This paper describes a novel version of the method of Lagrange multipliers for an improved modeling of multi-point constraints that emanate from contact-impact problems, partitioned structural analysis using parallel computers, and structural inverse problems. It is shown that the classical method of Lagrange multipliers can lead to a non-unique set of constraint conditions for the modeling of interfaces involving more than two or multi-point substructural interface nodes. The proposed version of the method of Lagrange multipliers leads not only to unique construction of constraints but also encounters no singularity in modeling an arbitrary number of multi-point constraints. An important utilization of the present method is in the regularized modeling of interfaces whose rigidities are radically different from one to another. The present approach is demonstrated via several examples for its simplicity in modeling constraints, ease of implementation and computational advantages.