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Distributionally robust joint chance constraints with second-order moment information

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We develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove that this approximation is exact for robust individual chance constraints with concave or (not necessarily concave) quadratic constraint functions, and we demonstrate that the Worst-Case CVaR can be computed efficiently for these classes of constraint functions. Next, we study the Worst-Case CVaR approximation for joint chance constraints. This approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations. The tightness depends on a set of scaling parameters, which can be tuned via a sequential convex optimization algorithm. We show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the Worst-Case CVaR can be evaluated efficiently if the scaling parameters are kept constant. We evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations.

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  1. Alizadeh F., Goldfarb D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrieu L., Henrion R., Römisch W.: A model for dynamic chance constraints in hydro power reservoir management. Eur. J. Oper. Res. 207(2), 579–589 (2010)

    Article  MATH  Google Scholar 

  3. Ben-Tal A., El Ghaoui L., Nemirovski A.: Robust Optimization. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  4. Bertsimas D., Doan X., Natarajan K., Teo C.P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Calafiore G., El Ghaoui L.: Distributionally robust chance-constrained linear programs with applications. J. Optim. Theory Appl. 130(1), 1–22 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Calafiore G., Topcu U., El Ghaoui L.: Parameter estimation with expected and residual-at-risk criteria. Syst. Control Lett. 58(1), 39–46 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Charnes A., Cooper W.W., Symonds G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4(3), 235–263 (1958)

    Article  Google Scholar 

  9. Chen W., Sim M., Sun J., Teo C.P.: From CVaR to uncertainty set: implications in joint chance-constrained optimization. Oper. Res. 58(2), 470–485 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen X., Sim M., Sun P.: A robust optimization perspective on stochastic programming. Oper. Res 55(6), 1058–1071 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. (2010)

  12. Erdoǧan E., Iyengar G.: Ambiguous chance constrained problems and robust optimization. Math. Program. Series B 107, 37–61 (2006)

    Article  MathSciNet  Google Scholar 

  13. Goh J., Sim M.: Robust optimization made easy with ROME. Oper. Res 59(4), 973–985 (2010)

    Article  MathSciNet  Google Scholar 

  14. Isii K.: The extrema of probability determined by generalized moments (i) bounded random variables. Ann. Inst. Stat. Math. 12(2), 119–134 (1960)

    Article  MathSciNet  Google Scholar 

  15. Kuhn D., Wiesemann W., Georghiou A.: Primal and dual linear decision rules in stochastic and robust optimization. Math. Program 130(1), 177–209 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Löfberg, J.: YALMIP : a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei (2004)

  17. 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 

  18. Miller L.B., Wagner H.: Chance-constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)

    Article  MATH  Google Scholar 

  19. Natarajan K., Pachamanova D., Sim M.: Constructing risk measures from uncertainty sets. Oper. Res. 57(5), 1129–1141 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nemirovski A., Shapiro A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pólik I., Terlaky T.: A survey of the S-lemma. SIAM Rev. 49(3), 371–481 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Prékopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming, pp. 113–138. Princeton University Press, Princeton (1970)

  24. Rockafellar R.T., Uryasev S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2002)

    Google Scholar 

  25. Shapiro A.: On duality theory of conic linear problems. In: Goberna, M.A., Lopez, M.A. (eds) Semi-Infinite Programming: Recent Advances, Kluwer, Dordrecht (2001)

    Google Scholar 

  26. Shapiro A., Kleywegt A.J.: Minimax analysis of stochastic problems. Optim. Methods Softw 17(3), 523–542 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Toker, O., Ozbay, H.: On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: Proceedings of the American Control Conference, pp. 2525–2526. Seatle (1995)

  28. Vandenberghe L., Boyd S., Comanor K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49, 52–64 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zymler, S., Kuhn, D., Rustem, B.: Worst-case value-at-risk of non-linear portfolios. (2009)

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Correspondence to Steve Zymler.

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Zymler, S., Kuhn, D. & Rustem, B. Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137, 167–198 (2013).

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