Abstract
A branch and bound global optimization method,αBB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bilinear, fractional, signomial) with customized tight convex lower bounding functions and (ii) by utilizing the α parameter as defined in [17] to underestimate nonconvex terms of generic structure. The proposed branch and bound type algorithm attains finiteε-convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems. The global optimization method,αBB, is implemented in C and tested on a variety of example problems.
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References
F.A. Al-Khayyal. Jointly Constrained Bilinear Programs and Related Problems.Int. J. Comp. Math., 19(11):53, 1990.
F.A. Al-Khayyal and J.E. Falk. Jointly constrained biconvex programming.Math. Opers. Res., 8:523, 1983.
I.P. Androulakis and V. Venkatasubramanian. A Genetic Algorithmic Framework for Process Design and Optimization.Comp. chem. engng., 15(4):217, 1991.
A. Ben-Tal, G. Eiger, and V. Gershovitz. Global Optimization by Reducing the Duality Gap.in press, Math. Prog., 1994.
C.A. Floudas and P.M. Pardalos.Recent advances in global optimization Princeton Series in Computer Science. Princeton University Press, Princeton, New Jersey, 1992.
C.A. Floudas and V. Visweswaran. A global optimization algorithm (GOP) for certain classes of nonconvex NLPs: I. Theory.Comput. chem. Engng., 14(12):1397, 1990.
C.A. Floudas and V. Visweswaran. A primal-relaxed dual global optimization approach.J. Opt. Th. Appl., 78(2):187, 1993.
E.R. Hansen.Global Optimization Using Interval Analysis. Marcel Dekkar, New York, NY, 1992.
C.A. Haverly. Studies of the Behavior of Recursion for the Pooling Problem.SIGMAP Bulletin, 25:19, 1978.
D.E. Goldberg.Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, New York, NY, 1989.
R. Horst and P.M. Pardalos.Handbook of Global Optimization: Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, 1994.
P. Hansen, B. Jaumard, and S. Lu. Global Optimization of Univariate Lipschitz Functions: I. Survey and Properties.Math. Prog., 55:251, 1992a.
P. Hansen, B. Jaumard, and S. Lu. Global Optimization of Univariate Lipschitz Functions: II. New Algorithms and Computational Comparison.Math. Prog., 55:273, 1992b.
R. Horst, N.V. Thoai, and J. De Vries. A New Simplicial Cover Technique in Constrained Global Optimization.J. Global Opt., 2:1, 1992.
R. Horst and H. Tuy. On the Convergence of Global Methods in Multiextremal Optimization.Opt. Th. Appl., 54:283, 1987.
R. Horst and H. Tuy.Global Optimization. Springer-Verlag, Berlin, Germany, 1990.
C.D. Maranas and C.A. Floudas. Global Minimum Potential Energy Conformations of Small Molecules.J. Glob. Opt., 4:135, 1994a.
C.D. Maranas and C.A. Floudas. Global Optimization in Generalized Geometric Porgramming.submitted to Comp. chem. Engnr., 1994c.
C.D. Maranas and C.A. Floudas. Finding All Solutions of Nonlinearly Constrained Systems of Equations.submitted to J. Global Opt., 1995a.
C.M. McDonald and C.A. Floudas. Decomposition Based and Branch and Bound Global Optimization Approaches for the Phase Equilibrium Problem.J. Global Opt., 5:205, 1994.
R. Moore, E. Hansen and A. Leclerc. Rigorous Methods for Global Optimization. InRecent advances in global optimization Princeton Series in Computer Science. Princeton University Press, Princeton, New Jersey, 1992.
B.A. Murtagh and M.A. Saunders.MINOS5.0 Users Guide. Systems Optimization Laboratory, Dept. of Operations Research, Stanford University, CA., 1983. Appendix A: MINOS5.0, Technical Report SOL 83-20.
P.M. Pardalos and J.B. Rosen.Constrained global optimization: Algorithms and applications, volume 268of Lecture Notes in Computer Science. Springer Verlag, Berlin, Germany, 1987.
H. Ratschek and J. Rokne.New Computer Methods for Global Optimization. Halsted Press, Chichester, Great Britain, 1988.
A.H.G. Rinnoy-Kan and G.T. Timmer. Stochastic Global Optimization Methods. Part I: Clustering Methods.Math. Prog., 39:27, 1987.
C.D. Gelatt, S. Kirkpatrick and M.P. Vecchi.Science, 220:671, 1983.
H.D. Sherali and A. Alameddine. A New Reformulation Linearization Technique for Bilinear Programming Problems.J. Global Opt., 2(4):379, 1992.
H.D. Sherali and C.H. Tuncbilek. A Global Optimization Algorithm for Polynomial Programming Using a Reformulation-Linearization Technique.J. Global Opt., 2:101, 1992.
N.Z. Shor. Dual Quadratic Estimates in Polynomial and Boolean Programming.Annals of Operations Research, 25:163, 1990.
A. Torn and A. Zilinskas.Global Optimization, volume 350 ofLecture Notes in Computer Science. Springer-Verlag, Berlin, Germany, 1989.
H. Tuy. Global Minimum of the Difference of Two Convex Functions.Mathematical Programming Study, 30:150, 1987.
H. Tuy, T.V. Thieu, and N.Q. Thai. A Conical Algorithm for Globally Minimizing A Concave Function over a Closed Convex Set.Mathematics of Operations Research, 10:498, 1985.
V. Visweswaran and C.A. Floudas. New properties and computational improvement of the GOP algorithm for problems with quadratic objective function and constraints.J. Global Opt., 3(3):439, 1993.
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Androulakis, I.P., Maranas, C.D. & Floudas, C.A. αBB: A global optimization method for general constrained nonconvex problems. J Glob Optim 7, 337–363 (1995). https://doi.org/10.1007/BF01099647
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DOI: https://doi.org/10.1007/BF01099647