Abstract
Since the work of Zwart, it is known that cycling may occur in the cone splitting algorithm proposed by Thy in 1964 to minimize a concave function over a polytope. In this paper, we show that despite this fact, Tuy’s algorithm is convergent in the sense that it always finds an optimal solution. This result also holds for a variant of Tuy’s algorithm proposed by Gallo, in which a cone is split into a smaller subset of subcones (in term of inclusion). As shown by an example, this variant may also cycle. The transformation of these two algorithms into finite step procedures is discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bali, S. (1973). Minimization of a Concave Function on a Bounded Convex Polyhedron. Ph.D. Dissertation, University of California, Los Angeles.
Benson, H. P. (1996). Concave minimization: Theory, applications and algorithms In Handbook of Global Optimization, R. Horst and P. M. Pardalos, Eds. Kluwer Academic Publishers, Dordrecht, pp. 43–148.
Gallo, G. (1975). On Hoang Tui’s concave programming algorithm. Nota scientifica S-76–1, Instituto di Scienze dell’Informazione, University of Pisa, Italie.
Gallo, G., and Ulkücü, A. (1977). Bilinear programming: an exact algorithm. Mathematical Programming, 12, 173–194.
Hansen, P., Jaunard, B., Meyer, C., and Tuy, H. (1996). Best simplicial and double-simplicial bounds for concave minimization. Les Cahiers du GERAD G-96–17, GERAD, Montréal, Canada.
Hamami, M. and Jacobsen, S.E. (1988). Exhaustive Nondegenerate Conical Processes for Concave Minimization on Convex Polytopes. Mathematics of Operations Research, 13, 479–487.
Horst, R., and Thoai, N. V. (1990). On solving general reverse convex programming problems by a sequence of linear programs and line searches. Annals of Operations Research 25, 1–18.
Horst, R., and Thoai, N. V. (1989). Modification, implementation and comparison of three algorithms for globally solving linearly constrained concave minimization problems. Computing, 42, 271–289.
Horst, R., and Tuy, H. (1991). The Geometric Complementarity Problem and Transcending Stationarity in Global Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 4, 341–354.
Horst, R., and Tuy, H. (1996). Global Optimization. Deterministic Approaches, third, revised and enlarged edition. Springer-Verlag, Berlin.
Horst, R., Tiioai, N. V., and de Bneson, H. P. (1991). Concave minimization via conical partitions and polyhedral outer approximation. Mathematical Programming, 50, 259–274.
Horst, R., Tiioai, N. V., and de Vries, J. (1992). On geometry and convergence of a class of simplicial covers. Optimization, 25, 53–64.
Horst, R., Tiioai, N. V., and de Vries, J. (1992). A new simplicial cover technique in constrained global optimization. Journal of Global Optimization, 2, 1–19.
Jaumard, B., and Meyer, C. (1996). On the convergence of cone splitting algorithms with c.)-subdivisions. Les Cahiers du GERAD G-96–36, GERAD, Montréal, Canada.
Meyer, C. (1996). Algorithmes coniques pour la minimisation concave. PhD thesis, Ecole Polytechnique de Montréal, Montréal, Canada.
Thoai, H., and Tuy, H. (1980). Convergent algorithms for minimizing a concave function. Mathematics of Operations Research, 5, 556–566.
Tuy, H. (1964). Concave programming under linear constraints, Soviet Mathematics, 5, 1437–1440.
Tuy, H. (1990). On Polyhedral Annexation Method for Concave Minimization, in Lev J. Leifman (ed.), Functional Analysis, Optimization, and Mathematical Economics, Oxford University Press, New-York, 248–260.
Tuy, H. (1991). Normal Conical Algorithm for Concave Minimization. Mathematical Programming, 51, 229–245.
Tuy, H. (1991). Effect of the Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms. Journal of Global Optimization, 1, 23–36.
Tuy, H. (1996). D.c. optimization: Theory, methods and algorithms In Handbook of Global Optimization, R. Horst and P. M. Pardalos, Eds. Kluwer Academic Publishers, pp. 149–216.
Tuy, H., Khatchaturov, V., and Utkin, S. (1987). A class of exhaustive cone splitting procedures in conical algorithms for concave minimization. Optimization, 18 (6), 791–807.
Tuy, H., Thieu, T. V., and Thai, N. Q. (1985). A conical algorithm for globally minimizing a concave function over a closed convex set. Mathematics of Operations Research, 10, 498–514.
Vaish, H. (1974). Nonconvex Programming with Applications to Production and Location Problems. PhD thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia.
Zwart, P. B. (1973). Nonlinear Programming: counterexamples to two global optimization algorithms. Operations Research, 21, 1260–1266.
Zwart, P. B. (1974). Global maximization of a convex function with linear inequality constraints. Operations Research, 22, 602–609.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Meyer, C. (2001). On Tuy’s 1964 Cone Splitting Algorithm for Concave Minimization. In: Migdalas, A., Pardalos, P.M., Värbrand, P. (eds) From Local to Global Optimization. Nonconvex Optimization and Its Applications, vol 53. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5284-7_7
Download citation
DOI: https://doi.org/10.1007/978-1-4757-5284-7_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4852-6
Online ISBN: 978-1-4757-5284-7
eBook Packages: Springer Book Archive