This paper proposes a new approach to solve the problem of designing optimal proportional-integral-derivative (PID) controllers that minimize an H₂-norm associated with the set-point response while subjecting to an H_∞-norm on the load disturbance rejection. The proposed design approach consists of constructing the feasible domain in the controller gain space and searching over the domain the optimal gain values of minimizing the H₂-norm objective. The construction of the feasible domain that satisfies the requirements of closed-loop stability and H_∞-norm constraint on disturbance-rejection is achieved through analytically characterizing the domain boundary with the notion of principal points associated with the value set of differentiable mapping. The feasible domain boundary construction greatly save the computational effort required to search for the H₂-optimal controller gain values that satisfy both the stability and disturbance rejection requirements.