Abstract
In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.
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Frenk, J.B.G., Scheible, S.: Fractional programming. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, vol. II, pp. 162–172. Kluwer, Dordrecht (2001)
Rao, M.R.: Cluster analysis and mathematical programming. J. Am. Stat. Assoc. 66, 622–626 (1971)
Almogy, Y., Levin, O.: Parametric analysis of a multi-stage stochastic shipping problem. In: Proceedings of the 5th IFORS Conference (IFORS), pp. 359–370 (1964)
Colantoni, C.S., Manes, R.P., Whinston, A.: Programming, profit rates, and pricing decisions. Account. Rev. 44, 467–481 (1969)
Konno, H., Inori, M.: Bond portfolio optimization by bilinear fractional programming. J. Oper. Res. Soc. Jpn. 32, 143–158 (1989)
Cambini, A., Martein, L., Schaible, S.: On maximizing a sum of ratios. J. Inf. Optim. Sci. 10, 65–79 (1989)
Konno, H., Yajima, Y., Matsui, T.: Parametric simplex algorithms for solving a special class of nonconvex minimization problems. J. Glob. Optim. 1, 65–81 (1991)
Falk, J.E., Palocsay, S.W.: Image space analysis of generalized fractional programs. J. Glob. Optim. 4, 63–88 (1994)
Konno, H., Fukaishi, K.: A branch-and-bound algorithm for solving low-rank linear multiplicative and fractional programming problems. J. Glob. Optim. 18, 283–299 (2000)
Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Glob. Optim. 22, 155–174 (2002)
Konno, H., Yamashita, H.: Minimizing sums and products of linear fractional functions over a polytope. Nav. Res. Logist. 46, 583–596 (1999)
Phuong, N.T.H., Tuy, H.: A unified monotonic approach to generalized linear fractional programming. J. Glob. Optim. 26, 229–259 (2003)
Konno, H., Kuno, T., Yajima, Y.: Global minimization of a generalized convex multiplicative function. J. Glob. Optim. 4, 47–62 (1994)
Benson, H.P.: Global optimization of nonlinear sums of ratios. J. Math. Anal. Appl. 263, 301–315 (2001)
Benson, H.P.: Global optimization algorithm for the nonlinear sum of ratios problem. J. Optim. Theory Appl. 112, 1–29 (2002)
Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problem. J. Glob. Optim. 22, 343–364 (2002)
Dur, M., Horst, R., Thoai, N.V.: Solving sum-of-ratios fractional programs using efficient points. Optimization 49, 447–466 (2001)
Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Optim. 19, 83–102 (2001)
Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 495–608. Kluwer, Dordrecht (1995)
Schaible, S.: Fractional programming with sums of ratios. In: Castagnoli, E., Giorgi, G. (eds.) Scalar and Vector Optimization in Economic and Financial Problems, Proceedings of the Workshop in Milan, 1995, pp. 163–175. Elioprint, Modena (1995)
Benson, H.P.: On the global optimization of sums of linear fractional functions over a convex set. J. Optim. Theory Appl. 121, 19–39 (2004)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 2nd edn. Springer, Berlin (1993)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems Math. Program. 10, 147–175 (1976)
Veinott, A.F.: The supporting hyperplane method for unimodal programming. Oper. Res. 15, 147–152 (1967)
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Benson, H.P. Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs. J Optim Theory Appl 146, 1–18 (2010). https://doi.org/10.1007/s10957-010-9647-8
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DOI: https://doi.org/10.1007/s10957-010-9647-8