, Volume 23, Issue 2, pp 591–616 | Cite as

Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints

  • Peter Kirst
  • Oliver Stein
  • Paul Steuermann
Original Paper


We discuss some difficulties in determining valid upper bounds in spatial branch-and-bound methods for global minimization in the presence of nonconvex constraints. In fact, two examples illustrate that standard techniques for the construction of upper bounds may fail in this setting. Instead, we propose to perturb infeasible iterates along Mangasarian–Fromovitz directions to feasible points whose objective function values serve as upper bounds. These directions may be calculated by the solution of a single linear optimization problem per iteration. Preliminary numerical results indicate that our enhanced algorithm solves optimization problems where a standard branch-and-bound method does not converge to the correct optimal value.


Branch-and-bound Convergence Consistency  Mangasarian–Fromovitz constraint qualification 

Mathematics Subject Classification




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


  1. Adjiman CS, Dallwig S, Floudas CA, Neumaier A (1998) A global optimization method, \(\alpha \text{ BB }\), for general twice-differentiable constrained NLPs-I. Theoretical advances. Comput Chem Eng 22:1137–1158CrossRefGoogle Scholar
  2. Adjiman CS, Androulakis IP, Floudas CA (1998) A global optimization method, \(\alpha \text{ BB }\), for general twice-differentiable constrained NLPs-II. Implementation and computational results. Comput Chem Eng 22:1159–1179CrossRefGoogle Scholar
  3. Androulakis IP, Maranas CD, Floudas CA (1995) \(\alpha \text{ BB }\): a global optimization method for general constrained nonconvex problems. J Glob Optim 7:337–363CrossRefGoogle Scholar
  4. Baumann E (1988) Optimal centered forms. BIT Numer Math 28:80–87CrossRefGoogle Scholar
  5. Belotti P, Lee J, Liberti L, Margot F, Wächter A (2009) Branching and bounds tightening techniques for non-convex MINLP. Optim Methods Softw 24:597–634CrossRefGoogle Scholar
  6. Berthold T, Gleixner AM (2014) Undercover: a primal MINLP heuristic exploring a largest sub-MIP. Math Program 144:315–346CrossRefGoogle Scholar
  7. Berthold T, Heinz S, Pfetsch ME, Vigerske S (2011) Large neighborhood search beyond MIP. In: di Gaspar L (eds) Proceedings of the 9th metaheuristics international conference, pp 51–60Google Scholar
  8. Bonami P, Biegler LT, Conn AR, Cornuéjols G, Grossmann IE, Laird CD, Lee J, Lodi A, Margot F, Sawaya N, Wächter A (2008) An algorithmic framework for convex mixed integer nonlinear programs. Discret Optim 5:186–204CrossRefGoogle Scholar
  9. Cafieri S, Lee J, Liberti L (2010) On convex relaxations of quadrilinear terms. J Glob Optim 47:661–685CrossRefGoogle Scholar
  10. D’Ambrosio C, Frangioni A, Liberti L, Lodi A (2012) A storm of feasibility pumps for nonconvex MINLP. Math Program 136:375–402CrossRefGoogle Scholar
  11. Domes F, Neumaier A (2015) Rigorous verification of feasibility. J Glob Optim 61:255–278CrossRefGoogle Scholar
  12. Dür M (2001) Dual bounding procedures lead to convergent branch-and-bound algorithms. Math Program 91:117125Google Scholar
  13. Dür M (2002) A class of problems where dual bounds beat underestimation bounds. J Glob Optim 22:4957CrossRefGoogle Scholar
  14. Floudas CA (2000) Deterministic global optimization. Theory, methods, and applications. Kluwer, DordrechtCrossRefGoogle Scholar
  15. Geißler B, Martin A, Morsi A, Schewe L (2012) Using piecewise linear functions for solving MINLPs. In: Lee J, Leyffer S (eds) Mixed integer nonlinear programming. Springer, Berlin, pp 287–314CrossRefGoogle Scholar
  16. Hock W, Schittkowski K (1981) Test examples for nonlinear programming codes. Springer, BerlinCrossRefGoogle Scholar
  17. Horst R, Tuy H (1996) Global optimization. Deterministic approaches. Springer, BerlinCrossRefGoogle Scholar
  18. Jongen HTh, Jonker P, Twilt F (1986) Critical sets in parametric optimization. Math Program 34:333–353CrossRefGoogle Scholar
  19. Kearfott RB (1998) On proving existence of feasible points in equality constrained optimization problems. Math Program 83:89–100Google Scholar
  20. Kearfott RB (2014) On rigorous upper bounds to a global optimum. J Glob Optim 59:459–476CrossRefGoogle Scholar
  21. Kelley Jr JE (1960) The cutting-plane method for solving convex programs. J Soc Ind Appl Math 8:703–712CrossRefGoogle Scholar
  22. Knüppel O (1994) PROFIL/BIAS-a fast interval library. Computing 53:277–287CrossRefGoogle Scholar
  23. Krawczyk R, Nickel K (1982) Die zentrische Form in der Intervallarithmetik, ihre quadratische Konvergenz und ihre Inklusionsisotonie. Computing 28:117–137CrossRefGoogle Scholar
  24. Liberti L, Pantelides CC (2003) Convex envelopes of monomials of odd degree. J Glob Optim 25:157–168CrossRefGoogle Scholar
  25. Makhorin A (2010) GNU linear programming kit. Department for Applied Informatics, Moscow Aviation Institute, MoscowGoogle Scholar
  26. McCormick GP (1983) Nonlinear programming: theory, algorithms and applications. Wiley, New YorkGoogle Scholar
  27. Neumaier A, Shcherbina O (2015) The COCONUT benchmark. Accessed 22 July 2014
  28. Misener R, Floudas CA (2010) Piecewise-linear approximations of multidimensional functions. J Optim Theory Appl 145:120–147CrossRefGoogle Scholar
  29. Misener R, Floudas CA (2013) GloMIQO: global mixed-integer quadratic optimizer. J Glob Optim 57:3–30CrossRefGoogle Scholar
  30. Misener R, Floudas CA (2013) Mixed-integer nonlinear optimization problems: ANTIGONE 1.0 test suite. Accessed 22 July 2014
  31. Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, CambridgeGoogle Scholar
  32. Ninin J, Messine F (2014) A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms. Glob Optim 50:629–644CrossRefGoogle Scholar
  33. Pintér J (1988) Branch and bound algorithms for solving global optimization problems with Lipschitzian structure. Optimization 19:101–110CrossRefGoogle Scholar
  34. Paulavičius R, Žilinskas J (2014) Simplicial global optimization. Springer, BerlinCrossRefGoogle Scholar
  35. Rockafellar RT, Wets RJB (1998) Variational analysis. Springer, BerlinCrossRefGoogle Scholar
  36. Sahinidis NV (1996) BARON: a general purpose global optimization software package. J Glob Optim 8:201–205CrossRefGoogle Scholar
  37. Smith EM, Pantelides CC (1997) Global optimization of nonconvex MINLPs. Comput Chem Eng 21:791–796CrossRefGoogle Scholar
  38. Smith EM, Pantelides CC (1999) A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput Chem Eng 23:457–478CrossRefGoogle Scholar
  39. Tawarmalani M, Sahinidis NV (2004) Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math Program 99:563–591CrossRefGoogle Scholar
  40. Tuy H (2005) Polynomial optimization: a robust approach. Pac J Optim 1:357–374Google Scholar
  41. Tuy H (2010) \({\cal D\cal ({\cal C}})\)-optimization and robust global optimization. J Glob Optim 47:485–501Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2015

Authors and Affiliations

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

Personalised recommendations