Functionally graded materials (FGMs) are a means to remove discrete material interfaces which lead to high local stress concentrations, such as the tungsten–copper (W–Cu) interface of the current ITER monoblock design. This paper employs adjoint-based optimization methods to identify the highest potential reduction of stresses that could be reached with these materials, while ensuring that the local temperature does not exceed the material temperature operational window. The cheap sensitivity evaluation inherent to the adjoint approach enables the optimization of the detailed 3D material distribution. Furthermore, a novel optimization method based on an augmented Lagrangian formulation is proposed that allows accurate treatment of the material temperature window constraints. The temperature and stresses are modelled by the steady heat conduction and Navier's equation, respectively. We compare the results of different optimization formulations, with cost functions based on the von Mises stress and corresponding yield criterion and considering different values of the stress free temperature. To assess the performance under off-design conditions, two optimized designs were chosen and compared to the ITER and flat tile (FT) design, which consists of a copper block protected by a tungsten layer on top. The optimized designs lead to a factor 2–4 decrease in maximal stress near the original W–Cu interface of the FT design and a factor 10 decrease in yield criterion measure near the cooling duct. Under off-design conditions, they realized a factor 2–10 decrease in yield criterion in the upper part of the monoblock. This confirms numerically that FGMs can lead to significant design improvements. Finally, the inclusion of the material temperature operation window constraints leads to a decrease of 30–55 vol% W compared to the unconstrained cases, thus profoundly influencing the final design. The stress free temperature was found to have a comparably weaker influence on the final design with differences of 5–30 vol% W.