Abstract
Monotonic optimization is concerned with optimization problems dealing with multivariate monotonic functions and differences of monotonic functions. For the study of this class of problems a general framework (Tuy, 2000a) has been earlier developed where a key role was given to a separation property of solution sets of monotonic inequalities similar to the separation property of convex sets. In the present paper the separation cut is combined with other kinds of cuts, called reduction cuts, to further exploit the monotonic structure. Branch and cuts algorithms based on an exhaustive rectangular partition and a systematic use of cuts have proved to be much more efficient than the original polyblock and copolyblock outer approximation algorithms.
Keywords
- Feasible Solution
- Global Optimization
- Separation Property
- Outer Approximation
- Approximate Optimal Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Floudas, C.A. (2000). Deterministic Global Optimization. Theory, Methods and Applications. Nonconvex Optimization and its Applications, vol. 37. Kluwer Academic Publishers, Dordrecht.
Floudas, C.A. et al. (1999). Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and its Applications, vol. 33, Kluwer Academic Publishers, Dordrecht.
Luc L.T. (2001). Reverse polyblock approximation for optimization over the weakly efficient set and efficient set. Acta Mathematica Vietnamica, 26(1):65–80.
Murtagh, B.A. and Saunders, S.N. (1983). MINOS 5.4 User's Guide. Systems Optimization Laboratory, Department of Operations research, Stanford University.
Phuong, Ng.T.H. and H. Tuy (2002). A monotonicity based approach to nonconvex quadratic minimization. Vietnam Journal of Mathematics 30(4):373–393.
Phuong, Ng.T.H. and H. Thy, H. (2003). A unified approach to generalized fractional programming. Journal of Global Optimization, 26:229–259.
Rubinov, A., Tuy, H., and Mays, H. (2001). Algorithm for a monotonic global optimization problem. Optimization, 49:205–221.
Tuy, H. (1987). Convex programs with an additional reverse convex constraint. Journal of Optimization Theory and Applications, 52:463–486.
Tuy, H. (1995). D.C. optimization: Theory, methods and algorithms. In: R. Horst and P. Pardalos (eds.), Handbook of Global Optimization, pp. 149–216. Kluwer Academic Publishers, Dordrecht.
Tuy, H. (1998). Convex Analysis and Global Optimization. Nonconvex Optimization and its Applications, vol. 22. Kluwer Academic Publishers, Dordrecht.
Tuy, H. (1999). Normal sets, polyblocks and monotonic optimization. Vietnam Journal of Mathematics, 27:277–300.
Tuy, H. (2000a). Monotonic optimization: Problems and solution approaches. SIAM Journal on Optimization, 11(2):464–494.
Tuy, H. (2000b). Which Functions Are D.M.? Preprint, Institute of Mathematics, Hanoi.
Tuy, H. (2001) Convexity and monotonicity in global optimization. In: N. Hadjisavvas and P.M. Pardalos (eds.), Advances in Convex Analysis and Global Optimization, pp. 569–594. Kluwer Academic Publishers, Dordrecht.
Tuy, H. and Luc, L.T. (2000). A new approach to optimization under monotonic constraint. Journal of Global Optimization, 18:1–15.
Tuy, H. and Nghia, Ng.D. (2003). Reverse polyblock approximation for generalized multiplicative/fractional programming. Vietnam Journal of Mathematics, 31:391–402.
Tuy, H., Thach, P.T., and Konno, H. (2004). Optimization of polynomial fractional functions. Journal of Global Optimization, 29(1):19–44.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Tuy, H., Al-Khayyal, F., Thach, P.T. (2005). Monotonic Optimization: Branch and Cut Methods. In: Audet, C., Hansen, P., Savard, G. (eds) Essays and Surveys in Global Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25570-2_2
Download citation
DOI: https://doi.org/10.1007/0-387-25570-2_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-25569-9
Online ISBN: 978-0-387-25570-5
eBook Packages: Business and EconomicsBusiness and Management (R0)