EURO Journal on Computational Optimization

, Volume 4, Issue 1, pp 93–121 | Cite as

A modification of the \(\alpha \hbox {BB}\) method for box-constrained optimization and an application to inverse kinematics

  • Gabriele EichfelderEmail author
  • Tobias Gerlach
  • Susanne Sumi
Original Paper


For many practical applications it is important to determine not only a numerical approximation of one but a representation of the whole set of globally optimal solutions of a non-convex optimization problem. Then one element of this representation may be chosen based on additional information which cannot be formulated as a mathematical function or within a hierarchical problem formulation. We present such an application in the field of robotic design. This application problem can be modeled as a smooth box-constrained optimization problem. We extend the well-known \(\alpha \hbox {BB}\) method such that it can be used to find an approximation of the set of globally optimal solutions with a predefined quality. We illustrate the properties and give a proof for the finiteness and correctness of our modified \(\alpha \hbox {BB}\) method.


Non-convex programming Global optimization Optimal solution set \(\alpha \hbox {BB}\) method Robotic design 

Mathematics Subject Classification

90C26 90C30 90C90 



The authors thank the two anonymous referees for their careful reading and their helpful comments on the first version of this manuscript.


  1. Adjiman CS, Androulakis IP, Floudas CA (1998) A global optimization method, \(\alpha \text{ BB }\), for general twice-differeentiable constrained NLPs—II. Implementation and computational results. Comput Chem Eng 22(9):1159–1179CrossRefGoogle Scholar
  2. Adjiman CS, Dallwig S, Floudas CA, Neumaier A (1998) A global optimization method, \(\alpha \text{ BB }\), for general twice-differeentiable constrained NLPs–I. Theoretical advances. Comput Chem Eng 22(9):1137–1158CrossRefGoogle Scholar
  3. Androulakis IP, Floudas CA (2004) Computational experience with a new class of convex underestimators: box-constrained NLP problems. J Glob Optim 29:249–264CrossRefGoogle Scholar
  4. Androulakis IP, Floudas CA (2004) A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J Glob Optim 30:367–390CrossRefGoogle Scholar
  5. 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
  6. Al-Khayyal FA, Falk JE (1983) Jointly constrained biconvex programming. Math Oper Res 8:273–286CrossRefGoogle Scholar
  7. Csendes T, Ratz D (1997) Subdivision direction selection in interval methods for global optimization. SIAM J Numer Anal 34(3):922–938CrossRefGoogle Scholar
  8. Csendes T, Klatte R, Ratz D (2000) A posteriori direction selection rules for interval optimization methods. Cent Eur J Oper Res 8(3):225–236Google Scholar
  9. Easom EE (1990) A survey of global optimization techniques. M. Eng. thesis, University of Louisville, Louisville, KYGoogle Scholar
  10. Epitropakis MG, Plagianakos VP, Vrahatis MN (2011) Finding multiple global optima exploiting differential evolution’s niching capability. In: Proceedings of IEEE SDE, Paris, France, April 2011, pp 80–87Google Scholar
  11. Floudas CA (2000) Deterministic global optimization. Kluwer, DordrechtCrossRefGoogle Scholar
  12. Gerschgorin S (1931) Über die Abgrenzung der Eigenwerte einer Matrix. Izv Akad Nauk SSSR Ser Fiz Mat 6:749–754Google Scholar
  13. Hammer R, Hocks M, Kulisch U, Ratz D (1995) C++ toolbox for verified computing I. Springer, BerlinGoogle Scholar
  14. Hansen E (1980) Global optimization using interval analysis-the multi-dimensional case. Numer Math 34:247–270CrossRefGoogle Scholar
  15. Hansen E, Walster GW (2004) Global optimization using interval analysis, 2nd edn., revised and expanded, Marcel Dekker, New YorkGoogle Scholar
  16. Hertz D (1992) The extreme eigenvalues and stability of real symmetric interval matrices. IEEE Trans Autom Control 37:532–535CrossRefGoogle Scholar
  17. Hladík M, Daney D, Tsigaridas E (2010) Bounds on real eigenvalues and singular values of interval matrices. SIAM J Matrix Anal Appl 31(4):2116–2129CrossRefGoogle Scholar
  18. Hladík M (2015a) An extension of the \(\alpha \text{ BB }\)-type underestimation to linear parametric hessian matrices. J Glob Optim. doi: 10.1007/s10898-015-0304-5
  19. Hladík M (2015b) On the efficient Gerschgorin inclusion usage in the global optimization \(\alpha \text{ BB }\) method. J Global Optim 61(2):235–253Google Scholar
  20. Jamil M, Yang X-S, Zepernick H-J (2013) Test functions for global optimization: a comprehensive survey. In: Yang X-S et al (eds) Swarm intelligence and bio-inspired computation: theory and applications. Elsevier, London, pp 194–222Google Scholar
  21. Liu WB, Floudas CA (1993) A remark on the GOP algorithm for global optimization. J Glob Optim 3:519–521CrossRefGoogle Scholar
  22. Maranas CD, Floudas CA (1992) A global optimization approach for Lennard–Jones microclusters. J Chem Phys 97(10):7667–7678CrossRefGoogle Scholar
  23. Maranas CD, Floudas CA (1994) A deterministic global optimization approach for molecular structure determination. J Chem Phys 100(2):1247–1261CrossRefGoogle Scholar
  24. Maranas CD, Floudas CA (1994) Global minimum potential energy conformations of small molecules. J Glob Optim 4:135–170CrossRefGoogle Scholar
  25. Maranas CD, Floudas CA (1995) Finding all solutions of nonlinearly constrained systems of equitations. J Glob Optim 7:143–183CrossRefGoogle Scholar
  26. McCormick GP (1976) Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math Program 10:147–175CrossRefGoogle Scholar
  27. Mönningmann M (2008) Efficient calculations of bounds on spectra of Hessian matrices. SIAM J Sci Comput 30:2340–2357CrossRefGoogle Scholar
  28. Mönningmann M (2011) Fast calculations of spectral bounds for Hessian matrices on hyperrectangles. SIAM J Matrix Anal Appl 32(4):1351–1366CrossRefGoogle Scholar
  29. Montaz Ali M, Khomatraporn C, Zabinsky ZB (2005) A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J Glob Optim 31:635–672CrossRefGoogle Scholar
  30. Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM, Philadelphia, PACrossRefGoogle Scholar
  31. Paul RP (1982) Robot manipulators: mathematics, programming, and control. MIT Press, Cambridge, MAGoogle Scholar
  32. Rastrigin LA (1974) Systems of extremal control. Nauka, MoscowGoogle Scholar
  33. Ratz D (1992) Automatische Ergebnisverifikation bei globalen Optimierungsproblemen. Diss., Univ. KarlsruheGoogle Scholar
  34. Rohn J (1994) Positive definiteness and stability of interval matrices. SIAM J Matrix Anal Appl 15(1):175–184CrossRefGoogle Scholar
  35. Rump SM (1999) INTLAB—INTerval LABoratory. In: Csendes T (ed) Developments in reliable computing. Kluwer Academic Publishers, Dordrecht, pp 77–104CrossRefGoogle Scholar
  36. Schulze Darup M, Kastsian M, Mross S, Mönningmann M (2014) Efficient computation of spectral bounds for Hessian matrices on hyperrectangles for global optimization. J Glob Optim 58:631–652CrossRefGoogle Scholar
  37. Skjäl A, Westerlund T, Misener R, Floudas CA (2012) A generalization of the classical \(\alpha \text{ BB }\) convex underestimation via diagonal and non-diagonal quadratic terms. J Optim Theory Appl 154(2):462–490CrossRefGoogle Scholar
  38. Skjäl A, Westerlund T (2014) New methods for calculating \(\alpha \text{ BB }\)-type underestimators. J Glob Optim 58:411–427CrossRefGoogle Scholar

Copyright information

© EURO - The Association of European Operational Research Societies 2015

Authors and Affiliations

  • Gabriele Eichfelder
    • 1
    Email author
  • Tobias Gerlach
    • 1
  • Susanne Sumi
    • 2
  1. 1.Institute for MathematicsTechnische Universität IlmenauIlmenauGermany
  2. 2.Technical Mechanics GroupTechnische Universität IlmenauIlmenauGermany

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