Equivalent Lagrangians for generalized fractional programming (Q688521)

Consider the generalized fractional programming problem \[ \min_{x\in S} \max_{1\leq i\leq p} (f_ i(x)/h_ i(x)), \] where \(S= \{x: x\in X,\;g_ j(x)\leq 0,\;j=1,2,\dots,m\}\). A Lagrangian \[ L(x,w,u)= {w^ T f(x)+ u^ T g(x)\over w^ T h(x)}, \] is introduced and appropriate saddlepoint and duality results established. The relationship of \(L(x,w,u)\) to another Lagrangian \[ GX(x,u)= \max_{1\leq i\leq p} (f_ i(x)/h_ i(x))+ \sum^ m_{i=1} u_ j \max_{1\leq i\leq p} (g_ i(x)/h_ i(x)), \] recently introduced in the literature is discussed. The advantages of \(L(x,w,u)\) over \(GX(x,u)\) is pointed out.