Abstract
A new conical algorithm is developed for finding the global minimum of a concave function over a polytope. To ensure faster convergence and overcome some major drawbacks of previous conical algorithms, a normal (rather than exhaustive) subdivision process is used.
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References
S. Bali, “Minimization of a concave function on a bounded convex polyhedron,” Ph.D. Dissertation, University of California (Los Angeles, A, 1973).
V.P. Bulatov,Metody Pogrujeniia v Zadatchak Optimizatsii (Nauka, Sibirskoe Otdelenie, Novosibirsk, 1977).
M. Hamami and S.E. Jacobsen, “Exhaustive nondegenerate conical processes for concave minimization on convex polytopes,”Mathematics of Operations Research 13 (1988) 479–487.
R. Horst, “An algorithm for nonconvex programming problems,”Mathematical Programming 10 (1976) 312–321.
R. Horst, “On the global minimization of concave functions: Introduction and survey,”Operations Research Spektrum 6 (1984) 195–205.
R. Horst and Ng.V. Thoai, “Implementation, modification and comparison of some algorithms for concave minimization problems,” Preprint, Department of Mathematics, University of Trier (1988), to appear in:Computing.
S.E. Jacobsen, “Convergence of a Tuy-type algorithm for concave minimization subject to linear constraints,”Applied Mathematics and Optimization 7 (1981) 1–9.
P.M. Pardalos and J.B. Rosen, “Methods for global concave minimization: a bibliographic survey,”SIAM Review 26 (1986) 367–379.
Ng.V. Thoai and H. Tuy, “Convergent algorithm for minimizing a concave function,”Mathematics of Operations Research 5 (1980) 556–566.
Ng.V. Thoai and J. de Vries, “Numerical experiments on concave minimization algorithms,” to appear in:Methods of Operations Research.
H. Tuy, “Vognutoe programmirovaniie pri lineinyk ogranitchenyakh,”Doklady AN SSSR 158 (1964) 32–35. [Translated as: “Concave programming under linear constraints,”Soviet Mathematics 5 (1964) 1437–1440.]
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) 498–514.
H. Tuy, V. Khachaturov and S. Utkin, “A class of exhaustive cone splitting procedures in conical algorithms for concave minimization,”Optimization 18 (1987) 791–808.
H. Tuy, “A general deterministic approach to global optimization via D.C. programming,” in: J.B. Hiriari-Urruty, ed.,Fermat Days 1985: Mathematics for Optimization. Mathematics Studies Series (North-Holland, Amsterdam, 1986).
H. Tuy, “On polyhedral annexation method for concave minimization,” submitted for publication.
H. Tuy and R. Horst, “Convergence and restart in branch and bound algorithms for global optimization algorithms,”Mathematical Programming 41 (1988) 161–184.
P.B. Zwart, “Nonlinear Programming: counterexamples to two global optimization algorithms,”Operations Research 21 (1973) 1260–1266.
P.B. Zwart, “Global maximization of a convex function with linear inequality constraints,”Operations Research 22 (1974) 602–609.
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Tuy, H. Normal conical algorithm for concave minimization over polytopes. Mathematical Programming 51, 229–245 (1991). https://doi.org/10.1007/BF01586935
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DOI: https://doi.org/10.1007/BF01586935