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Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming

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Abstract

In this paper, we present a new hybrid algorithm for convex Mixed Integer Nonlinear Programming (MINLP). The proposed hybrid algorithm is an improved version of the classical nonlinear branch-and-bound (BB) procedure, where the enhancements are obtained with the application of the outer approximation algorithm on some nodes of the enumeration tree. The two methods are combined in such a way that each one collaborates to the convergence of the other. Computational experiments with benchmark instances of the MINLP problem show the good performance of the proposed algorithm, which is compared to the outer approximation algorithm, the nonlinear BB algorithm and the hybrid algorithm implemented in the solver Bonmin.

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Acknowledgments

The authors would like to thank the anonymous referees, whose comments helped to improve the paper, and CAPES and CNPq for the financial support.

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Authors and Affiliations

  1. Institute for Graduate Studies and Research in Engineering (COPPE), Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Wendel Melo

  2. Institute of Mathematics and COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Marcia Fampa

  3. Department of Industrial Engineering, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

    Fernanda Raupp

  4. National Laboratory for Scientific Computing, Petrópolis, Brazil

    Fernanda Raupp

Authors
  1. Wendel Melo
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  2. Marcia Fampa
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  3. Fernanda Raupp
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Corresponding author

Correspondence to Wendel Melo.

Appendices

Appendix A

1.1 Information about the test problems

Table 1 describes the characteristics of the selected test-instances. The columns of the table provide for each instance: name, objective type, number of continuous variables (\(n_x\)), number of integer (binary) variables (\(n_y\)), number of nonlinear constraints (\(m_{nl}\)), number of quadratic constraints (\(m_q\)), number of linear constraints (\(m_l\)), number of nonzeros in objective function quadratic matrix (\(\#Q\)), number of nonzeros in the Jacobian of the nonlinear constraints (excluding linear and quadratic ones) (\(\#{\nabla g}\)), number of nonzeros in the nonlinear Hessian of the Lagrangian (excluding quadratic expressions) (\(\#{\nabla ^2 H}\)).

Appendix B

1.1 Detailed computational results

Table 2 shows the computational results of OA, BB, our hybrid approach, and Bonmin’s hybrid approach for the 50 test problems selected. The columns on the table present, for each approach: final status (St), best solution (Sol), and CPU time in seconds (Time). The possible values for the final status (St) are: Optimal Solution (OS), Maximum Time Achieved (MT), and Solver Error (SE). Maximum Time was set at 14,400 s (4 h).

Table 2 Computational results for OA, BB, our Hybrid and Bonmin
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Melo, W., Fampa, M. & Raupp, F. Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming. J Glob Optim 60, 373–389 (2014). https://doi.org/10.1007/s10898-014-0217-8

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