The maximum cable tension is a crucial parameter in the design of a cable-driven parallel robot (CDPR) since the various mechanical components of the CDPR must be designed to safely withstand the loads induced by this maximum tension. For CDPRs having a number of cables at least equal to its number of DOFs, this paper deals with the determination of the smallest maximum cable tension vectors allowing a required wrench set to be feasible. The problem is formulated as the minimization of the maximum cable tension infinity norm under linear inequality constraints which include the wrench-feasibility constraints. The solution to this minimization problem is not unique, and the solution set is shown to be a convex polytope in the maximum tension space. Hence, various smallest maximum tension vectors generally exist and the computations of two different solution vectors are introduced. The first vector has all its components equal to the minimum infinity norm which can be directly obtained from the minimization problem inequality constraints. An algorithm is proposed to determine the second vector as the solution vector having the least possible value for each of its components. The computation of the smallest maximum tension vectors for general required wrench sets are then presented. The cases of particular wrench set definitions relevant to heavy payload manipulation applications are also introduced. Finally, these contributions are applied to the configuration (geometry) optimization of a large-dimension 6-DOF CDPR installed on a building facade to manipulate heavy payloads.