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A hierarchy of relaxations for nonlinear convex generalized disjunctive programming


  • Ruiz, Juan P.
  • Grossmann, Ignacio E.


We propose a framework to generate alternative mixed-integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations. We extend the concept of “basic steps” defined for disjunctive linear programs to the nonlinear case. A basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts. We show that the strength of the relaxations increases as the number of conjuncts decreases, leading to a hierarchy of relaxations. We prove that the tightest of these relaxations, allows in theory the solution of the disjunctive convex program as a nonlinear programming problem. We present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed-integer program. Finally, we apply the theory developed to improve the computational efficiency of solution methods for nonlinear convex generalized disjunctive programs (GDP). This methodology is validated through a set of numerical examples.

Suggested Citation

  • Ruiz, Juan P. & Grossmann, Ignacio E., 2012. "A hierarchy of relaxations for nonlinear convex generalized disjunctive programming," European Journal of Operational Research, Elsevier, vol. 218(1), pages 38-47.
  • Handle: RePEc:eee:ejores:v:218:y:2012:i:1:p:38-47 DOI: 10.1016/j.ejor.2011.10.002

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    References listed on IDEAS

    1. Omprakash K. Gupta & A. Ravindran, 1985. "Branch and Bound Experiments in Convex Nonlinear Integer Programming," Management Science, INFORMS, vol. 31(12), pages 1533-1546, December.
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    Cited by:

    1. Juan P. Ruiz & Ignacio E. Grossmann, 2017. "Global optimization of non-convex generalized disjunctive programs: a review on reformulations and relaxation techniques," Journal of Global Optimization, Springer, vol. 67(1), pages 43-58, January.


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