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ce: affine functionals are linear functionals on a higher dimensional space with fixed variable (e.g. x_1=1), so no need to specify affine. Also LP ignored affine, so discussing affine was more complicated |
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For example, 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). 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, which requires only $50 of investment however.
Formally, a linear-fractional program is defined as
:<math>\text{maximize}_{x \in X}f(x)</math>
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>
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