Journal of Global Optimization

, Volume 22, Issue 1–4, pp 49–57 | Cite as

A class of problems where dual bounds beat underestimation bounds

  • Mirjam Dür
Article

Abstract

We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality.

Nonconvex duality Dual bounds Convex underestimation 

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References

  1. Bomze, I.M. (1998), On Standard Quadratic Optimization Problems. Journal of Global Optimization, 13: 369–387.Google Scholar
  2. Dür, M., Dual Bounding Procedures Lead to Convergent Branch-and-Bound Algorithms. Forthcoming in Mathematical Programming, 2001.Google Scholar
  3. Dür, M. and Horst, R. (1997), Lagrange-Duality and Partitioning Techniques in Nonconvex Global Optimization. Journal of Optimization Theory and Applications, 95: 347–369.Google Scholar
  4. Falk, J.E. (1969), Lagrange Multipliers and Nonconvex Programs. SIAM Journal on Control, 7: 534–545, 1969.Google Scholar
  5. Geoffrion, A.M. (1971), Duality in Nonlinear Programming: A Simplified Application-Oriented Development. SIAM Review, 13: 1–37.Google Scholar
  6. Horst, R. and Tuy, H. (1996), Global Optimization. Springer, Berlin.Google Scholar
  7. Nowak, I. (2000), Dual Bounds and Optimality Cuts for All-Quadratic Programs with Convex Constraints. Journal of Global Optimization 18: 337–356.Google Scholar
  8. Thoai, N.V. On Convergence and application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization. Forthcoming in Journal of Optimization Theory and Applications, 2001.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Mirjam Dür
    • 1
  1. 1.Department of StatisticsVienna University of Economics and Business AdministrationViennaAustria

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