“Fractional programming”的意思、由来-开放百科全书

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Fractional programming

释义

Definition
Concave fractional programs
Properties Transformation to a concave program Duality
Notes
References

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

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

where on , is called a fractional program.

Concave fractional programs

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

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

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 f is nonnegative may be dropped.

Duality

The Lagrangian dual of the equivalent concave program is

Notes

1. ^{{cite journal|last1=Schaible |first1=Siegfried |title=Parameter-free Convex Equivalent and Dual Programs|journal=Zeitschrift für Operations Research |volume=18 |year=1974 |number=5 |pages=187–196|ref=harv|doi=10.1007/BF02026600|mr=351464}}

References

{{cite book |last1=Avriel |first1=Mordecai |last2=Diewert |first2=Walter E. |last3=Schaible |first3=Siegfried |last4=Zang |first4=Israel |title=Generalized Concavity |publisher=Plenum Press |year=1988}}
{{cite journal |title=Fractional programming |last1=Schaible |first1=Siegfried |journal=Zeitschrift für Operations Research |volume=27 |year=1983 |pages=39–54 |doi=10.1007/bf01916898}}

1 : Optimization algorithms and methods

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