In this paper the optimization of the thermal conductivity distribution for heat conduction enhancement is discussed. Different optimization objectives are considered which include the conductivity weighted quadratic temperature gradient and the weighted average temperature in the whole region or on the heat flux boundary. The adjoint state equations and gradient relations for the optimizations are obtained by the variational method and the 1D and 2D optimization problems are solved to demonstrate the analyses. The analyses show that different objectives are not generally equivalent to each other. When all the temperature boundaries have a same constant temperature, the optimization of the conductivity weighted quadratic temperature gradient has the following equivalences: 1) it is equivalent to the constant temperature gradient relation; 2) it is equivalent to the optimization of heat source averaged temperature in the domain when the heat flux boundaries are adiabatic; 3) it is equivalent to the heat flux averaged temperature on the heat flux boundary when the heat source intensity is zero. Otherwise, the configurations of the optimized temperature and thermal conductivity distributions for different objectives can have large differences. Therefore, the objectives should be carefully chosen when dealing with the heat conduction optimizations. (C) 2017 Elsevier Ltd. All rights reserved.