Journal of Global Optimization

, Volume 70, Issue 4, pp 737–756 | Cite as

An interleaved depth-first search method for the linear optimization problem with disjunctive constraints

  • Yinrun Lyu
  • Li Chen
  • Changyou Zhang
  • Dacheng Qu
  • Nasro Min-Allah
  • Yongji Wang
Article
  • 73 Downloads

Abstract

Being an extension of classical linear programming, disjunctive programming has the ability to express the problem constraints as combinations of linear equalities and inequalities linked with logic AND and OR operations. All the existing theories such as generalized disjunctive programming, optimization modulo theories, linear optimization over arithmetic constraint formula, and mixed logical linear programming pose one commonality of branching among different solving techniques. However, branching constructs a depth-first search which may traverse a whole bad subtree when the branching makes a mistake ordering a bad successor. In this paper, we propose the interleaved depth-first search with stochastic local optimal increasing (IDFS-SLOI) method for solving the linear optimization problem with disjunctive constraints. Our technique searches depth-first several subtrees in turn, accelerates the search by subtree splitting, and uses efficient backtracking and pruning among the subtrees. Additionally, the local optimal solution is improved iteratively by constructing and solving a stochastic linear programming problem. We evaluate our approach against existing counterparts on the rate-monotonic optimization problem (RM-OPT) and the linear optimization with fuzzy relation inequalities problem (LOFRI). Experimental results show that for the tested instances, the IDFS-SLOI method performs better from performance perspective, especially promising results have been obtained for the larger three groups where the execution time is reduced by 85.6 and 51.6% for RM-OPT and LOFRI, respectively.

Keywords

Global optimization Disjunctive programming Optimization modulo theories Search algorithm 

Notes

Acknowledgements

This work is jointly supported by the CAS/SAFEA International Partnership Program for Creative Research Teams, and the Natural Science Foundation of China under Grants 61379048 and 61672508.

References

  1. 1.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1998)MATHGoogle Scholar
  2. 2.
    Lodi, A.: Mixed integer programming computation. In: Jnger, M., Liebling, TM., Naddef, D., Nemhauser, GL., Pulleyblank, WR., Reinelt, G., Rinaldi, G., Wolsey, LA. (eds.) 50 Years of Integer Programming 1958–2008, pp. 619–645. Springer (2010)Google Scholar
  3. 3.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Hooker, J.N., Osorio, M.A.: Mixed logical-linear programming. Discrete Appl. Math. 96, 395–442 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, L., Lyu, Y., Wang, C., Wu, J., Zhang, C., Min-Allah, N., Alhiyafi, J., Wang, Y.: Solving linear optimization over arithmetic constraint formula. J. Glob. Optim. 69, 1–34 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Sebastiani, R., Tomasi, S.: Optimization modulo theories with linear rational costs. ACM Trans. Comput. Log. (TOCL) 16(2), 12 (2015)MathSciNetMATHGoogle Scholar
  7. 7.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. MSRR No. 330 (1974)Google Scholar
  8. 8.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discrete Appl. Math. 89(1), 3–44 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Raman, R., Grossmann, I.E.: Modelling and computational techniques for logic based integer programming. Comput. Chem. Eng. 18(7), 563–578 (1994)CrossRefGoogle Scholar
  10. 10.
    Vecchietti, A., Grossmann, I.: Computational experience with LogMIP solving linear and nonlinear disjunctive programming problems. In: Proceeding of FOCAPD, pp. 587–590. Citeseer (2004)Google Scholar
  11. 11.
    Sawaya, N., Grossmann, I.: A hierarchy of relaxations for linear generalized disjunctive programming. Eur. J. Oper. Res. 216(1), 70–82 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Trespalacios, F., Grossmann, I.E.: Algorithmic approach for improved mixed-integer reformulations of convex generalized disjunctive programs. INFORMS J. Comput. 27(1), 59–74 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Trespalacios, F., Grossmann, I.E.: Cutting plane algorithm for convex generalized disjunctive programs. INFORMS J. Comput. 28(2), 209–222 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kirst, P., Rigterink, F., Stein, O.: Global optimization of disjunctive programs. J. Glob. Optim. 69, 1–25 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ruiz, J.P., Grossmann, I.E.: Global optimization of non-convex generalized disjunctive programs: a review on reformulations and relaxation techniques. J. Glob. Optim. 67(1), 43–58 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Barrett, C., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M., van Maaren, H. (eds.) Handbook of Satisfiability, vol. 185, pp. 825–885. IOS Press (2009)Google Scholar
  17. 17.
    De Moura, L., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Commun. ACM 54(9), 69–77 (2011)CrossRefGoogle Scholar
  18. 18.
    Monniaux, D.: A survey of satisfiability modulo theory. In: International Workshop on Computer Algebra in Scientific Computing, pp. 401–425. Springer (2016)Google Scholar
  19. 19.
    Silva, J.P.M., and Sakallah, K.A.: GRASP: a new search algorithm for satisfiability. In: Proceedings of the 1996 IEEE/ACM International Conference on Computer-Aided Design, pp. 220–227. IEEE Computer Society (1997)Google Scholar
  20. 20.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient sat solver. In: Proceedings of the 38th Annual Design Automation Conference, pp. 530–535. ACM (2001)Google Scholar
  21. 21.
    Gomes, C.P., Selman, B., Kautz, H., et al.: Boosting combinatorial search through randomization. In: AAAI/IAAI, vol. 98, pp. 431–437 (1998)Google Scholar
  22. 22.
    Goldberg, E., Novikov, Y.: BerkMin: a fast and robust sat-solver. Discrete Appl. Math. 155(12), 1549–1561 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hooker, J.N.: Logic, optimization, and constraint programming. INFORMS J. Comput. 14(4), 295–321 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Meseguer, P.: Interleaved depth-first search. In: IJCAI, vol. 97, pp. 1382–1387 (1997)Google Scholar
  25. 25.
    Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1986)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    de la Tour, T.B.: An optimality result for clause form translation. J. Symb. Comput. 14(4), 283–301 (1992)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Li, Y., Albarghouthi, A., Kincaid, Z., Gurfinkel, A., Chechik, M.: Symbolic optimization with SMT solvers. In: ACM SIGPLAN Notices, vol. 49, pp. 607–618. ACM (2014)Google Scholar
  28. 28.
    Sebastiani, R., Tomasi, S.: Optimization in SMT with \({\cal{LA}}({\mathbb{Q}})\) cost functions. In: Gramlich, B., Miller, D., Sattler, U. (eds.) Automated Reasoning, pp. 484–498. Springer (2012)Google Scholar
  29. 29.
    Sawaya, N.W., Grossmann, I.E.: A cutting plane method for solving linear generalized disjunctive programming problems. Comput. Chem. Eng. 29(9), 1891–1913 (2005)CrossRefGoogle Scholar
  30. 30.
    Trespalacios, F., Grossmann, I.E.: Improved Big-M reformulation for generalized disjunctive programs. Comput. Chem. Eng. 76, 98–103 (2015)CrossRefGoogle Scholar
  31. 31.
    Vielma, J.P.: Mixed integer linear programming formulation techniques. SIAM Rev. 57(1), 3–57 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Liu, J., Wang, Y., Wang, Y., Xing, J., Zeng, H.: Real-time system design based on logic or constrained optimization. J. Softw. 17(7), 1641–1649 (2006)CrossRefMATHGoogle Scholar
  33. 33.
    Liu, C.L., Layland, J.W.: Scheduling algorithms for multiprogramming in a hard-real-time environment. J. ACM (JACM) 20(1), 46–61 (1973)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Min-Allah, N., Khan, S.U., Yongji, W.: Optimal task execution times for periodic tasks using nonlinear constrained optimization. J. Supercomput. 59(3), 1120–1138 (2012)CrossRefGoogle Scholar
  35. 35.
    Bini, E., Buttazzo, G.C.: The space of rate monotonic schedulability. In: 23rd IEEE on Real-Time Systems Symposium, 2002. RTSS 2002, pp. 169–178. IEEE (2002)Google Scholar
  36. 36.
    Fang, S., Li, G.: Solving fuzzy relation equations with a linear objective function. Fuzzy Sets Syst. 103(1), 107–113 (1999)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Ghodousian, A., Khorram, E.: Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max–min composition. Inf. Sci. 178(2), 501–519 (2008)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Su, C., Guo, F.: Solving interval-valued fuzzy relation equations with a linear objective function. In: International Conference on Fuzzy Systems and Knowledge Discovery, pp. 380–385 (2009)Google Scholar
  39. 39.
    Guo, F., Pang, L., Meng, D., Xia, Z.: An algorithm for solving optimization problems with fuzzy relational inequality constraints. Inf. Sci. 252, 20–31 (2013)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Miyagi, H., Kinjo, I., Fan, Y.: Qualified decision-making using the fuzzy relation inequalities. In: IEEE International Conference on Systems, Man, and Cybernetics, vol. 2, pp. 2014–2018 (1998)Google Scholar
  41. 41.
    Wang, H., Wang, C.H.: A fixed-charge model with fuzzy inequality constraints composed by max-product operator. Comput. Math. Appl. 36(7), 23–29 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Yinrun Lyu
    • 3
    • 4
  • Li Chen
    • 3
    • 4
  • Changyou Zhang
    • 3
  • Dacheng Qu
    • 1
  • Nasro Min-Allah
    • 2
  • Yongji Wang
    • 3
    • 4
  1. 1.Beijing Institute of TechnologyBeijingChina
  2. 2.CCSITUniversity of DammamDammamSaudi Arabia
  3. 3.State Key Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingChina
  4. 4.University of Chinese Academy of SciencesBeijingChina

Personalised recommendations