Lagrange duality is used in order to obtain a novel and useful state-feedback exponential stabilisability characterisation for discrete-time switched positive linear systems. A switched positive linear system is proved to be state-feedback exponentially stabilisable if and only if there is a set of Lagrange multipliers whose image, via a linear mapping of the system's modes, is a Schur matrix. This characterisation directly leads to a simple and novel method for the synthesis of exponentially stabilising state-feedback mappings. A particular set of Lagrange multipliers is explicitly obtained as the result of constructively establishing an upper bound on the minimum number of Lagrange multipliers required to attain (through the aforementioned linear mapping) a Schur matrix. This construction of a particular set of Lagrange multipliers can always be used to complete the first step of the obtained synthesis method. It is further shown that for a (stabilisable) switched positive linear system with rank-one modes, the number of modes that constitute the system is an upper bound on the minimum number of Lagrange multipliers required to generate a Schur matrix, as well as an upper bound on the minimum number of linear functionals required to represent an exponentially stabilising state-feedback mapping. Examples illustrating the obtained method for the synthesis of stabilising state-feedback mappings are included.