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A primal-dual aggregation algorithm for minimizing conditional value-at-risk in linear programs

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Abstract

Recent years have seen growing interest in coherent risk measures, especially in Conditional Value-at-Risk (\(\mathrm {CVaR}\)). Since \(\mathrm {CVaR}\) is a convex function, it is suitable as an objective for optimization problems when we desire to minimize risk. In the case that the underlying distribution has discrete support, this problem can be formulated as a linear programming (LP) problem. Over more general distributions, recent techniques, such as the sample average approximation method, allow to approximate the solution by solving a series of sampled problems, although the latter approach may require a large number of samples when the risk measures concentrate on the tail of the underlying distributions. In this paper we propose an automatic primal-dual aggregation scheme to exactly solve these special structured LPs with a very large number of scenarios. The algorithm aggregates scenarios and constraints in order to solve a smaller problem, which is automatically disaggregated using the information of its dual variables. We compare this algorithm with other common approaches found in related literature, such as an improved formulation of the full problem, cut-generation schemes and other problem-specific approaches available in commercial software. Extensive computational experiments are performed on portfolio and general LP instances.

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References

  1. Avriel, M., Williams, A.: The value of information and stochastic programming. Oper. Res. 18(5), 947–954 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dantzig, G.B.: The diet problem. Interfaces 20(4), 43–47 (1990)

    Article  Google Scholar 

  3. Stigler, G.J.: The cost of subsistence. J. Farm Econ. 27(2), 303–314 (1945)

    Article  Google Scholar 

  4. Tintner, G.: Stochastic linear programming with applications to agricultural economics. In: Proceedings of the Second Symposium in Linear Programming, vol. 1, pp. 197–228. National Bureau of Standards Washington, DC (1955)

  5. Wets, R.J.B.: Programming under uncertainty: the equivalent convex program. SIAM J. Appl. Math. 14(1), 89–105 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  6. Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)

    MATH  Google Scholar 

  7. Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  8. Yaari, M.E.: Some remarks on measures of risk aversion and on their uses. J. Econ. Theory 1(3), 315–329 (1969). doi:10.1016/0022-0531(69)90036-2

    Article  MathSciNet  Google Scholar 

  9. Yaari, M.E.: The dual theory of choice under risk. Econometrica 55(1), 95–115 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Thinking coherently: generalised scenarios rather than var should be used when calculating regulatory capital. Risk 10, 68–71 (1997)

    Google Scholar 

  11. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rockafellar, R., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Banking Finance 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  13. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Google Scholar 

  14. Kusuoka, S.: On law invariant coherent risk measures. In: Advances in Mathematical Economics, pp. 83–95. Springer, Berlin (2001)

  15. Bertsimas, D., Brown, D.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57(6), 1483–1495 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. MPS-SIAM Series on Optimization. SIAM-Society for Industrial and Applied Mathematics, Philadelphia (2009)

  17. Kleywegt, A., Shapiro, A., Homem-de Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2), 479–502 (2002)

    Article  MathSciNet  Google Scholar 

  18. Linderoth, J., Shapiro, A., Wright, S.: The empirical behavior of sampling methods for stochastic programming. Ann. Oper. Res. 142(1), 215–241 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lim, A.E., Shanthikumar, J.G., Vahn, G.Y.: Conditional value-at-risk in portfolio optimization: coherent but fragile. Oper. Res. Lett. 39(3), 163–171 (2011). doi:10.1016/j.orl.2011.03.004

    Article  MATH  MathSciNet  Google Scholar 

  20. Lim, C., Sherali, H.D., Uryasev, S.: Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Comput. Optim. Appl. 46(3), 391–415 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ogryczak, W., Śliwiński, T.: On solving the dual for portfolio selection by optimizing conditional value at risk. Comput. Optim. Appl. 50, 591–595 (2011). doi:10.1007/s10589-010-9321-y

    Article  MATH  MathSciNet  Google Scholar 

  22. Künzi-Bay, A., Mayer, J.: Computational aspects of minimizing conditional value-at-risk. Comput. Manag. Sci. 3(1), 3–27 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Van Slyke, R.M., Wets, R.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17(4), 638–663 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zabarankin, M., Uryasev, S.: Statistical Decision Problems, Selected Concepts and Portfolio Safeguard Case Studies, Springer Optimization and Its Applications, vol. 85. Springer, New York (2014)

  25. Ogryczak, W., Śliwiński, T.: On dual approaches to efficient optimization of lp computable risk measures for portfolio selection. Asia-Pac. J. Oper. Res. 28(01), 41–63 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gay, D.M.: Electronic mail distribution of linear programming test problems. Math. Program. Soc, COAL Bull. 13, 10–12 (1985). http://www.netlib.org/lp/. Accessed 23 Aug 2014

  27. Bixby, R.E.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50, 3–15 (2002). doi:10.1287/opre.50.1.3.17780

    Article  MATH  MathSciNet  Google Scholar 

  28. Dantzig, G.B.: Linear Programming and Extensions. Princeton landmarks in mathematics and physics. Princeton University Press, Princeton (1963)

  29. Todd, M.J.: The many facets of linear programming. Math. Program. 91(3), 417–436 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Gu, Z.: Private Communication (2013)

  31. Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W., Espinoza, D.G., Goycoolea, M., Helsgaun, K.: Certification of an optimal tsp tour through 85,900 cities. Oper. Res. Lett. 37(1), 11–15 (2009). doi:10.1016/j.orl.2008.09.006

    Article  MATH  MathSciNet  Google Scholar 

  32. Kiwiel, K.C.: Methods of descent for nondifferentiable optimization. Lecture Notes in Mathematics, vol. 1133. Springer-Verlag, Berlin (1985)

  33. Portfolio Safeguard. version 2.1. http://www.aorda.com/aod/psg.action (2009). Accessed 23 Aug 2014

  34. Bienstock, D., Zuckerberg, M.: Solving lp relaxations of large-scale precedence constrained problems. Integer Programming and Combinatorial Optimization 6080, 1–14 (2010)

    Article  MathSciNet  Google Scholar 

  35. Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5(2), 266–288 (1957)

    Article  MathSciNet  Google Scholar 

  36. Gassmann, H.I.: Mslip: a computer code for the multistage stochastic linear programming problem. Math. Program. 47(1–3), 407–423 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wets, R.: Stochastic programming: Solution techniques and approximation schemes. Springer, Berlin (1983)

  38. Young, M.R.: A minimax portfolio selection rule with linear programming solution. Manag. Sci. 44(5), 673–683 (1998)

    Article  MATH  Google Scholar 

  39. Konno, H., Yamazaki, H.: Mean-absolute deviation portfolio optimization model and its applications to tokyo stock market. Manag. Sci. 37(5), 519–531 (1991)

    Article  Google Scholar 

  40. Fleming, P.J., Wallace, J.J.: How not to lie with statistics: the correct way to summarize benchmark results. Commun. ACM 29(3), 218–221 (1986)

    Article  Google Scholar 

  41. Vielma, J.P., Ahmed, S., Nemhauser, G.L.: A lifted linear programming branch-and-bound algorithm for mixed-integer conic quadratic programs. INFORMS J. Comput. 20(3), 438–450 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  42. Ledoit, O., Wolf, M.: Honey, I shrunk the sample covariance matrix. UPF Economics and Business Working Paper 691 (2003)

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Acknowledgments

Daniel Espinoza was partially funded by the FONDECYT Grant 1110024 and Millenium Nucleus Information and Coordination in Networks ICM/FIC P10-024F. Eduardo Moreno acknowledges the financial support of the FONDECYT Grant 1130681 and Basal Center CMM-UCh. Both authors acknowledge Stan Uryasev for providing AORDA software for benchmarking purposes.

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Authors and Affiliations

  1. Department of Industrial Engineering, Universidad de Chile, Santiago, Chile

    Daniel Espinoza

  2. Faculty of Engineering and Science, Universidad Adolfo Ibañez, Santiago, Chile

    Eduardo Moreno

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  1. Daniel Espinoza
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  2. Eduardo Moreno
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Correspondence to Eduardo Moreno.

Appendix: Detailed tables of results

Appendix: Detailed tables of results

See Tables 3 and 4.

Table 3 Results of all algorithms on running Netlib instances
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Table 4 Results of PDAgg algorithm on Netlib and portfolio instances with 100,000 scenarios
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Espinoza, D., Moreno, E. A primal-dual aggregation algorithm for minimizing conditional value-at-risk in linear programs. Comput Optim Appl 59, 617–638 (2014). https://doi.org/10.1007/s10589-014-9692-6

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