The rigorous optimization of thermoelectric (TE) devices with inhomogeneous material properties is still an unsettled problem. With the help of a newly introduced method of generalized Lagrange multiplier functions a solution is presented. The method leads to a coupled system of ordinary differential equations, which can be solved swiftly numerically. That may also be of interest in other branches of physics. For TE-generators the achieved enhancements by optimized thermo-power distribution for constant figure of merit Z are small compared to the uniform case. In the refrigerator mode exists a higher potential for optimization. A drastic increase in the maximum temperature difference is possible at the expense of largely reduced cooling efficiency. It is shown that the maximum TE generator and cooling efficiencies are independent of the thermal conductivity profile, which limits the optimization potential for graded devices.
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