Advertisement

Mathematical Programming Computation

, Volume 11, Issue 4, pp 755–789 | Cite as

An active set algorithm for robust combinatorial optimization based on separation oracles

  • Christoph Buchheim
  • Marianna De SantisEmail author
Full Length Paper
  • 144 Downloads

Abstract

We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi.

Keywords

Robust optimization Active set methods SOCP 

Mathematics Subject Classification

90C25 90C27 90C57 

Notes

Acknowledgements

The first author acknowledges support within the project “Mixed-Integer Non Linear Optimisation: Algorithms and Applications”, which has received funding from the Europeans Union’s EU Framework Programme for Research and Innovation Horizon 2020 under the Marie Skłodowska-Curie Actions Grant Agreement No 764759. The second author acknowledges support within the project “Nonlinear Approaches for the Solution of Hard Optimization Problems with Integer Variables”(No RP11715C7D8537BA) which has received funding from Sapienza, University of Rome. Moreover, the authors are grateful to Philipp Speckenmeyer for developing and implementing the special treatment of equations in our code (and for the careful reading of the manuscript).

References

  1. 1.
    Atamtürk, A., Gómez, A.: Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra. Math. Prog. Comp. (2018).  https://doi.org/10.1007/s12532-018-0152-7 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. SIAM, Philadelphia (2001)CrossRefGoogle Scholar
  5. 5.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15, 780–804 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Buchheim, C., Kurtz, J.: Robust combinatorial optimization under convex and discrete cost uncertainty. Technical report, Optimization Online (2017)zbMATHGoogle Scholar
  7. 7.
    Buchheim, C., De Santis, M., Lucidi, S., Rinaldi, F., Trieu, L.: A feasible active set method with reoptimization for convex quadratic mixed-integer programming. SIAM J. Optim. 26(3), 1695–1714 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Buchheim, C., De Santis, M., Rinaldi, F., Trieu, L.: A Frank-Wolfe based branch-and-bound algorithm for mean-risk optimization. J. Glob. Optim. 70(3), 625–644 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gurobi Optimization, Inc.: Gurobi optimizer reference manual (2016)Google Scholar
  11. 11.
    Kouvelis, P., Yu, G.: Robust Discrete Optimization and Its Applications. Springer, Dordrecht (1996)zbMATHGoogle Scholar
  12. 12.
    Meyer Jr., C.D.: Generalized inversion of modified matrices. SIAM J. Appl. Math. 24(3), 315–323 (1973)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  14. 14.
    Mittelmann, H.D.: Latest benchmark results—informs annual conference. http://plato.asu.edu/talks/informs2018.pdf. Accessed 4–7 November 2018
  15. 15.
    MOSEK ApS: The MOSEK optimization toolbox for MATLAB manual. Version 8.0.0.81 (2017)Google Scholar
  16. 16.
    Nesterov, Y., Nemirovski, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1993)Google Scholar
  17. 17.
    Nikolova, E.: Approximation algorithms for offline risk-averse combinatorial optimization. Technical report (2010)Google Scholar
  18. 18.
    Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer-Verlag, New York (2006)zbMATHGoogle Scholar
  19. 19.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999). Version 1.05 available from http://fewcal.kub.nl/sturm MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikTU DortmundDortmundGermany
  2. 2.Dipartimento di Ingegneria Informatica, Automatica e GestionaleSapienza Università di RomaRomaItaly

Personalised recommendations