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. 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. Let f , g , h j , j = 1 , … , m {displaystyle f,g,h_{j},j=1,ldots ,m} be real-valued functions defined on a set S 0 ⊂ R n {displaystyle mathbf {S} _{0}subset mathbb {R} ^{n}} . Let S = { x ∈ S 0 : h j ( x ) ≤ 0 , j = 1 , … , m } {displaystyle mathbf {S} ={{oldsymbol {x}}in mathbf {S} _{0}:h_{j}({oldsymbol {x}})leq 0,j=1,ldots ,m}} . The nonlinear program where g ( x ) > 0 {displaystyle g({oldsymbol {x}})>0} on S {displaystyle mathbf {S} } , is called a fractional program. 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 f , g , h j , j = 1 , … , m {displaystyle f,g,h_{j},j=1,ldots ,m} are affine. The function q ( x ) = f ( x ) / g ( x ) {displaystyle q({oldsymbol {x}})=f({oldsymbol {x}})/g({oldsymbol {x}})} 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. By the transformation y = x g ( x ) ; t = 1 g ( x ) {displaystyle {oldsymbol {y}}={frac {oldsymbol {x}}{g({oldsymbol {x}})}};t={frac {1}{g({oldsymbol {x}})}}} , any concave fractional program can be transformed to the equivalent parameter-free concave program If g is affine, the first constraint is changed to t g ( y t ) = 1 {displaystyle tg({frac {oldsymbol {y}}{t}})=1} and the assumption that f is nonnegative may be dropped.