Journal of Global Optimization

, Volume 30, Issue 4, pp 367–390 | Cite as

A New Class of Improved Convex Underestimators for Twice Continuously Differentiable Constrained NLPs

  • Ioannis G. Akrotirianakis
  • Christodoulos A. Floudas


We present a new class of convex underestimators for arbitrarily nonconvex and twice continuously differentiable functions. The underestimators are derived by augmenting the original nonconvex function by a nonlinear relaxation function. The relaxation function is a separable convex function, that involves the sum of univariate parametric exponential functions. An efficient procedure that finds the appropriate values for those parameters is developed. This procedure uses interval arithmetic extensively in order to verify whether the new underestimator is convex. For arbitrarily nonconvex functions it is shown that these convex underestimators are tighter than those generated by the αBB method. Computational studies complemented with geometrical interpretations demonstrate the potential benefits of the proposed improved convex underestimators.

αBB convex underestimators global optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adjiman, C.S., Androulakis, I. and Floudas, C.A. (1998a), A global optimization method, αBB, for general twice-differentiable constrained NLPs-II. Implementation and computational results, Computers and Chemical Engineering, 22, 1159–1179.Google Scholar
  2. Adjiman, C.S., Dallwig, S., Floudas, C.A. and Neumaier, A. (1998b), A global optimization method, αBB, for general twice-differentiable constrained NLPs-I: Theoretical aspects, Computers and Chemical Engineering, 22(9), 1137–1158.Google Scholar
  3. Akrotirianakis, I.G. and Floudas, C.A. (2004), Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP problems, Submitted for publication, Journal of Global Optimization.Google Scholar
  4. Al-Khayyal, F.H. and Falk, J.E. (1983), Jointly constrained biconvex programming, Mathematics of Operations Research, 8, 523.Google Scholar
  5. Boggs, P.T. and Tolle, J.W. (1995), Sequential quadratic programming, Acta Numerica, 4, 1–52.Google Scholar
  6. Dixon, L.C.W. and Szego, G.P. (1975), Towards global optimization, In: Proceedings of a Workshop at the University of Cagliari, Italy: North-Holland.Google Scholar
  7. Floudas, C.A. (2000), Deterministic Global Optimization, Theory, Methods and Applications, Kluwer Academic Publishers.Google Scholar
  8. Floudas, C.A., Pardalos, P.M., Adjiman C.S., Esposito W.R., Gumus Z.H., Harding S.T., Klepeis J.L., Meyer C.A. and Schweiger, C.A. (1999), Handbook of Test Problems in Local and Global Optimization, Dordrecht, The Netherlands, Kluwer Academic Publishers.Google Scholar
  9. Gelatt, C.D., Kirkpatric, S. and Vecchi, M.P. (1983), Optimization by simulated annealing, Science, 220, 671.MathSciNetGoogle Scholar
  10. Goldberg, D.E. (1987), Genetic Algorithms in Search, Optimization and Machine Learning, New York, NY, Addison-Welsey.Google Scholar
  11. Goldstein, A. and Price, J. (1971), On descent from local minima, Mathematics of Computation, 25, 569–574.Google Scholar
  12. Hansen, E. (1992), Global Optimization using Interval Analysis, New York, M. Dekker.Google Scholar
  13. Horst, R. and Tuy, H. (1987), On the convergence of global methods in multiextrimal optimization, Journal of Optimization Theory and Applications, 54, 283.Google Scholar
  14. Maranas, C.D. and Floudas, C.A. (1994a), A deterministic global optimization approach for molecular structure determination, Journal of Chemical Physics, 100(2), 1247–1261.Google Scholar
  15. Maranas, C.D. and Floudas, C.A. (1994b), Global minimum potential energy conformations for small molecules, Journal of Global Optimization, 4, 135–170.Google Scholar
  16. Murty, K.G. and Kabadi S.N. (1987), Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39, 117–129.Google Scholar
  17. Neumaier, A. (1990), Interval Methods for Systems of Equations, Cambridge University Press.Google Scholar
  18. Pardalos, P.M. and Schnitger, G. (1988), Checking local optimality in constrained quadratic programming, Operations Research Letters, 7, 33–35.Google Scholar
  19. Porn, R., Harjunkoski, I. and Westerlund, T. (1999), Convexification of different classes of nonconvex MINLP problems, Computers and Chemical Engineering, 23, 439–448.Google Scholar
  20. Rinnoy-Kan, A.H.G. and Timmer, G.T. (1987a), Stochastic global optimization methods. Part I: Clustering methods, Mathematical Programming, 39, 27–56.Google Scholar
  21. Rinnoy-Kan, A.H.G. and Timmer, G.T. (1987b), Stochastic global optimization, Part II: Multi-livel Methods, Mathematical Programming, 39, 57–78.Google Scholar
  22. Ryoo, H.S. and Sahinidis, N.V. (1996), A branch-and-reduce approach to global optimization, Journal of Global Optimization, 8(2), 107–139.Google Scholar
  23. Schoen, F. (1991), Stochastic techniques for global optimization: A survey of recent advances, Journal of Global Optimization, 1(3), 207–228.Google Scholar
  24. Sherali, H.D. and Alameddine, A. (1992), A new reformulation linearization technique for bilinear programming problems, Journal of Global Optimization, 2(4), 379.Google Scholar
  25. Smith, E.M.B. and Pantelides C.C. (1996), Global optimization for general process models. In: Grossmann I.E. (ed.), Global Optimization in Engineering Design, Kluwer Academic Publishers, pp. 355–386.Google Scholar
  26. Tuy, H. (1987), Global minimum of the difference of two convex functions, Mathematical Programming Study, 30, 150.Google Scholar
  27. Wright, M.H. (1992), Interior point methods for constrained optimization, Acta Numerica, 1, 341–407.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Ioannis G. Akrotirianakis
    • 1
  • Christodoulos A. Floudas
    • 1
  1. 1.Department of Chemical EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations