Abstract
We propose unified branch-and-bound and cutting plane algorithms for global minimization of a functionf(x, y) over a certain closed set. By formulating the problem in terms of two groups of variables and two groups of constraints we obtain new relaxation bounding and adaptive branching operations. The branching operation takes place in y-space only and uses the iteration points obtained through the bounding operation. The cutting is performed in parallel with the branch-and-bound procedure. The method can be applied implementably for a certain class of nonconvex programming problems.
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References
Al-Khayyal, F. A. and Falk, J. E. (1983), Jointly constrained biconvex programming,Mathematics of Operations Research 8, 273–286.
Benders, J. F. (1962), Partitioning procedures for solving mixed-variables programming problems,Numerische Mathematik 4, 238–252.
Benson, H. P. (1982), On the convergence of two branch-and-bound algorithms for nonconvex programming problems,Journal of Optimization Theory and Applications 36, 129–134.
Blum, E. and Oettli, W. (1975),Mathematische Optimierung, Springer, Berlin.
Cheney, E. W. and Goldstein, A. A. (1959), Newton's method for convex programming and Tchebycheff approximation,Numerische Mathematik 1, 253–268.
Dantzig, G. B. and Wolfe, P. (1960), Decomposition principle for linear programs,Operations Research 8, 101–111.
Falk, J. E. and Soland, R. M. (1969), An algorithm for separable nonconvex programming problems,Management Science 15, 550–569.
Fukushima, M. (1983), An outer approximation algorithm for solving general convex programs,Operations Research 31, 101–113.
Fukushima, M. (1984), On the convergence of a class of outer approximation algorithms for convex programs,Journal of Computational and Applied Mathematics 10, 147–156.
Fukushima, M. (1984), A descent algorithm for nonsmooth convex optimization,Mathematical Programming 30, 163–175.
Geoffrion, A. (1970), Elements of large-scale mathematical programming, Part I: Concepts,Management Science 16, 652–675.
Horst, R. (1976), An algorithm for nonconvex programming problems,Mathematical Programming 10, 312–321.
Horst, R., de Vries, J., and Thoai, N. V. (1988), On finding new vertices and redundant constraints in cutting plane algorithms for global optimization,Operations Research Letters 7, 85–90.
Horst, R. (1988), Deterministic global optimization with partition sets whose feasibility is not known. Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization,Journal of Optimization Theory and Applications 58, 11–37.
Horst, R. and Tuy, H. (1987), On the convergence of global methods in multiextremal optimization,Journal of Optimization Theory and Applications 54, 253–271.
Horst, R. and Tuy, H. (1990),Global Optimization, Springer, Berlin; second edition 1992.
Kelley, J. E. (1960), The cutting-plane method for solving convex programs,Journal of the SIAM 8, 703–712.
Kiwiel, K. C. (1985),Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics1133, Springer, Berlin.
Mayne, D. Q. and Polak, E. (1984), Outer approximation algorithm for nondifferentiable optimization problems,Journal of Optimization Theory and Applications 42, 19–30.
McCormick, G. P. (1976), Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems,Mathematical Programming 10, 147–175.
Muu, L. D. (1985), A convergent algorithm for solving linear programs with an additional reverse convex constraint,Kybernetika (Prague) 21, 428–435.
Muu, L. D. and Oettli, W. (1991), An algorithm for indefinite quadratic programming with convex constraints,Operations Research Letters 10, 323–327.
Muu, L. D. and Oettli, W. (1991), A method for minimizing a convex-concave function over a convex set,Journal of Optimization Theory and Applications 70, 377–384.
Pardalos, P. M. (1987), Generation of large-scale quadratic programs for use as global optimization test problems,ACM Transactions on Mathematical Software 13, 133–137.
Pardalos, P. M. and Rosen, J. B. (1987),Constrained Global Optimization: Algorithms and Applications. Lecture Notes in Computer Science268.
Rosen, J. B. and Pardalos, P. M. (1986), Global minimization of large-scale constrained concave quadratic problems by separable programming,Mathematical Programming 34, 163–174.
Soland, R. M. (1971), An algorithm for separable nonconvex programming problems II: Nonconvex constraints,Management Science 17, 759–773.
Topkis, D. M. (1970), Cutting-plane methods without nested constraint sets,Operations Research 18, 404–413.
Thoai, N. V. and Tuy, H. (1980), Convergent algorithms for minimizing a concave function,Mathematics of Operations Research 5, 556–566.
Tuy, H. (1987), Global minimization of a difference of two convex functions,Mathematical Programming Study 30, 150–182.
Tuy, H. and Horst, R. (1988), Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and d.c. optimization problems,Mathematical Programming 41, 161–183.
Tuy, H., Thieu, T. V., and Thai, N. Q. (1985), A conical algorithm for globally minimizing a concave function over a closed convex set,Mathematics of Operations Research 10, 498–514.
Veinott, A. F. (1967), The supporting hyperplane method for unimodal programming,Operations Research 15, 147–152.
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On leave from Institute of Mathematics, Hanoi, by a grant from Alexander-von-Humboldt-Stiftung.
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Muu, L.D., Oettli, W. Combined branch-and-bound and cutting plane methods for solving a class of nonlinear programming problems. J Glob Optim 3, 377–391 (1993). https://doi.org/10.1007/BF01096777
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DOI: https://doi.org/10.1007/BF01096777
Key words
- Branch-and-bound
- cutting plane
- decomposition
- convex-concave function
- global
- optimization