control on the other side of the coin, the feasibility of the obtained solution. In this way, trading feasibility for performance is put on solid mathematical grounds in this paper.

The feasibility result here obtained applies to a vast class of chance-constrained optimization problems, and has the distinctive feature that it holds true irrespective of the algorithm used to discard k constraints in the SP problem. For constraints discarding, one can thus, e.g., resort to one of the many methods introduced in the literature to solve chance-constrained problems with discrete distribution, or even use a greedy algorithm, which is computationally very low-demanding, and the feasibility result remains intact.

We further prove that, if constraints in the SP problem are optimally removed—i.e., one deletes those constraints leading to the largest possible cost improvement—, then a precise optimality link to the original chance-constrained problem CCP in addition holds.

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References

  1. Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4, 235–263 (1958)

    Article  Google Scholar 

  2. Miller, L.B., Wagner, H.: Chance-constrained programming with joint constraints. Oper. Res. 13, 930–945 (1965)

    Article  MATH  Google Scholar 

  3. Prèkopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming. Princeton University Press, Princeton (1970)

    Google Scholar 

  4. Prèkopa, A.: Contributions to the theory of stochastic programming. Math. Program. 4, 202–221 (1973)

    Article  MATH  Google Scholar 

  5. Prèkopa, A.: Stochastic Programming. Kluwer, Boston (1995)

    Google Scholar 

  6. Prèkopa, A.: Probabilistic programming. In: Ruszczyǹski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10. Elsevier, London (2003)

    Chapter  Google Scholar 

  7. Dentcheva, D.: Optimization models with probabilistic constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty. Springer, London (2006)

    Google Scholar 

  8. Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41, 263–276 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88, 411–424 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ben-Tal, A., Nemirovski, A.: Robust optimization—methodology and applications. Math. Program. Ser. B 92, 453–480 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henrion, R., Römisch, W.: Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints. Math. Program. 100, 589–611 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henrion, R., Römisch, W.: Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Math. Program. 84, 55–88 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Prèkopa, A.: Sharp bound on probabilities using linear programming. Oper. Res. 38, 227–239 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sen, S.: Relaxations for the probabilistically constrained programs with discrete random variables. Oper. Res. Lett. 11, 81–86 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dentcheva, D., Prèkopa, A., Ruszczyǹski, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dentcheva, D., Prèkopa, A., Ruszczyǹski, A.: On convex probabilistic programming with discrete distribution. Nonlinear Anal. 47, 1997–2009 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Beraldi, P., Ruszczyǹski, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Softw. 17, 359–382 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dentcheva, D., Lai, B., Ruszczyǹski, A.: Dual methods for probabilistic optimization problems. Math. Methods Oper. Res. 60, 331–346 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Prèkopa, A.: Dual method for a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution. Z. Oper. Res. 34, 441–461 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Ruszczyǹski, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. 93(2), 195–215 (2002)

    Article  MathSciNet  Google Scholar 

  23. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122, 247–272 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vogel, S.: A stochastic approach to stability in stochastic programming. J. Comput. Appl. Math. 56(1–2), 65–96 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tempo, R., Bai, E.W., Dabbene, F.: Probabilistic robustness analysis: explicit bounds for the minimum number of samples. Syst. Control Lett. 30, 237–242 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bai, E.W., Tempo, R., Fu, M.: Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis. Math. Control Signals Syst. 11, 183–196 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Calafiore, G., Dabbene, F., Tempo, R.: Randomized algorithms for probabilistic robustness with real and complex structured uncertainty. IEEE Trans. Autom. Control 45(12), 2218–2235 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shapiro, A.: Monte-Carlo sampling methods. In: Ruszczyǹski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10. Elsevier, London (2003)

    Chapter  Google Scholar 

  29. Pflug, G.C.: Stochastic optimization and statistical inference. In: Ruszczyǹski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10. Elsevier, London (2003)

    Chapter  Google Scholar 

  30. de Farias, D.P., Van Roy, B.: On constraints sampling in the linear programming approach to approximate dynamic programming. Math. Oper. Res. 29(3), 462–478 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. Ser. B 107(1–2), 37–61 (2006)

    Article  MATH  Google Scholar 

  32. Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty. Springer, London (2006)

    Google Scholar 

  33. Polyak, B.T., Scherbakov, P.S.: Randomized methods for solving semidefinite programming. In: Granichin, O.N. (ed.) Stochastic Optimization in Informatics, vol. 10, pp. 38–70. St.-Petersburg University Press, London (2006)

    Google Scholar 

  34. Shapiro, A., Dentcheva, D., Ruszczyǹski, A.: Lectures on Stochastic Programming—Modeling and Theory. MPS-SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  35. Alamo, T., Tempo, R., Camacho, E.F.: Statistical learning theory: a pack-based strategy for uncertain feasibility and optimization problems. In: Blondel, V.D., Boyd, S.P., Kimura, H. (eds.) Recent Advances in Learning and Control, vol. 10. Springer, London (2008)

    Google Scholar 

  36. Polyak, B.T., Tempo, R.: Probabilistic robust design with linear quadratic regulators. Syst. Control Lett. 43, 343–353 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Fujisaki, Y., Dabbene, F., Tempo, R.: Probabilistic robust design of LPV control systems. Automatica 39, 1323–1337 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ishii, H., Basar, T., Tempo, R.: Randomized algorithms for quadratic stability of quantized sampled-data systems. Automatica 40, 839–846 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  39. Alpcan, T., Basar, T., Tempo, R.: Randomized algorithms for stability and robustness of high-speed communication networks. IEEE Trans. Neural Netw. 16, 1229–1241 (2005)

    Article  Google Scholar 

  40. Tempo, R., Calafiore, G., Dabbene, F.: Randomized Algorithms for Analysis and Control of Uncertain Systems. Springer, London (2005)

    MATH  Google Scholar 

  41. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Alamo, T., Tempo, R., Camacho, E.F.: Randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems. IEEE Trans. Autom. Control 54, 2545–2559 (2009)

    Article  MathSciNet  Google Scholar 

  43. Calafiore, G., Campi, M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102(1), 25–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Calafiore, G., Campi, M.C.: The scenario approach to robust control design. IEEE Trans. Autom. Control 51(5), 742–753 (2006)

    Article  MathSciNet  Google Scholar 

  45. Campi, M.C., Garatti, S.: The exact feasibility of randomized solutions of uncertain convex programs. SIAM J. Optim. 19(3), 1211–1230 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Vidyasagar, M.: A Theory of Learning and Generalization: with Applications to Neural Networks and Control Systems. Springer, London (1997)

    MATH  Google Scholar 

  47. Shiryaev, A.N.: Probability. Springer, New York (1996)

    Google Scholar 

  48. Matoušek, J.: On geometric optimization with few violated constraints. Discrete Comput. Geom. 14, 365–384 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Informazione, Università di Brescia, via Branze 38, 25123, Brescia, Italy

    M. C. Campi

  2. Dipartimento di Elettronica e Informazione, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133, Milano, Italy

    S. Garatti

Authors
  1. M. C. Campi
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  2. S. Garatti
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Correspondence to M. C. Campi.

Additional information

Communicated by B.T. Polyak.

This paper was supported by the MIUR project “Identification and adaptive control of industrial systems”.

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Campi, M.C., Garatti, S. A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality. J Optim Theory Appl 148, 257–280 (2011). https://doi.org/10.1007/s10957-010-9754-6

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