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On Valid Inequalities for Mixed Integer p-Order Cone Programming

Author

Listed:
  • Alexander Vinel

    (University of Iowa)

  • Pavlo Krokhmal

    (University of Iowa)

Abstract

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.

Suggested Citation

  • Alexander Vinel & Pavlo Krokhmal, 2014. "On Valid Inequalities for Mixed Integer p-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 439-456, February.
    • Handle: RePEc:spr:joptap:v:160:y:2014:i:2:d:10.1007_s10957-013-0315-7
    DOI: 10.1007/s10957-013-0315-7
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    References listed on IDEAS

    as
    1. PAVLO A. Krokhmal, 2007. "Higher moment coherent risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 373-387.
    2. Juan Pablo Vielma & Shabbir Ahmed & George L. Nemhauser, 2008. "A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed-Integer Conic Quadratic Programs," INFORMS Journal on Computing, INFORMS, vol. 20(3), pages 438-450, August.
    3. Thierry Bracke & Reiner Martin, 2012. "Introduction," Palgrave Macmillan Books, in: Thierry Bracke & Reiner Martin (ed.), From Crisis to Recovery, pages 1-5, Palgrave Macmillan.
    4. P. Bonami & M. A. Lejeune, 2009. "An Exact Solution Approach for Portfolio Optimization Problems Under Stochastic and Integer Constraints," Operations Research, INFORMS, vol. 57(3), pages 650-670, June.
    5. Andre F. Perold, 1984. "Large-Scale Portfolio Optimization," Management Science, INFORMS, vol. 30(10), pages 1143-1160, October.
    6. Pierre Bonami & Miguel A. Lejeune, 2009. "An Exact Solution Approach for Integer Constrained Portfolio Optimization Problems Under Stochastic Constraints," Post-Print hal-00421756, HAL.
    7. Krokhmal, Pavlo A. & Soberanis, Policarpio, 2010. "Risk optimization with p-order conic constraints: A linear programming approach," European Journal of Operational Research, Elsevier, vol. 201(3), pages 653-671, March.
    8. Aharon Ben-Tal & Arkadi Nemirovski, 2001. "On Polyhedral Approximations of the Second-Order Cone," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 193-205, May.
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    Cited by:

    1. Baha Alzalg, 2016. "The Algebraic Structure of the Arbitrary-Order Cone," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 32-49, April.
    2. Manish Bansal & Yingqiu Zhang, 2021. "Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs," Journal of Global Optimization, Springer, vol. 81(2), pages 391-433, October.

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