Fractional programming

In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear. The ratio to be optimized often describes some kind of efficiency of a system.

Definition

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Let   be real-valued functions defined on a set  . Let  . The nonlinear program

 

where   on  , is called a fractional program.

Concave fractional programs

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A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program. If g is affine, f does not have to be restricted in sign. The linear fractional program is a special case of a concave fractional program where all functions   are affine.

Properties

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The function   is semistrictly quasiconcave on S. If f and g are differentiable, then q is pseudoconcave. In a linear fractional program, the objective function is pseudolinear.

Transformation to a concave program

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By the transformation  , any concave fractional program can be transformed to the equivalent parameter-free concave program[1]

 

If g is affine, the first constraint is changed to   and the assumption that g is positive may be dropped. Also, it simplifies to  .

Duality

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The Lagrangian dual of the equivalent concave program is

 

Notes

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  1. ^ Schaible, Siegfried (1974). "Parameter-free Convex Equivalent and Dual Programs". Zeitschrift für Operations Research. 18 (5): 187–196. doi:10.1007/BF02026600. MR 0351464. S2CID 28885670.

References

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  • Avriel, Mordecai; Diewert, Walter E.; Schaible, Siegfried; Zang, Israel (1988). Generalized Concavity. Plenum Press.
  • Schaible, Siegfried (1983). "Fractional programming". Zeitschrift für Operations Research. 27: 39–54. doi:10.1007/bf01916898. S2CID 28766871.