Abstract
We consider the problem of minimizing the sum of a convex function and of p≥1 fractions subject to convex constraints. The numerators of the fractions are positive convex functions, and the denominators are positive concave functions. Thus, each fraction is quasi-convex. We give a brief discussion of the problem and prove that in spite of its special structure, the problem is \(\mathcal{N}\mathcal{P}\)-complete even when only p=1 fraction is involved. We then show how the problem can be reduced to the minimization of a function of p variables where the function values are given by the solution of certain convex subproblems. Based on this reduction, we propose an algorithm for computing the global minimum of the problem by means of an interior-point method for convex programs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aurenhammer, F. (1991), Voronoi diagrams-a survey of a fundamental geometric data structure, ACM Computing Surveys 23: 345-405.
Cambini, A., Martein, L. and Schaible, S. (1989), On maximizing a sum of ratios, J. Information and Optimization Sciences 10: 65-79.
Charnes, A. and Cooper, W.W. (1962), Programming with linear fractional functionals,Naval Research Logistics Quarterly 9: 181-186.
Chen, D.Z., Daescu, O., Dai, Y., Katoh, N., Wu, X. and Xu, J. (1998), Efficient algorithms and implementations for optimizing the sum of linear fractional functions, with applications, Technical Report, Department of Computer Science, University of Notre Dame, Notre Dame, IN.
Falk, J.E. and Palocsay, S.W. (1992), Optimizing the sum of linear fractional functions, in: C.A. Floudas and P.M. Pardalos (eds.), Recent Advances in Global Optimization, Princeton University Press, Princeton, NJ, pp. 221-258.
Fortune, S.J. (1997), Voronoi diagrams and Delaunay triangulations, in: J.E. Goodman and J. O'Rourke (eds.), Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, pp. 377-388.
Freund, R.W. and Jarre, F. (1994), An interior-point method for fractional programs with convex constraints, Mathematical Programming 67: 407-440.
Freund, R.W. and Jarre, F. (1995), An interior-point method for multifractional programs with convex constraints, J. Optimization Theory and Applications 85: 125-161.
Freund, R.W., Jarre, F. and Schaible, S. (1996), On self-concordant barrier functions for conic hulls and fractional programming, Mathematical Programming 74: 237-246.
Garey, M.R. and Johnson, D.S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, NY.
Hirche, J. (1985), Some remarks on generalized convexity of sums and products, Zeitschrift für Angewandte Mathematik und Mechanik 65: 62-63.
Hiriart-Urruty, J.-B. and Lemarechal, C. (1993), Convex Analysis and Minimization Algorithms I, Grundlehren der Mathematischen Wissenschaften 305, Springer-Verlag, Berlin, Germany.
Jarre, F. and Saunders, M.A. (1995), A practical interior-point method for convex programming, SIAM J. Optimization 5: 149-171.
Konno, H. and Kuno, T. (1990), Generalized linear multiplicative and fractional programming, Annals of Operations Research 25: 147-161.
Konno, H. and Yamashita, H. (1998), Minimization of the sum and the product of several linear fractional functions over a polytope, Manuscript, Department of Industrial Engineering and Management, Tokyo Institute of Technology, Tokyo, Japan.
Nemirovskii, A. (1996), On polynomiality of the method of analytic centers for fractional problems, Mathematical Programming 73: 175-198.
Nesterov, Y. and Nemirovskii, A. (1994), Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics Vol. 13, SIAM, Philadelphia, PA.
Ritter, K. (1967), A parametric method for solving certain nonconcave maximization problems, J. Computer and System Sciences 1: 44-54.
Schaible, S. (1984), Simultaneous optimization of absolute and relative terms, Zeitschrift für Angewandte Mathematik und Mechanik 64: 363-364.
Schaible, S. (1995), Fractional programming, in: R. Horst and P.M. Pardalos (eds.), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 495-608.
Schaible, S. (1996), Fractional programming with sums of ratios, Working Paper Series No. 96-04, Graduate School of Management, University of California, Riverside, CA.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Freund, R.W., Jarre, F. Solving the Sum-of-Ratios Problem by an Interior-Point Method. Journal of Global Optimization 19, 83–102 (2001). https://doi.org/10.1023/A:1008316327038
Issue Date:
DOI: https://doi.org/10.1023/A:1008316327038