This paper is related to an application of infinite-dimensional nonlinear programming: the max-min fractional optimal control problem with linear state constraints. Such a problem is called the continuous generalized fractional programming problem (CGFP). The author develops a parametric approach for the generic instance of (CGFP) by converting the problem (CFGP) into a family \{(CLP\(_{\lambda}\))\(:\lambda\in \mathbb{R} \)\} of continuous-time linear programs. By establishing the strong duality between (CLP\(_{\lambda}\)) and its dual problem, the author shows that for all \(\lambda\in \mathbb{R} ,\) the program (CLP\(_{\lambda}\)) is solvable, and then considers the function \(Q(\cdot): \mathbb{R} \rightarrow \mathbb{R} ,\) where the value \(Q(\lambda)\) is defined as the optimal value of the problem (CLP\(_{\lambda}\)). The general properties of \(Q(\cdot)\) and the equivalence between the problems (CFGP) and (CLP\(_{\lambda}\)) is studied. These properties ensure that solving the problem (CFGP) is equivalent to finding the root of the nonlinear equation \(Q(\lambda)=0.\) However, it is notoriously difficult to find the exact solution of every (CLP\(_{\lambda}\)). The author announces that in an accompanying paper, he shall refine the discrete approximation method developed in this paper and extend the interval-type algorithm by \textit{C.-F. Wen} [Taiwanese J. Math. 16, No. 4, 1423--1452 (2012; Zbl 1286.90147)] to solve the problem (CFGP). In the present paper, by using different step sizes of discretization problems, the author constructs a sequence of feasible solutions for (CLP\(_{\lambda}\)) and its dual problem (DCLP\(_{\lambda}\)), respectively. It is shown the convergent property of the constructed feasible solutions.