Optimization Letters

, Volume 13, Issue 1, pp 95–111 | Cite as

Unified approach for solving box-constrained models with continuous or discrete variables by non monotone direct search methods

  • Ubaldo M. García-PalomaresEmail author
  • Pedro S. Rodríguez-Hernández
Original Paper


This paper is an outgrowth of previous works devoted to the application of non monotone direct search methods (DSMs) to locate the global minimum of an objective function subjected to bounds on its variables defined in the Euclidean space. This paper proves that DSMs can be easily adapted for solving models with discrete variables, as long as these variables are regularly spaced along each coordinate. Convergence is established without imposing additional conditions on the objective function, but the construction of the search directions plays a fundamental role. Moreover, function evaluations are carried out only on discrete feasible points, avoiding spurious computations on non feasible points. The paper describes a pseudo code and a preliminary code written in C, which is applied to solve a discretized version of a continuous problem with encouraging results.


Direct search methods Discrete variable Non monotone 



Authors’ research is being partly funded by Grants TEC2016-76465-C2-2-R (Mineco, Spain) and GRC2014/046 (Xunta de Galicia, Spain).


  1. 1.
    Abramson, M.A., Audet, C., Chrissis, J.W., Walston, J.G.: Mesh adaptive direct search algorithms for mixed variable optimization. Optim. Lett. 3(1), 35–47 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Audet, C., Dennis, J.: Pattern search algorithms for mixed variable programming. SIAM J. Optim. 11(3), 573–594 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Audet, C., Kokkolaras, M.: Blackbox and derivative-free optimization: theory, algorithms and applications. Optim. Eng. 17(1), 1–2 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Custodio, A.L., Scheinberg, K., Nunes Vicente, L.: Methodologies and software for derivative-free optimization. Internal Report (2017).
  5. 5.
    García-Palomares, U.M.: Software companion for DFO approach for bounded and discrete variables. Researchgate. (2016). Google Scholar
  6. 6.
    García-Palomares, U.M., Costa-Montenegro, E., Asorey-Cacheda, R., González-Castaño, F.J.: Adapting derivative free optimization methods to engineering. Models with discrete variables. Optim. Eng. 13(4), 579–594 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    García-Palomares, U.M., García-Urrea, I.J., Rodríguez-Hernández, P.S.: On sequential and parallel non monotone derivative free algorithms for box constrained optimization. Optim. Methods Softw. 28(6), 1233–1261 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    García-Palomares, U.M., González-Castaño, F.J., Burguillo-Rial, J.C.: A combined global & local search (CGLS) approach to global optimization. J. Global Optim. 34, 409–426 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    García-Palomares, U.M., Rodríguez, J.F.: New sequential and parallel derivative-free algorithms for unconstrained minimization. SIAM J. Optim. 13(1), 79–96 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gratton, S., Toint, P.L., Troltzsch, A.: An active-set trust-region method for derivative-free nonlinear bound-constrained optimization. Optim. Methods Softw. 26(4), 873–894 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
  12. 12.
    Lewis, R.M., Torczon, V.J.: Pattern search algorithms for bound constrained minimization. SIAM J. Optim. 9, 1082–1099 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liuzzi, G., Lucidi, S., Rinaldi, F.: Derivative-free methods for bound constrained mixed-integer optimization. Comput. Optim. Appl. 53(2), 505–526 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Moré, J.J., Wild, S.M.: Benchmarking derivative-free optimization algorithms. SIAM J. Optim. 20(1), 172–191 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Newby, E.: General solution methods for mixed integer quadratic programming and derivative free mixed integer non-linear programming problems. Dissertation Thesis, University of the Witwatersrand (2013)Google Scholar
  16. 16.
    Newby, E., Ali, M.: A trust region based derivative free algorithm for mixed integer programs. Comput. Optim. Appl. 60(1), 199–229 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ng, K.-M.: A continuation approach for solving nonlinear optimization problems with discrete variables. Dissertation Thesis, Stanford University (2002)Google Scholar
  18. 18.
    Pintér, J.: Global optimization software, test problems, and applications. In: Pardalos, P., Romeijn, H. (eds.) Handbook of Global Optimization, vol. 2, pp. 515–569. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
  19. 19.
    Powell, M.J.D.: The BOBYQA algorithm for bound constrained optimization without derivatives. Cambridge Report NA2009/06, University of Cambridge (2009)Google Scholar
  20. 20.
    Ríos, L.M., Sahidinis, N.V.: Derivative-free optimization: a review of algorithms and comparison of software implementations. J. Global Optim. 56, 1247–1293 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rodríguez-Hernández, P.S.: (2016). Accessed 5 July 2016
  22. 22.
    Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  23. 23.
    Torczon, V.J.: Multi-directional search: a direct search algorithm for parallel machines. PhD Thesis, Rice University, Houston, TX (1989)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Information Technology Group, AtlanticUniversidade de VigoVigoSpain

Personalised recommendations