This paper presents a fractional programming formulation and its solution strategy for design of control Lyapunov function (CLF) to guarantee the closed-loop stability of a control affine system for the states in a specified region. Without restrictive assumptions found in previous approaches, the fractional programming problem is reformulated as a recursive optimization problem to solve for a CLF with basis functions. A computationally effective derivative-free coordinate search method is proposed to find the solution, where the search space is confined by a piecewise linear function that approximates the lower bound of objective function. A CLF-based controller design is also proposed to handle infinity-norm input constraints. Two examples with actuator saturation and state constraints demonstrate the efficacy of the proposed approach.