Abstract
Based on a review of existing algorithms, a general branch-and-bound concept in global optimization is presented. A sufficient and necessary convergence condition is established, and a broad class of realizations is derived that include existing and several new approaches for concave minimization problems.
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Horst, R.,On the Global Minimization of Concave Functions: Introduction and Survey, Operations Research Spektrum, Vol. 6, pp. 195–205, 1984.
Thoai, N., andTuy, H.,Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research, Vol. 5, pp. 556–566, 1980.
Falk, I., andSoland, R. M.,An Algorithm for Separable Nonconvex Programming Problems, Management Science, Vol. 15, pp. 550–569, 1969.
Soland, R. M.,An Algorithm for Separable Nonconvex Programming Problems, II: Nonconvex Constraints, Management Science, Vol. 17, pp. 759–773, 1971.
McCormick, G. P.,Computability of Global Solutions to Factorable Nonconvex Programs, Part I: Convex Underestimating Problems, Mathematical Programming, Vol. 10, pp. 147–175, 1976.
Grotzinger, S. I.,Supports and Convex Envelopes, Mathematical Programming, Vol. 31, pp. 339–347, 1985.
Horst, R.,An Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 10, pp. 312–321, 1976.
Horst, R.,A Note on the Convergence of an Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 19, pp. 237–238, 1980.
Horst, R.,A New Branch-and-Bound Approach for Concave Minimization Problems, Lecture Notes in Computer Sciences, Vol. 41, pp. 330–337, 1976.
Horst, R.,Nichtlineare Optimierung, Carl Hanser Verlag, München, West Germany, 1979.
Benson, H. P.,On the Convergence of Two Branch-and-Bound Algorithms for Nonconvex Programming Problems, Journal of Optimization Theory and Applications, Vol. 36, pp. 129–134, 1982.
Tuy, H., Thieu, T. V., andThai, N. Q.,A Conical Algorithm for Globally Minimizing a Concave Function over a Closed Convex Set, Mathematics of Operations Research, Vol. 10, pp. 498–514, 1985.
Berge, C.,Topological Spaces, Macmillan Company, New York, New York, 1963.
Kummer, B.,Global Stability of Optimization Problems, Mathematische Operationsforschung und Statistik, Serie Optimization, Vol. 8, pp. 367–382, 1977.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Horst, R., Zur Charakterisierung Affin-Linearer Hüllfunktionale, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 56, 347–348, 1976.
Falk, I.,Lagrange Multipliers and Nonconvex Programs, SIAM Journal on Control and Optimization, Vol. 7, pp. 534–545, 1969.
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Communicated by G. Leitmann
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Horst, R. A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization. J Optim Theory Appl 51, 271–291 (1986). https://doi.org/10.1007/BF00939825
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DOI: https://doi.org/10.1007/BF00939825