Duality for Fractional Programming with Generalized Convexity Conditions

Abstract

Consider the problem of minimizing f(x)/g(x) subject to h(x)≤0 where f and g are differentiable functions from Rn into R and h is a differentiable function from Rn into Rm . Under the assumptions that f is convex and nonnegative, g is concave and positive (or fconvex and g linear and positive) and h is convex, a number of duality results can be found in the literature. Here we give a number of different duals that allow the weakening of these convexity requirements. A specific example is given where h is quasi-convex and not convex for which standard fractional programming duality does not apply but for which the duals proposed here do hold. Weak, strong and converse dual theorems are given for these new dual programs.