Abstract
A general branch-and-bound conceptual scheme for global optimization is presented that includes along with previous branch-and-bound approaches also grid-search techniques. The corresponding convergence theory, as well as the question of restart capability for branch-and-bound algorithms used in decomposition or outer approximation schemes are discussed. As an illustration of this conceptual scheme, a finite branch-and-bound algorithm for concave minimization is described and a convergent branch-and-bound algorithm, based on the previous one, is developed for the minimization of a difference of two convex functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V.T. Ban, “A finite algorithm for minimizing a concave function under linear constraints and its application,” IFIP Working Conference on Recent Advances on System Modeling and Optimization (Hanoi, 1983).
C. Berge,Théorie des Graphes et ses Applications (Dunod, Paris, 1958).
M.C. Böhringer and S.E. Jacobsen, “Convergent Cutting Planes for Linear Programs with Additional Reverse Convex Constraints,” Lecture Notes in Control and Information Science 59 (Springer, Verlag, Berlin, Heidelberg, New York, Tokyo, 1984) pp. 263–272.
V.P. Bulatov,Embedding Methods in Optimization Problems (in Russian) (Nauka, Novosibirsk, 1977).
J.E. Falk and K.R. Hoffman, “A successive underestimation method for concave minimization problems,”Mathematics of Operations Research 1 (1976) 251–259.
J.E. Falk and R.M. Soland, “An algorithm for separable nonconvex programming problems,”Management Science 15 (1969) 550–569.
E.A. Galperin, “The cubic algorithm,”Journal of Mathematical Analysis and Applications 112 (1985) 635–640.
R.J. Hillestad and S.E. Jacobsen, “Reverse convex programming,”Applied Mathematics and Optimization 6 (1980) 63–78.
R.J. Hillestad and S.E. Jacobsen, “Linear programs with an additional reverse convex constraint,”Applied Mathematics and Optimization 6 (1980) 257–269.
K.L. Hoffmann, “A method for globally minimizing concave functions over convex sets,”Mathematical Programming 20 (1981) 22–32.
R. Horst, “An algorithm for nonconvex programming problems,”Mathematical Programming 10 (1976) 312–321.
R. Horst, “A note on the convergence of an algorithm for nonconvex programming problems,”Mathematical Programming 19 (1980) 237–238.
R. Horst, “On the global minimization of concave functions-introduction and survey,”Operations Research Spektrum 6 (1984) 195–205.
R. Horst, “A general class of branch-and-bound-methods in global optimization with some new approaches for concave minimization,”Journal of Optimization Theory and Applications 51 (1986) 271–291.
R. Horst, “Deterministic global optimization with partition sets whose feasibility is not known. Application to concave minimization, reverse convex constraints, d.c. programming and Lipschitzian optimization,”Journal of Optimization Theory and Applications (forthcoming).
R. Horst and N.V. Thoai, “Branch-and-bound methods for solving systems of Lipschitzian equations and inequalities,”Journal of Optimization Theory and Applications (forthcoming).
R. Horst and H. Tuy, “On the convergence of global methods in multiextremal optimization,”Journal of Optimization Theory and Applications 54 (1987), 253–271.
D.Q. Mayne and E. Polak, “Outer approximation algorithm for nondifferentiable optimization problems,”Journal of Optimization Theory and Applications 42 (1984) 19–30.
P.M. Pardalos and J.B. Rosen, “Constrained Global Optimization: Algorithms and Applications,” Lecture Notes in Computer Science 258 (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1987).
J. Pinter, “Globally convergent methods forn-dimensional multiextremal optimization,”Optimization 17 (1986) 187–202.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, New Jersey, 1970).
J.B. Rosen, “Global minimization of a linearly constrained concave function by partition of feasible domain,”Mathematics of Operations Research 8 (1983) 215–230.
R.G. Strongin,Numerical Methods for Multiextremal Problems (in Russian) (Nauka, Moscow 1978).
P.T. Thach and H. Tuy, “Global optimization under Lipschitzian constraints,”Japanese Journal of Applied Mathematics 4 (1987) 205–217.
Ng.V. Thoai and H. Tuy, “Convergent algorithms for minimizing a concave function,”Mathematics of Operations Research 5 (1980) 556–566.
T.V. Thieu, B.T. Tam and V.T. Ban, “An outer approximation method for globally minimizing a concave function over a compact convex set,”IFIP Working Conference on Recent Advances in System Modelling and Optimization (Hanoi, 1983).
Ng.V. Thuong and H. Tuy,A Finite Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint,” Lecture Notes in Economics and Mathematical Systems 225 (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984) pp. 291–302.
H. Tuy and Ng. V. Thuong, “Minimizing a convex function over the complement of a convex set,”Methods of Operations Research 49 (1985) 85–99.
H. Tuy, “Concave programming under linear constraints,”Doklady Akademic Nauk 159 (1964) 32–35. TranslatedSoviet Mathematics 5 (1964) 1437–1440.
H. Tuy, “On outer approximation methods for solving concave minimization problems,”Acta Mathematica Vietnamica 8 (2) (1983) 3–34.
H. Tuy, “Convex programs with an additional reverse convex constraint,”Journal of Optimization Theory and Applications 52 (1987) 463–486.
H. Tuy, T.V. Thieu and Ng. Q. Thai, “A conical algorithm for globally minimizing a concave function over a closed convex set,”Mathematics of Operations Research 10 (1985) 489–514.
H. Tuy, “Global minimization of a difference of two convex functions,” in: G. Hammer and D. Pallaschke, eds.Selected Topics in Operations Research and Mathematical Economics Lecture Notes in Economics and Mathematical Systems 226 (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985) pp. 98–118.
H. Tuy, “A general deterministic approach to global optimization via d.c. programming,” in: J.B. Hiriart-Urruty, ed.Fermat Days 1985: Mathematics for Optimization (North-Holland, Amsterdam, 1986) pp. 137–162.
H. Tuy, “Concave Programming and extensions: a survey,”Optimization (forthcoming).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tuy, H., Horst, R. 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 (1988). https://doi.org/10.1007/BF01580762
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580762