Advertisement

Distributionally robust optimization with polynomial densities: theory, models and algorithms

  • Etienne de Klerk
  • Daniel KuhnEmail author
  • Krzysztof Postek
Full Length Paper Series B
First Online:
  • 205 Downloads

Abstract

In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage. We thus introduce a new class of ambiguity sets that contain only distributions with sum-of-squares (SOS) polynomial density functions of known degrees. We show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher-order moments, conditional probabilities, conditional moments or marginal distributions. Exploiting the theoretical properties of a measure-based hierarchy for polynomial optimization due to Lasserre (SIAM J Optim 21(3):864–885, 2011), we prove that certain worst-case expectation constraints are polynomial-time solvable under these new ambiguity sets. We also show how SOS densities can be used to approximately solve the general problem of moments. We showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company.

Keywords

Distributionally robust optimization Semidefinite programming Sum-of-squares polynomials Generalized eigenvalue problem 

Mathematics Subject Classification

90C22 90C26 90C15 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

Etienne de Klerk would like to thank Dorota Kurowicka and Jean Bernard Lasserre for valuable discussions and references. Daniel Kuhn gratefully acknowledges financial support from the Swiss National Science Foundation under grant BSCGI0_157733.

References

  1. 1.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bertsimas, D., Popescu, I.: On the relation between option and stock prices: a convex optimization approach. Oper. Res. 50(2), 358–374 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Belmont (2009)zbMATHGoogle Scholar
  5. 5.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)zbMATHGoogle Scholar
  6. 6.
    Cont, R.: Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223–236 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    den Ben-Tal, A., den Hertog, D., de Waegenaere, A., Melenberg, B., Rennen, G.: Robust solutions of optimization problems affected by uncertain probabilities. Manag. Sci. 59(2), 341–357 (2013)CrossRefGoogle Scholar
  8. 8.
    De Klerk, E., Laurent, M.: Comparison of Lasserre’s measure-based bounds for polynomial optimization to bounds obtained by simulated annealing. Math. Oper. Res. 43(4), 1317–1325 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    De Klerk, E., Laurent, M.: Worst-case examples for Lasserre’s measure–based hierarchy for polynomial optimization on the hypercube (2018). Preprint available at arXiv:1804.05524
  10. 10.
    De Klerk, E., Laurent, M., Sun, Z.: Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization. Math. Program. Ser. A 162(1), 363–392 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci. 1(3–4), 197–206 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Doan, X.V., Li, X., Natarajan, K.: Robustness to dependency in portfolio optimization using overlapping marginals. Oper. Res. 63(6), 1468–1488 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Doan, X.V., Natarajan, K.: On the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems. Oper. Res. 60(1), 138–49 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fiala, J., Kočvara, M., Michael Stingl, M.: PENLAB: a MATLAB solver for nonlinear semidefinite optimization. http://arxiv.org/pdf/1311.5240v1.pdf (2013)
  16. 16.
    Genz, A., Cools, R.: An adaptive numerical cubature algorithm for simplices. ACM Trans. Math. Softw. 29(3), 297–308 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4), 902–917 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  19. 19.
    Grant, M., Boyd, S.: CVX: Matlab Software for Disciplined Convex Programming. http://cvxr.com/cvx (2014)
  20. 20.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)zbMATHCrossRefGoogle Scholar
  21. 21.
    Grundmann, A., Moeller, H.M.: Invariant integration formulas for the \(n\)-simplex by combinatorial methods. SIAM J. Numer. Anal. 15, 282–290 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math. Program. Ser. B 151(1), 35–62 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hanasusanto, G.A., Roitch, V., Kuhn, D., Wiesemann, W.: Ambiguous joint chance constraints under mean and dispersion information. Oper. Res. 65(3), 751–767 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hanasusanto, G.A., Kuhn, D., Wallace, S.W., Zymler, S.: Distributionally robust multi-item newsvendor problems with multimodal demand distributions. Math. Program. Ser. A 152(1), 1–32 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kroo, A., Szilárd, R.: On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies. J. Approx. Theory 99(1), 134–152 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lasserre, J.B., Zeron, E.S.: Solving a class of multivariate integration problems via Laplace techniques. Appl. Math. 28(4), 391–405 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lasserre, J.B.: A semidefinite programming approach to the generalized problem of moments. Math. Program. Ser. B 112, 65–92 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Lasserre, J.B.: A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21(3), 864–885 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Lasserre, J.B.: The \({\mathbf{K}}\)-moment problem for continuous linear functionals. Trans. Am. Math. Soc. 365(5), 2489–2504 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lasserre, J.B., Weisser, T.: Representation of distributionally robust chance-constraints (2018). Preprint available at arXiv:1803.11500
  31. 31.
    Li, B., Jiang, R., Mathieu, J.L.: Ambiguous risk constraints with moment and unimodality information. Mathematical Programming Series A (to appear) (2017). Preprint available at http://www.optimization-online.org/DB_FILE/2016/09/5635.pdf
  32. 32.
    Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference (2004)Google Scholar
  33. 33.
    McNeil, A., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton (2015)zbMATHGoogle Scholar
  34. 34.
    Marichal, J.-L., Mossinghof, M.J.: Slices, slabs, and sections of the unit hypercube. Online J. Anal. Comb. 3, 1–11 (2008)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mevissen, M., Ragnoli, E., Yu, J.Y.: Data-driven distributionally robust polynomial optimization. In: Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 26, pp. 37–45. Curran Associates, Inc. (2013)Google Scholar
  36. 36.
    Mohajerin Esfahani, P., Kuhn, D.: Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Math. Program. Ser. A 171(1–2), 115–166 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Natarajan, K., Pachamanova, D., Sim, M.: Constructing risk measures from uncertainty sets. Oper. Res. 57(5), 1129–1141 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Pflug, G.C., Pichler, A., Wozabal, D.: The \(1/N\) investment strategy is optimal under high model ambiguity. J. Bank. Finance 36(2), 410–417 (2012)CrossRefGoogle Scholar
  39. 39.
    Pflug, G.C., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7, 435–442 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Popescu, I.: A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Math. Oper. Res. 30(3), 632–657 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Prékopa, A.: Stochastic Programming. Kluwer Academic Publishers, Berlin (1995)zbMATHCrossRefGoogle Scholar
  42. 42.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHCrossRefGoogle Scholar
  43. 43.
    Rogosinski, W.W.: Moments of non-negative mass. Proc. R. Soc. A 245, 1–27 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Scarf, H.: A min–max solution of an inventory problem. In: Scarf, H., Arrow, K., Karlin, S. (eds.) Studies in the Mathematical Theory of Inventory and Production, vol. 10, pp. 201–209. Stanford University Press, Redwood City (1958)Google Scholar
  45. 45.
    Sklar, A.: Fonctions de répartition à \(n\) dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris 8, 229–231 (1959)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.Á., López, M.A. (eds.) Semi-Infinite Programming: Recent Advances, pp. 135–165. Springer, New York (2001)CrossRefGoogle Scholar
  47. 47.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)zbMATHCrossRefGoogle Scholar
  48. 48.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. In: Optimization Methods and Software, pp. 11–12, 625–653 (1999)Google Scholar
  49. 49.
    Van Parys, B.P.G., Goulart, P.J., Embrechts, P.: Fréchet inequalities via convex optimization (2016). Preprint available at http://www.optimization-online.org/DB_FILE/2016/07/5536.pdf
  50. 50.
    Van Parys, B.P.G., Goulart, P.J., Kuhn, D.: Generalized Gauss inequalities via semidefinite programming. Math. Program. Ser. A 156(1–2), 271–302 (2016)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Žáčková, J.: On minimax solutions of stochastic linear programming problems. Časopis pro pěstování matematiky 91, 423–430 (1966)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Zuluaga, L., Peña, J.F.: A conic programming approach to generalized Tchebycheff inequalities. Math. Oper. Res. 30(2), 369–388 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Tilburg UniversityTilburgThe Netherlands
  2. 2.EPFLLausanneSwitzerland
  3. 3.Erasmus University RotterdamRotterdamThe Netherlands

Personalised recommendations