This study presents a systematic approach to engineering the dispersion relation of one-dimensional monoatomic lattices through the use of long-range interactions constrained to strictly positive stiffness values. Departing from classical cosine fitting techniques, the proposed framework employs interpolation at prescribed frequency–wave number pairs, leading to a linear system formulation. The stiffness coefficients are then determined using non-negative least squares, ensuring the physical admissibility of the design. This method enables the realization of dispersion curves featuring non-standard characteristics, such as negative group velocity, roton-like stationary points, or locally flat bands, without resorting to non-physical parameters. The concept of admissibility domains is introduced to characterize the feasible set of target values under positivity constraints, and several numerical examples illustrate the trade-offs between design flexibility and physical realizability. The framework provides a versatile tool for the inverse design of dispersion in non-local lattices, with potential extensions to higher-dimensional and multi-physical systems.