Journal of Optimization Theory and Applications

, Volume 160, Issue 2, pp 439–456 | Cite as

On Valid Inequalities for Mixed Integer p-Order Cone Programming

  • Alexander Vinel
  • Pavlo Krokhmal


We discuss two families of valid inequalities for linear mixed integer programming problems with cone constraints of arbitrary order, which arise in the context of stochastic optimization with downside risk measures. In particular, we extend the results of Atamtürk and Narayanan (Math. Program., 122:1–20, 2010, Math. Program., 126:351–363, 2011), who developed mixed integer rounding cuts and lifted cuts for mixed integer programming problems with second-order cone constraints. Numerical experiments conducted on randomly generated problems and portfolio optimization problems with historical data demonstrate the effectiveness of the proposed methods.


Valid inequalities Nonlinear cuts Mixed integer p-order cone programming Stochastic optimization Risk measures 



This work was supported in part by AFOSR grant FA9550-12-1-0142 and NSF grant EPS1101284.


  1. 1.
    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, 438–450 (2008) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: On polyhedral approximations of the second-order cone. Math. Oper. Res. 26, 193–205 (2001) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Vinel, A., Krokhmal, P.: On polyhedral approximations for solving p-order conic programming problems. Working paper (2012) Google Scholar
  4. 4.
    Drewes, S.: Mixed integer second order cone programming. Ph.D. Thesis, Technische Universität Darmstadt, Germany (2009) Google Scholar
  5. 5.
    Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5, 186–204 (2008) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Atamtürk, A., Narayanan, V.: Conic mixed-integer rounding cuts. Math. Program. 122, 1–20 (2010) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Atamtürk, A., Narayanan, V.: Lifting for conic mixed-integer programming. Math. Program. 126, 351–363 (2011) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Belotti, P., Goez, J., Polik, I., Ralphs, T., Terlaky, T.: A conic representation of the convex hull of disjunctive sets and conic cuts for integer second order cone optimization. Working paper (2012) Google Scholar
  10. 10.
    Dadush, D., Dey, S.S., Vielma, J.P.: The split closure of a strictly convex body. Oper. Res. Lett. 39, 121–126 (2011) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0-1 mixed convex programming. Math. Program. 86, 515–532 (1999) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Çezik, M.T., Iyengar, G.: Cuts for mixed 0-1 conic programming. Math. Program. 104, 179–202 (2005) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Krokhmal, P.A.: Higher moment coherent risk measures. Quant. Finance 7, 373–387 (2007) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Krokhmal, P., Zabarankin, M., Uryasev, S.: Modeling and optimization of risk. Surv. Oper. Res. Manag. Sci. 16, 49–66 (2011) Google Scholar
  15. 15.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (1988) MATHGoogle Scholar
  16. 16.
    Krokhmal, P.A., Soberanis, P.: Risk optimization with p-order conic constraints: a linear programming approach. Eur. J. Oper. Res. 201, 653–671 (2010) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Nesterov, Y.E., Nemirovski, A.: Interior Point Polynomial Algorithms in Convex Programming. Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994) CrossRefMATHGoogle Scholar
  18. 18.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Morenko, Y., Vinel, A., Yu, Z., Krokhmal, P.: On p-cone linear discrimination (2012). Submitted for publication Google Scholar
  20. 20.
    Perold, A.F.: Large-scale portfolio optimization. Manag. Sci. 30, 1143–1160 (1984) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Bonami, P., Lejeune, M.A.: An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Oper. Res. 57, 650–670 (2009) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Scherer, B., Martin, R.D.: Introduction to Modern Portfolio Optimization with NUOPT and S-PLUS. Springer, New York (2005) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of IowaIowa CityUSA

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