Journal of Global Optimization

, Volume 73, Issue 1, pp 1–25 | Cite as

Global optimization of generalized semi-infinite programs using disjunctive programming

  • Peter KirstEmail author
  • Oliver Stein


We propose a new branch-and-bound algorithm for global minimization of box-constrained generalized semi-infinite programs. It treats the inherent disjunctive structure of these problems by tailored lower bounding procedures. Three different possibilities are examined. The first one relies on standard lower bounding procedures from conjunctive global optimization as described in Kirst et al. (J Global Optim 69: 283–307, 2017). The second and the third alternative are based on linearization techniques by which we derive linear disjunctive relaxations of the considered sub-problems. Solving these by either mixed-integer linear reformulations or, alternatively, by disjunctive linear programming techniques yields two additional possibilities. Our numerical results on standard test problems with these three lower bounding procedures show the merits of our approach.


Generalized semi-infinite optimization Disjunctive optimization Global optimization Branch-and-bound 



We thank two anonymous referees for their precise and substantial remarks, which helped to significantly improve the paper.


  1. 1.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Guerra Vázquez, F., Rückmann, J.J., Stein, O., Still, G.: Generalized semi-infinite programming: a tutorial. J. Comput. Appl. Math. 217, 394–419 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Stein, O.: How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. 223, 312–320 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blankenship, J.W., Falk, J.W.: Infinitely constrained optimization problems. J. Optim. Theory Appl. 19, 261–281 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhattacharjee, B., Green, W.H., Barton, P.: Interval methods for semi-infinite programs. Comput. Optim. Appl. 30, 63–93 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bhattacharjee, B., Lemonidis, P., Green, W.H., Barton, P.: Global solution of semi-infinite programs. Math. Program. 103, 283–307 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lemonidis, P.: Global Optimization Algorithms for Semi-infinite and Generalized Semi-infinite Programs. Ph.D. Thesis. Massachusetts Institute of Technology (2008)Google Scholar
  9. 9.
    Mitsos, A.: Global optimization of semi-infinite programs via restriction of the right-hand side. Optimization 60, 1291–1308 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM J. Optim. 18, 1187–1208 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Stein, O., Steuermann, P.: The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets. Math. Program. 136, 183–207 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mitsos, A., Tsoukalas, A.: Global optimization of generalized semi-infinite programs via restriction of the right hand side. J. Global Optim. 61, 1–17 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schwientek, J., Seidel, T., Küfer, K.-H.: A Transformation-based Discretization Method for Solving General Semi-infinite Optimization Problems. Optimization Online Preprint-ID 2017-12-6380 (2017)Google Scholar
  14. 14.
    Günzel, H., Jongen, H.T., Stein, O.: On the closure of the feasible set in generalized semi-infinite programming. Central Eur. J. Oper. Res. 15, 271–280 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guerra Vázquez, F., Jongen, HTh, Shikhman, V.: General semi-infinite programming: symmetric Mangasarian–Fromovitz constraint qualification and the closure of the feasible set. SIAM J. Optim. 20, 2487–2503 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tsoulakas, A., Rustem, B., Pistikopoulos, E.N.: A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems. J. Global Optim. 44, 235–250 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Still, G.: Generalized semi-infinite programming: theory and methods. Eur. J. Oper. Res. 119, 301–313 (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Still, G.: Generalized semi-infinite programming: numerical aspects. Optimization 49, 223–242 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Diehl, M., Houska, B., Stein, O., Steuermann, S.: A lifting method for generalized semi-infinite programs based on lower level Wolfe duality. Comput. Optim. Appl. 54, 189–210 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stein, O., Winterfeld, A.: Feasible method for generalized semi-infinite programming. J. Optim. Theory Appl. 146, 419–443 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Levitin, E., Tichatschke, R.: A branch-and-bound approach for solving a class of generalized semi-infinite programming problems. J. Global Optim. 13, 299–315 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kirst, P., Rigterink, F., Stein, O.: Global optimization of disjunctive programs. J. Global Optim. 69, 283–307 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kirst, P., Stein, O., Steuermann, P.: Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints. TOP 23, 591–616 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Krawczyk, R., Nickel, K.: Die zentrische Form in der Intervallarithmetik, ihre quadratische Konvergenz und ihre Inklusionsisotonie. Computing 28, 117–137 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Baumann, E.: Optimal centered forms. BIT Numer. Math. 28, 80–87 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Belotti, P.: Disjunctive Cuts for Nonconvex MINLP. Mixed Integer Nonlinear Programming, pp. 117–144. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  28. 28.
    Smith, E.M., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, S791–S796 (1997)CrossRefGoogle Scholar
  29. 29.
    Smith, E.M., Pantelides, C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimization of nonconvex MINLPs. Comput. Chem. Eng. 23, 457–478 (1999)CrossRefGoogle Scholar
  30. 30.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs. Part I—convex underestimating problems. Math. Program. 10, 145–175 (1976)CrossRefzbMATHGoogle Scholar
  31. 31.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software and Applications. Springer Science and Business Media, Berlin (2002)CrossRefzbMATHGoogle Scholar
  32. 32.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  33. 33.
    Williams, H.P.: Model Building in Mathematical Programming. Wiley, Chichester (1978)zbMATHGoogle Scholar
  34. 34.
    Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebraic Discrete Methods 6, 466–486 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Balas, E.: A note on duality in disjunctive programming. J. Optim. Theory Appl. 21, 523–528 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Knüppel, O.: PROFIL/BIAS-a fast interval library. Computing 53, 277–287 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Makhorin, A.: GNU Linear Proramming Kit. Department for Applied Informatics, Moscow Aviation Institute, Moscow (2010)Google Scholar
  39. 39.
    Jongen, H.T., Rückmann, J.J., Stein, O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83, 145–158 (1998)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595–614 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86, 515–532 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

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

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