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Robust nonlinear optimization with conic representable uncertainty set

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  • Soleimanian, Azam
  • Salmani Jajaei, Ghasemali

Abstract

The robust optimization methodology is known as a popular method dealing with optimization problems with uncertain data and hard constraints. This methodology has been applied so far to various convex conic optimization problems where only their inequality constraints are subject to uncertainty. In this paper, the robust optimization methodology is applied to the general nonlinear programming (NLP) problem involving both uncertain inequality and equality constraints. The uncertainty set is defined by conic representable sets, the proposed uncertainty set is general enough to include many uncertainty sets, which have been used in literature, as special cases. The robust counterpart (RC) of the general NLP problem is approximated under this uncertainty set. It is shown that the resulting approximate RC of the general NLP problem is valid in a small neighborhood of the nominal value. Furthermore a rather general class of programming problems is posed that the robust counterparts of its problems can be derived exactly under the proposed uncertainty set. Our results show the applicability of robust optimization to a wider area of real applications and theoretical problems with more general uncertainty sets than those considered so far. The resulting robust counterparts which are traditional optimization problems make it possible to use existing algorithms of mathematical optimization to solve more complicated and general robust optimization problems.

Suggested Citation

  • Soleimanian, Azam & Salmani Jajaei, Ghasemali, 2013. "Robust nonlinear optimization with conic representable uncertainty set," European Journal of Operational Research, Elsevier, vol. 228(2), pages 337-344.
  • Handle: RePEc:eee:ejores:v:228:y:2013:i:2:p:337-344
    DOI: 10.1016/j.ejor.2013.02.018
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    References listed on IDEAS

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    1. Gregory, Christine & Darby-Dowman, Ken & Mitra, Gautam, 2011. "Robust optimization and portfolio selection: The cost of robustness," European Journal of Operational Research, Elsevier, vol. 212(2), pages 417-428, July.
    2. Ben-Tal, A. & den Hertog, D. & Vial, J.P., 2012. "Deriving Robust Counterparts of Nonlinear Uncertain Inequalities," Discussion Paper 2012-053, Tilburg University, Center for Economic Research.
    3. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    4. Zymler, Steve & Rustem, Berç & Kuhn, Daniel, 2011. "Robust portfolio optimization with derivative insurance guarantees," European Journal of Operational Research, Elsevier, vol. 210(2), pages 410-424, April.
    5. Gorissen, Bram L. & den Hertog, Dick, 2013. "Robust counterparts of inequalities containing sums of maxima of linear functions," European Journal of Operational Research, Elsevier, vol. 227(1), pages 30-43.
    6. Gorissen, B.L. & den Hertog, D., 2011. "Robust Counterparts of Inequalities Containing Sums of Maxima of Linear Functions," Discussion Paper 2011-115, Tilburg University, Center for Economic Research.
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