An optimal control problem of steady-state complex heat transfer with monotone objective functionals is under consideration. A coefficient function appearing in boundary conditions and reciprocally corresponding to the reflection index of the domain surface is considered as control. The concept of strong maximizing (resp. strong minimizing) optimal controls, i.e. controls that are optimal for all monotone objective functionals, is introduced. The existence of strong optimal controls is proven, and optimality conditions for such controls are derived. An iterative algorithm for computing strong optimal controls is proposed.