Optimality conditions and duality for a class of continuous-time generalized fractional programming problems (Q2639330)

Consider the following continuous-time fractional minmax programming problem with nonlinear operator inequality and affine equality constraints: \[ \inf \max_{1\leq r\leq m}(\int^{T}_{0}f_ r(x)(t)dt)/(\int^{T}_{0}k_ r(x)(t)dt) \] subject to \(x\in \Phi\), where \(\Phi =\{x\in W^ n[0,T]:\) g(x)(t)\(\leq 0\) for all \(t\in [0,T]\), \(h(x)(t)=0\) for all \(t\in [0,T]\}\). First necessary and sufficient conditions for optimality are obtained. Then a dual problem is formulated and appropriate weak, strong, and convese duality theorems are given.