SDP reformulation for robust optimization problems based on nonconvex QP duality

  • Ryoichi Nishimura
  • Shunsuke HayashiEmail author
  • Masao Fukushima


In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with “single” (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand’s Lorentz positivity.


Robust optimization Second-order cone programming Semidefinite programming Nonconvex quadratic programming 


  1. 1.
    Adida, E., Perakis, G.: A robust optimization approach to dynamic pricing and inventory control with no backorders. Math. Program. 107, 97–129 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aghassi, M., Bertsimas, D.: Robust game theory. Math. Program. 107, 231–273 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17, 844–860 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009) zbMATHGoogle Scholar
  6. 6.
    Ben-Tal, A., Margalit, T., Nemirovski, A.: Robust modeling of multi-stage portfolio problems. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization, pp. 303–328. Kluwer, Dordrecht (2000) CrossRefGoogle Scholar
  7. 7.
    Ben-Tal, A., Nemirovski, A.: Stable truss topology design via semidefinite programming. SIAM J. Optim. 7, 991–1016 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. Society for Industrial & Applied Mathematics, Philadelphia (2001) zbMATHGoogle Scholar
  11. 11.
    Ben-Tal, A., Nemirovski, A.: Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. 107, 63–89 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ben-Tal, A., Nemirovski, A.: Selected topics in robust convex optimization. Math. Program. 112, 125–158 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ben-Tal, A., Nemirovski, A., Roos, C.: Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM J. Optim. 13, 535–560 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bertsekas, D.P.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003) zbMATHGoogle Scholar
  15. 15.
    Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52, 35–53 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bertsimas, D., Thiele, A.: Robust optimization approach to inventory theory. Oper. Res. 54, 150–168 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Boni, O., Ben-Tal, A., Nemirovski, A.: Robust solutions to conic quadratic problems and their applications. Optim. Eng. 9, 1–8 (2008) MathSciNetCrossRefGoogle Scholar
  18. 18.
    El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problem with uncertain data. SIAM J. Matrix Anal. Appl. 18, 1035–1064 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    El Ghaoui, L., Oks, M., Oustry, F.: Worst-case Value-at-Risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51, 543–556 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    El Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9, 33–52 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Goldfarb, D., Iyengar, G.: Robust portfolio selection problems. Math. Oper. Res. 28, 1–37 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Harville, D.A.: Matrix Algebra from a Statistician’s Perspective. Springer, New York (2007) Google Scholar
  23. 23.
    Hayashi, S., Yamashita, N., Fukushima, M.: Robust Nash equilibria and second-order cone complementarity problems. J. Nonlinear Convex Anal. 6, 283–296 (2005) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hildebrand, R.: An LMI description for the cone of Lorentz-positive maps. Linear Multilinear Algebra 55, 551–573 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hildebrand, R.: An LMI description for the cone of Lorentz-positive maps II. Linear Multilinear Algebra 59, 719–731 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Huang, D.S., Fabozzi, F.J., Fukushima, M.: Robust portfolio selection with uncertain exit time using worst-case VaR strategy. Oper. Res. Lett. 35, 627–635 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Huang, D.S., Zhu, S.S., Fabozzi, F.J., Fukushima, M.: Portfolio selection with uncertain exit time: a robust CVaR approach. J. Econ. Dyn. Control 32, 594–623 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007) zbMATHCrossRefGoogle Scholar
  30. 30.
    Nishimura, R., Hayashi, S., Fukushima, M.: Robust Nash equilibria in N-person noncooperative games: uniqueness and reformulation. Pac. J. Optim. 5, 237–259 (2009) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) zbMATHGoogle Scholar
  32. 32.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95, 189–217 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Zhong, P., Fukushima, M.: Second-order cone programming formulations for robust multiclass classification. Neural Comput. 19, 258–282 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Zhu, S.S., Fukushima, M.: Worst-case conditional Value-at-Risk with application to robust portfolio management. Oper. Res. 57, 1155–1168 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Zymler, S., Rustem, B., Kuhn, D.: Robust portfolio optimization with derivative insurance guarantees. Eur. J. Oper. Res. 210, 410–424 (2011) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Ryoichi Nishimura
    • 1
  • Shunsuke Hayashi
    • 1
    Email author
  • Masao Fukushima
    • 1
  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

Personalised recommendations