Effective and efficient electricity load-and demand-side management depends on the transmission, distribution and interconnecting networks of properly designed and adequately sized conductors to carry the produced electrical power to the ultimate consumers. A two-way optimal conductor design using computable convex functions was investigated in this paper. Composite materials whose area approaches the minimum and for which both the maximum vertical and horizontal currents simultaneously satisfy the Laplace’s equation, was considered. The resulting variational problem was homogenised or relaxed and thence, made polyconvex through the Lagrangian multipliers and Green’s identity. The main reason for the convexification of this design problem is that over the interval of convexity, there is only one minimum. This is so because any polyconvex function, which satisfies the boundary conditions is always minimising. That fact can strengthen many of the results we might desire while using the developed computable convex functions to show that no conductor area can be lower than that of the optimal two-way conductor designed in this study. Although the conductor proposed in this study would be more expensive than the conventional steel-cored cables, the economics of much higher current carrying capacities makes it more attractive. Additionally, their light weight requires that no new transmission towers are installed; they will suffer less sag, able to operate at much higher temperatures than aluminium conductor steel reinforced (ACSR) and less blackouts.