Journal of Global Optimization

, Volume 59, Issue 2–3, pp 459–476 | Cite as

On rigorous upper bounds to a global optimum

Article

Abstract

In branch and bound algorithms in constrained global optimization, a sharp upper bound on the global optimum is important for the overall efficiency of the branch and bound process. Software to find local optimizers, using floating point arithmetic, often computes an approximately feasible point close to an actual global optimizer. Not mathematically rigorous algorithms can simply evaluate the objective at such points to obtain approximate upper bounds. However, such points may actually be slightly infeasible, and the corresponding objective values may be slightly smaller than the global optimum. A consequence is that actual optimizers are occasionally missed, while the algorithm returns an approximate optimum and corresponding approximate optimizer that is occasionally far away from an actual global optimizer. In mathematically rigorous algorithms, objective values are accepted as upper bounds only if the point of evaluation is proven to be feasible. Such computational proofs of feasibility have been weak points in mathematically rigorous algorithms. This paper first reviews previously proposed automatic proofs of feasibility, then proposes an alternative technique. The alternative technique is tried on a test set that caused trouble for previous techniques, and is also employed in a mathematically rigorous branch and bound algorithm on that test set.

Keywords

Automatic verification Branch and bound algorithms  Interval analysis Global optimization Feasibility 

References

  1. 1.
    Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)CrossRefGoogle Scholar
  2. 2.
    Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Prog. 91(2), 239–269 (2002)CrossRefGoogle Scholar
  3. 3.
    Floudas, C.: Deterministic global optimization: advances and challenges. In: Plenary Lecture, First World Congress on Global Optimization in Engineering and Sciences, WCGO (2009)Google Scholar
  4. 4.
    Kearfott, R.B.: Rigorous global search: continuous problems. Number 13 in nonconvex optimization and its applications. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  5. 5.
    Kearfott, R.B.: On proving existence of feasible points in equality constrained optimization problems. Math. Program. 83(1), 89–100 (1998)Google Scholar
  6. 6.
    Kearfott, R.B.: GlobSol user guide. Optim. Methods Softw. 24(4–5), 687–708 (2009)CrossRefGoogle Scholar
  7. 7.
    Kearfott, R.B.: Interval computations, rigour and non-rigour in deterministic continuous global optimization. Optim. Methods Softw. 26(2), 259–279 (2011)CrossRefGoogle Scholar
  8. 8.
    Kearfott, R.B., Castille, J.M., Tyagi, G.: Assessment of a non-adaptive deterministic global optimization algorithm for problems with low-dimensional non-convex subspaces, accepted for publication in Optim. Methods Softw. 29(2), 430–441 (2014)Google Scholar
  9. 9.
    Kearfott, R.B., Muniswamy, S., Wang, Y., Li, X., Wang, Q.: On smooth reformulations and direct non-smooth computations in global optimization for minimax problems. J. Global Optim. 57(4), 1091–1111 (2013)Google Scholar
  10. 10.
    Neumaier, A.: COCONUT Web page, 2001–2003. http://www.mat.univie.ac.at/~neum/glopt/coconut
  11. 11.
    Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization, Rapport de recherche, RT-APO-10-05, IRIT. ftp://ftp.irit.fr/IRIT/APO/RT-APO-10-05.pdf (2010). Accessed March 2010
  12. 12.
    Ninin, J., Messine, F.: A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms. J. Global Optim. 50(4), 629–644 (2011)CrossRefGoogle Scholar
  13. 13.
    Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996)CrossRefGoogle Scholar
  14. 14.
    Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.-H., Nguyen, T.-V.: Benchmarking global optimization and constraint satisfaction codes. In: Bliek, C., Jermann, C., Neumaier, A. (eds.) COCOS, Volume 2861 of Lecture Notes in Computer Science, pp. 211–222. Springer, Berlin (2003)Google Scholar
  15. 15.
    Wächter, A.: Homepage of IPOPT, 2002. https://projects.coin-or.org/Ipopt
  16. 16.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: local convergence. SIAM J. Optim. 16(1), 32–48 (2005)CrossRefGoogle Scholar
  17. 17.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Louisiana at LafayetteLafayetteUSA

Personalised recommendations