Abstract
A modification of Tuy's cone splitting algorithm for minimizing a concave function subject to linear inequality constraints is shown to be convergent by demonstrating that the limit of a sequence of constructed convex polytopes contains the feasible region. No geometric tolerance parameters are required.
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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 (1973).
S. Bali and S. Jacobsen, On the Convergence of the Modified Tuy Algorithm for Minimizing a Concave Function on a Bounded Convex Polyhedron,Proceedings of the 8th IFIP Conference on Optimization Techniques, Würzburg, 1977,Optimization Techniques (ed. J. Stoer), Springer-Verlag, 59–66 (1978).
C. Berge,Topological Spaces, Oliver and Boyd, Edinburgh and London (1963).
M. J. Carrillo, A Relaxation Algorithm for the Minimization of a Quasiconcave Function on a Convex Polyhedron,Mathematical Programming, 13, 69–80 (1977).
R. Carvajal-Moreno, Minimization of Concave Functions Subject to Linear Constraints, Operations Research Center, University of California, Berkeley, ORC 72-3 (1972).
G. B. Dantzig,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey (1963).
J. E. Falk, and K. R. Hoffman, A Successive Underestimation Method for Concave Minimization Problems,Mathematics of Operations Research, 1, 251–259 (1976).
M. Frank, and P. Wolfe, An Algorithm for Quadratic Programming,Naval Research Logistics Quarterly, 3, 95–110 (1956).
F. Hausdorff,Set Theory, 2nd edition, Chelsea, New York (1962).
H. Tuy, Concave Programming Under Linear Constraints,Dokl. Akad. Nauk SSR, 159, 32–35 (1964).
P. B. Zwart, Nonlinear Programming: Counterexamples to Global Optimization Algorithms by Ritter and Tuy,Opns. Res. 21, 1260–1266 (1973).
P. B. Zwart, Global Maximization of a Convex Function with Linear Inequality Constraints,Opns. Res. 22, 602–609 (1974).
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Communicated by J. Stoer
Research supported by National Science Foundation Grant ENG 76-12250
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Jacobsen, S.E. Convergence of a Tuy-type algorithm for concave minimization subject to linear inequality constraints. Appl Math Optim 7, 1–9 (1981). https://doi.org/10.1007/BF01442106
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DOI: https://doi.org/10.1007/BF01442106