In [[mathematical optimization]], '''linear-fractional programming (LFP)''' is a generalization of [[linear programming]] (LP). Whereas the objective function in linear programs are [[linear functional|linear functions]], the objective function in a linear-fractional program is a ratio of two linear[[affine functionsfunction]]s. InA otherlinear words,program can be regarded as a linearspecial programcase isof a linear-fractional program in which the denominator is the constant function having the value one.
Informally, if linear programming determines the way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model and given some list of requirements represented as linear equations,. inIn contrast, a linear-fractional programming model weis canused to achieve the highest ratio of outcome to cost, i.e. the highest efficiency.
For example, if in the context of LP we maximize the objective function '''profit = income − cost''' and might obtain maximal profit of $100 (= $1100 of income − $1000 of cost), then using LFP we can obtain only $10 of profit which requires only $50 of investment however. Thus, in LP we have an efficiency of $100/$1000 = 0.1,. whereasUsing LFP provideswe might obtain an efficiency of $10/$50 = 0.2 with a profit of only $10, which requires only $50 of investment however.
Linear-fractional programs are [[quasiconvex function|quasiconvex]] [[convex minimization|minimization]] problems with a [[monotonicity|monotone]] property, [[pseudoconvex function|pseudoconvexity]], which is a stronger property than [[quasiconvex function|quasiconvexity]]. A linear-fractional objective function is both pseudoconvex and pseudoconcave; these properties allow FLP problems to be solved by a variant of the [[simplex algorithm]] (of [[George B. Dantzig]]).<ref>
