Abstract
An algorithm is proposed for globally minimizing a concave function over a compact convex set. This algorithm combines typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided. In each step, essentially only few linear programming problems have to be solved. Some preliminary computational results are reported.
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References
H.P. Benson, “On the convergence of two branch-and bound algorithms for nonconvex programming problems,”Journal of Optimization Theory and Applications 36 (1982) 129–134.
K.L. Hoffman, “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, d.c. programming, reverse convex constraints and Lipschitzian optimization,”Journal of Optimization Theory and Applications 58 (1988) 11–37.
R. Horst, “On consistency of bounding operations in deterministic global optimization,”Journal of Optimization Theory and Applications 61 (1989) 143–146.
R. Horst, “Deterministic global optimization: some recent advances and new fields of applications,”Naval Research Logistics 37 (1990) 433–471.
R. Horst and N.V. Thoai, “Branch-and-bound methods for solving systems of Lipschitzian equations and inequalities,”Journal of Optimization Theory and Applications 58 (1988) 139–145.
R. Horst and N.V. Thoai, “Modification, implementation and comparison of three algorithms for globally solving linearly constrained concave minimization problems,”Computing 42 (1989) 271–289.
R. Horst, N.V. Thoai and J. de Vries, “On finding new vertices and redundant constraints in cutting plane algorithms for global optimization,”Operations Research Letters 7 (1988) 85–90.
R. Horst, N.V. Thoai and H. Tuy, “Outer approximation by polyhedral convex sets,”Operations Research Spektrum 9 (1987) 153–159.
R. Horst, N.V. Thoai and H. Tuy “On an outer approximation concept in global optimization,”Optimization 20 (1989) 255–264.
R. Horst and H. Tuy, “On the convergence of global methods in multi-extremal optimization,”Journal of Optimization Theory and Applications 54 (1987) 253–271.
P.M. Pardalos and J.B. Rosen, “Methods for global concave minimization: A bibliographic survey,”SIAM Review 28 (1986) 367–379.
P.M. Pardalos and J.B. Rosen, “Constrained global optimization: Algorithms and applications,”Lecture Notes in Computer Science No. 268 (Springer, Berlin, 1987).
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,”Acta Mathematica Vietnamica 8 (1983) 21–40.
N.V. Thoai, “Verfahren zur Lösung konkaver Minimierungsaufgaben auf der Basis eines verallgemeinerten Erweiterungsprinzips,” Thesis (Habilitation), Humboldt-Universität (Berlin, 1984).
N.V. Thoai and H. Tuy, “Convergent algorithms for minimizing a concave function,”Mathematics of Operations Research 5 (1980) 556–566.
H. Tuy, “Concave programming under linear constraints,”Soviet Mathematics Doklady 5 (1964) 1437–1440.
H. Tuy, V. Khatchaturov and S. Utkin, “A class of exhaustive cone splitting procedures in conical algorithms for concave minimization,”Optimization 18 (1987) 791–807.
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 (1985) 498–514.
J. de Vries, “Algorithmen zur konkaven Minimierung,” Thesis, Universität Oldenburg (Oldenburg, 1986).
R. Horst and H. Tuy,Global Optimization — Deterministic Approaches, No. XIV (Springer, Berlin, 1990).
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Horst, R., Thoai, N.V. & Benson, H.P. Concave minimization via conical partitions and polyhedral outer approximation. Mathematical Programming 50, 259–274 (1991). https://doi.org/10.1007/BF01594938
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DOI: https://doi.org/10.1007/BF01594938