Linear-fractional programming

In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function 1. Formally, a linear-fractional program is defined as the problem of maximizing (or minimizing) a ratio of affine functions over a polyhedron,

maximize𝐜T𝐱+α𝐝T𝐱+βsubject toA𝐱≤𝐛,

where 𝐱∈ℝn represents the vector of variables to be determined, 𝐜,𝐝∈ℝn and 𝐛∈ℝm are vectors of (known) coefficients, A∈ℝm×n is a (known) matrix of coefficients and α,β∈ℝ are constants. The constraints have to restrict the feasible region to {𝐱|𝐝T𝐱+β>0}, i.e. the region on which the denominator is positive.[1][2] Alternatively, the denominator of the objective function has to be strictly negative in the entire feasible region.

## Contents

* 1 Motivation by comparison to linear programming
* 2 Transformation to a linear program
* 3 Duality
* 4 Properties and algorithms
* 5 Notes
* 6 Sources
* 7 Further reading
* 8 Software

## Motivation by comparison to linear programming

Both linear programming and linear-fractional programming represent optimization problems using linear equations and linear inequalities, which for each problem-instance define a feasible set. Fractional linear programs have a richer set of objective functions. Informally, linear programming computes a policy delivering the best outcome, such as maximum profit or lowest cost. In contrast, a linear-fractional programming is used to achieve the highest ratio of outcome to cost, the ratio representing the highest efficiency. For example, in the context of LP we maximize the objective function profit = income − cost and might obtain maximum profit of $100 (= $1100 of income − $1000 of cost). Thus, in LP we have an efficiency of $100/$1000 = 0.1. Using LFP we might obtain an efficiency of $10/$50 = 0.2 with a profit of only $10, but only requiring $50 of investment.

## Transformation to a linear program

Any linear-fractional program can be transformed into a linear program, assuming that the feasible region is non-empty and bounded, using the Charnes-Cooper transformation.[1] The main idea is to introduce a new non-negative variable t to the program which will be used to rescale the constants involved in the program (α,β,𝐛). This allows us to require that the denominator of the objective function (𝐝T𝐱+β) equals 1. (To understand the transformation, it is instructive to consider the simpler special case with α=β=0.)

Formally, the linear program obtained via the Charnes-Cooper transformation uses the transformed variables 𝐲∈ℝn and t≥0:

maximize𝐜T𝐲+αtsubject toA𝐲≤𝐛t𝐝T𝐲+βt=1t≥0.

A solution 𝐱 to the original linear-fractional program can be translated to a solution of the transformed linear program via the equalities

𝐲=1𝐝T𝐱+β⋅𝐱andt=1𝐝T𝐱+β.

Conversely, a solution for 𝐲 and t of the transformed linear program can be translated to a solution of the original linear-fractional program via

𝐱=1t𝐲.

## Duality

Let the dual variables associated with the constraints A𝐲−𝐛t≤𝟎 and 𝐝T𝐲+βt−1=0 be denoted by 𝐮 and λ, respectively. Then the dual of the LFP above is [3][4]

minimizeλsubject toAT𝐮+λ𝐝=𝐜−𝐛T𝐮+λβ≥α𝐮∈ℝ+m,λ∈ℝ,

which is an LP and which coincides with the dual of the equivalent linear program resulting from the Charnes-Cooper transformation.

## Properties and algorithms

The objective function in a linear-fractional problem is both quasiconcave and quasiconvex (hence quasilinear) with a monotone property, pseudoconvexity, which is a stronger property than quasiconvexity. A linear-fractional objective function is both pseudoconvex and pseudoconcave, hence pseudolinear. Since an LFP can be transformed to an LP, it can be solved using any LP solution method, such as the simplex algorithm (of George B. Dantzig),[5][6][7][8] the criss-cross algorithm,[9] or interior-point methods.

## Notes

1. ↑ 1.0 1.1 Charnes, A.; Cooper, W. W. (1962). "Programming with Linear Fractional Functionals". Naval Research Logistics Quarterly 9 (3–4): 181–186. doi:10.1002/nav.3800090303.
2. ↑ Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. p. 151. ISBN 978-0-521-83378-3. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf. Retrieved October 15, 2011.
3. ↑ Schaible, Siegfried (1974). "Parameter-free Convex Equivalent and Dual Programs". Zeitschrift für Operations Research 18 (5): 187–196. doi:10.1007/BF02026600.
4. ↑ Schaible, Siegfried (1976). "Fractional programming I: Duality". Management Science 22 (8): 858–867. doi:10.1287/mnsc.22.8.858.
5. ↑ Chapter five: Craven, B. D. (1988). Fractional programming. Sigma Series in Applied Mathematics. 4. Berlin: Heldermann Verlag. pp. 145. ISBN 978-3-88538-404-5.
6. ↑ Kruk, Serge; Wolkowicz, Henry (1999). "Pseudolinear programming". SIAM Review 41 (4): 795–805. doi:10.1137/S0036144598335259.
7. ↑ Mathis, Frank H.; Mathis, Lenora Jane (1995). "A nonlinear programming algorithm for hospital management". SIAM Review 37 (2): 230–234. doi:10.1137/1037046.
8. ↑ (Murty 1983)
9. ↑ Illés, Tibor; Szirmai, Ákos; Terlaky, Tamás (1999). "The finite criss-cross method for hyperbolic programming". European Journal of Operational Research 114 (1): 198–214. doi:10.1016/S0377-2217(98)00049-6. Postscript preprint.

## Sources

* Murty, Katta G. (1983). "3.10 Fractional programming (pp. 160–164)". Linear programming. New York: John Wiley & Sons, Inc.. pp. xix+482. ISBN 978-0-471-09725-9.

## Further reading

* Bajalinov, E. B. (2003). Linear-Fractional Programming: Theory, Methods, Applications and Software. Boston: Kluwer Academic Publishers.
* Barros, Ana Isabel (1998). Discrete and fractional programming techniques for location models. Combinatorial Optimization. 3. Dordrecht: Kluwer Academic Publishers. pp. xviii+178. ISBN 978-0-7923-5002-6.
* Martos, Béla (1975). Nonlinear programming: Theory and methods. Amsterdam-Oxford: North-Holland Publishing Co.. pp. 279. ISBN 978-0-7204-2817-9.
* Schaible, S. (1995). "Fractional programming". in Reiner Horst and Panos M. Pardalos. Handbook of global optimization. Nonconvex optimization and its applications. 2. Dordrecht: Kluwer Academic Publishers. pp. 495–608. ISBN 978-0-7923-3120-9.
* Stancu-Minasian, I. M. (1997). Fractional programming: Theory, methods and applications. Mathematics and its applications. 409. Translated by Victor Giurgiutiu from the 1992 Romanian. Dordrecht: Kluwer Academic Publishers Group. pp. viii+418. ISBN 978-0-7923-4580-0.

## Software

* WinGULF – educational interactive linear and linear-fractional programming solver with a lot of special options (pivoting, pricing, branching variables, etc.) [|permanent dead link|dead link}}]
* JOptimizer – Java convex optimization library (open source)

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