Global Optimization of the Scenario Generation and Portfolio Selection Problems

  • Panos Parpas
  • Berç Rustem
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3982)


We consider the global optimization of two problems arising from financial applications. The first problem originates from the portfolio selection problem when high-order moments are taken into account. The second issue we address is the problem of scenario generation. Both problems are non-convex, large-scale, and highly relevant in financial engineering. For the two problems we consider, we apply a new stochastic global optimization algorithm that has been developed specifically for this class of problems. The algorithm is an extension to the constrained case of the so called diffusion algorithm. We discuss how a financial planning model (of realistic size) can be solved to global optimality using a stochastic algorithm. Initial numerical results are given that show the feasibility of the proposed approach.


Global Optimization Stochastic Programming Portfolio Selection Moment Problem Scenario Tree 
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.


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  1. 1.
    Markowitz, H.M.: Portfolio selection. J. Finance 7, 77–91 (1952)CrossRefGoogle Scholar
  2. 2.
    Markowitz, H.M.: The utility of wealth. J. Polit. Econom., 151–158 (1952)Google Scholar
  3. 3.
    Dupacova, J., Consigli, G., Wallace, S.: Scenarios for multistage stochastic programs. Ann. Oper. Res. 100, 25–53 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gülpınar, N., Rustem, B., Settergren, R.: Simulation and optimization approaches to scenario tree generation. J. Econom. Dynam. Control 28, 1291–1315 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Prékopa, A.: Stochastic programming. Mathematics and its Applications, vol. 324. Kluwer Academic Publishers Group, Dordrecht (1995)Google Scholar
  6. 6.
    Parpas, P., Rustem, B., Pistikopoulos, E.N.: Linearly constrained global optimization and stochastic differential equations. Accepted in the J. Global Optim. (2006)Google Scholar
  7. 7.
    Aluffi-Pentini, F., Parisi, V., Zirilli, F.: Global optimization and stochastic differential equations. J. Optim. Theory Appl. 47, 1–16 (1985)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chiang, T., Hwang, C., Sheu, S.: Diffusion for global optimization in Rn. SIAM J. Control Optim. 25, 737–753 (1987)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Geman, S., Hwang, C.: Diffusions for global optimization. SIAM J. Control Optim. 24, 1031–1043 (1986)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gidas, B.: The Langevin equation as a global minimization algorithm. In: Disordered systems and biological organization (Les Houches, 1985). In: NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 20, pp. 321–326. Springer, Berlin (1986)Google Scholar
  11. 11.
    Konno, H.: Applications of global optimization to portfolio analysis. In: Charles, A., Pierr, H., Gilles, J. (eds.) Essays and Surveys in Global Optimization, pp. 195–210. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Maranas, C.D., Androulakis, I.P., Floudas, C.A., Berger, A.J., Mulvey, J.M.: Solving long-term financial planning problems via global optimization. J. Econom. Dynam. Control 21, 1405–1425 (1997); Computational financial modelingGoogle Scholar
  13. 13.
    Steinbach, M.C.: Markowitz revisited: mean-variance models in financial portfolio analysis. SIAM Rev. 43, 31–85 (2001) (electronic)Google Scholar
  14. 14.
    Parpas, P., Rustem, B.: Entropic regularization of the moment problem (2005) (submitted)Google Scholar
  15. 15.
    Bertsimas, D., Sethuraman, J.: Moment problems and semidefinite optimization. In: Handbook of semidefinite programming. Internat. Ser. Oper. Res. Management Sci., vol. 27, pp. 469–509. Kluwer Acad. Publ., Boston (2000)Google Scholar
  16. 16.
    Klaassen, P.: Discretized reality and spurious profits in stochastic programming models for asset/liability management. E.J of Op. Res. 101, 374–392 (1997)MATHCrossRefGoogle Scholar
  17. 17.
    Harrison, J., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408 (1979)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zirilli, F.: The use of ordinary differential equations in the solution of nonlinear systems of equations. In: Nonlinear optimization, 1981 (Cambridge, 1981). NATO Conf. Ser. II: Systems Sci., pp. 39–46. Academic Press, London (1982)Google Scholar
  19. 19.
    Recchioni, M.C., Scoccia, A.: A stochastic algorithm for constrained global optimization. J. Global Optim. 16, 257–270 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Panos Parpas
    • 1
  • Berç Rustem
    • 1
  1. 1.Department of ComputingImperial CollegeLondon

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