This paper presents a new approach to search for the optimal PID controller parameters that minimize a quadratic cost functional. It is shown that by augmenting the real-valued quadratic cost functional into a complex-valued function, the recently proposed algorithm of constructing two-dimensional image of a box under a continuously differentiable complex-valued mapping can be applied to find the true optimal PID controller for a feedback control system. The proposed approach is promising because there exist efficient parametric algorithms for computing the quadratic cost functional and its gradients, and there are powerful personal computers for fast computation almost everywhere.