In system identification and adaptive control, the problem of designing strictly positive real (SPR) transfer functions in the presence of uncertain adaptation parameters is essential for stability and convergence in a group of parameter adaptation algorithms. This paper proposes a convex optimization approach to address the robust SPR problem. Besides achieving the robust SPR condition, the presented solution is optimal in the sense of minimizing the distance from the transfer function to unity. Such consideration is important for parameter convergence in practical applications. New topics such as minimum-order compensator and minimum high-frequency magnitude design are also introduced.