Optimization under Composite Monotonic Constraints and Constrained Optimization over the Efficient Set
We present a unified approach to a class of nonconvex global optimization problems with composite monotonic constraints. (By composite monotonic function is meant a function which is the composition of a monotonic function on ℝn with a mapping from ℝn → ℝm with m ≤ n.) This class includes problems with constraints involving products of linear functions, sums of ratio functions, etc., and also problems of constrained optimization over efficient/weakly efficient points. The approach is based on transforming the problem into a monotonic optimization problem in the space ℝp, which can then be efficiently solved by recently developed techniques. Nontrivial numerical examples are presented to illustrate the practicability of the approach.
Key wordsGlobal optimization Monotonic optimization, difference of monotonic (d.m.) optimization Composite monotonic constraint Nonconvex optimization Branch-reduce-and-bound method Constrained optimization over the efficient/weakly efficient set Multiplicative constraint Sum-of-ratio constraint
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- 1.L.T. Hoai-An, N.T. Bach-Kim and T.M. Thanh: “Optimizing a monotonic function over the Pareto set”, Preprint, Department of Applied Mathematics and Informatics, Institute of Technology, Hanoi, 2004.Google Scholar
- 14.H. Konno, P.T. Thach and H. Tuy: Optimization on Low Rank Nonconvex structures, Kluwer, 1997.Google Scholar
- 17.C. Malivert: “Multicriteria fractional optimization”, in Proc. of the 2nd Catalan days on applied mathematics, M. Sofonea and J.N. Corvellac eds, (1995), 189–198.Google Scholar
- 18.C. Malivert and N. Popovici: “Bicriteria Linear Fractional Optimization”, in Optimization, V.H. Nguyen, J-J. Strodiot and P. Tossings eds, Lecture Notes in Economics and Mathematical Systems 481, Springer 2000, 305–319.Google Scholar
- 25.H. Tuy: Convex Analysis and Global Optimization, Kluwer, 1998.Google Scholar
- 28.H. Tuy: “Convexity and Monotonicity in Global Optimization”, in Advances in Convex Analysis and Optimization, N. Hadjisavvas and P.M. Pardalos eds, Kluwer, 2001, 569–594.Google Scholar
- 29.H. Tuy: “Monotonicity in the framework of generalized convexity”, in Generalized Convexity, Generalized Monotonicity and Applications, A. Eberhard, N. Hadjisavvas and D.T. Luc eds, Springer, 2005, 61–85.Google Scholar
- 31.H. Tuy, F. Al-Khayyal and P.T. Thach: “Monotonic Optimization: Branch and Cuts Methods”, in Essays and Surveys on Global Optimization, eds. C. Audet, P. Hansen, G. Savard, Kluwer, to appear.Google Scholar
- 32.H. Tuy: “Partly convex and convex-monotonic optimization problems”, in Modelling, Simulation and Optimization of Complex Processes, eds. H. G Bock, E. Kostina, H. X. Phu, R. Rannacher, Springer, 2005.Google Scholar
- 33.H. Tuy: “Polynomial Optimization: A Robust Approach”, Pacific Journal of Optimization, 2005, to appearGoogle Scholar
- 34.H. Tuy, M. Minoux and N.T. Hoai Phuong: “Discrete Monotonic Optimization With Application to A Discrete Location Problem”, SIAM Journal of Optimization, to appear.Google Scholar
- 35.P.L. Yu: ‘Multicriteria Decision Making, Plenum Press, New York, 1995.Google Scholar