Journal of Optimization Theory and Applications

, Volume 169, Issue 3, pp 1079–1109 | Cite as

Solving Disjunctive Optimization Problems by Generalized Semi-infinite Optimization Techniques

  • Peter Kirst
  • Oliver SteinEmail author


We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods in disjunctive programming, our approach does not expect any special formulation of the underlying logical expression. Applying existing lower-level reformulations for the corresponding semi-infinite program, we derive conjunctive nonlinear problems without any logical expressions, which can be locally solved by standard nonlinear solvers. Our preliminary numerical results on some small-scale examples indicate that our reformulation procedure is a reasonable method to solve disjunctive optimization problems.


Disjunctive optimization Generalized semi-infinite optimization Lower-level duality Mathematical program with complementarity constraints Smoothing 

Mathematics Subject Classification

90C34 90C30 



We thank the anonymous referees, the associate editor and the editor-in-chief for their precise and substantial remarks, which helped to significantly improve the paper. This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under Grant STE 772/14-1.


  1. 1.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hettich, R., Zencke, P.: Numerische Methoden der Approximation und Semi-infiniten Optimierung. Teubner, Stuttgart (1982)CrossRefzbMATHGoogle Scholar
  3. 3.
    Reemtsen, R., Görner, S.: Numerical methods for semi-infinite programming: a survey. In: Reemtsen, R., Rückmann, J.-J. (eds.) Semi-Infinite Programming, pp. 195–275. Kluwer, Boston (1998)CrossRefGoogle Scholar
  4. 4.
    Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, Chichester (1998)zbMATHGoogle Scholar
  5. 5.
    Polak, E.: Optimization. Algorithms and Consistent Approximations. Springer, Berlin (1997)zbMATHGoogle Scholar
  6. 6.
    Stein, O.: Bi-level Strategies in Semi-infinite Programming. Kluwer, Boston (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Vázquez, F.G., Rückmann, J.-J., Stein, O., Still, G.: Generalized semi-infinite programming: a tutorial. J. Comput. Appl. Math. 217, 394–419 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Stein, O.: How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. 223, 312–320 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)CrossRefzbMATHGoogle Scholar
  11. 11.
    Williams, H.P.: Model Building in Mathematical Programming. Wiley, Chichester (1978)zbMATHGoogle Scholar
  12. 12.
    Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8, 250–255 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Balas, E.: Disjunctive programming. Ann. Discrete Math. 5, 3–51 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discrete Appl. Math. 89, 3–44 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Balas, E., Perregaard, M.: Lift-and-project for mixed 0–1 programming: recent progress. Discrete Appl. Math. 123, 129–154 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Beaumont, N.: An algorithm for disjunctive programs. Eur. J. Oper. Res. 46, 362–371 (1990)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hooker, J.N., Osorio, M.A.: Mixed logical-linear programming. Discrete Appl. Math. 96–97, 395–442 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Raman, R., Grossmann, I.E.: Relations between MILP modelling and logical inference for chemical process synthesis. Comput. Chem. Eng. 15, 73–84 (1991)CrossRefGoogle Scholar
  19. 19.
    Raman, R., Grossmann, I.E.: Symbolic integration of logic in mixed-integer linear programming techniques for process synthesis. Comput. Chem. Eng. 17, 909–927 (1993)CrossRefGoogle Scholar
  20. 20.
    Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Westerlund, T., Pettersson, F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, 131–136 (1995)CrossRefGoogle Scholar
  24. 24.
    Grossmann, I.E., Lee, S.: Generalized convex disjunctive programming: nonlinear convex hull relaxation. Comput. Optim. Appl. 26, 83–100 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lee, S., Grossmann, I.E.: New algorithms for nonlinear generalized disjunctive programming. Comput. Chem. Eng. 24, 2125–2141 (2000)CrossRefGoogle Scholar
  26. 26.
    Jongen, HTh, Rückmann, J.-J., Stein, O.: Disjunctive optimization: critical point theory. J. Optim. Theory Appl. 93, 321–336 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lee, S., Grossmann, I.E.: A global optimization algorithm for nonconvex generalized disjunctive programming and applications to process systems. Comput. Chem. Eng. 25, 1675–1697 (2001)CrossRefGoogle Scholar
  28. 28.
    Levitin, E., Tichatschke, R.: A branch-and-bound approach for solving a class of generalized semi-infinite programming problems. J. Glob. Optim. 13, 299–315 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Diehl, M., Houska, B., Stein, O., Steuermann, P.: A lifting method for generalized semi-infinite programs based on lower level Wolfe duality. Comput. Optim. Appl. 54, 189–210 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Stein, O., Still, G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142, 444–462 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Bard, J.F.: Practical Bilevel Optimization. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  32. 32.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer, Dordrecht (2002)zbMATHGoogle Scholar
  33. 33.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42, 769–788 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Stein, O., Winterfeld, A.: A feasible method for generalized semi-infinite programming. J. Optim. Theory Appl. 146, 419–443 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Wiley, New York (2013)zbMATHGoogle Scholar
  38. 38.
    Pang, J.-S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)zbMATHGoogle Scholar
  40. 40.
    Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of 2003 MODE-SMAI Conference, EDP Sci., Les Ulis, pp. 1–17 (2003)Google Scholar
  41. 41.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Operations ResearchKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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