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Maximizing the ratio of two convex functions over a convex set

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  • Harold P. Benson

Abstract

The purpose of this article is to present an algorithm for globally maximizing the ratio of two convex functions f and g over a convex set X. To our knowledge, this is the first algorithm to be proposed for globally solving this problem. The algorithm uses a branch and bound search to guarantee that a global optimal solution is found. While it does not require the functions f and g to be differentiable, it does require that subgradients of g can be calculated efficiently. The main computational effort of the algorithm involves solving a sequence of subproblems that can be solved by convex programming methods. When X is polyhedral, these subproblems can be solved by linear programming procedures. Because of these properties, the algorithm offers a potentially attractive means for globally maximizing ratios of convex functions over convex sets. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Suggested Citation

  • Harold P. Benson, 2006. "Maximizing the ratio of two convex functions over a convex set," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(4), pages 309-317, June.
    • Handle: RePEc:wly:navres:v:53:y:2006:i:4:p:309-317
    DOI: 10.1002/nav.20143
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    1. Lo, Andrew W. & Mackinlay, A. Craig, 1997. "Maximizing Predictability In The Stock And Bond Markets," Macroeconomic Dynamics, Cambridge University Press, vol. 1(1), pages 102-134, January.
    2. James E. Falk & Karla R. Hoffman, 1976. "A Successive Underestimation Method for Concave Minimization Problems," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 251-259, August.
    3. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
    4. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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