Finding two link-disjoint QoS paths (primary and backup) between source-destination pairs is one of the most significant schemes to provide reliable QoS routing. Current algorithms for seeking multi-constrained link-disjoint path pair (MCLPP) can not always make sure to find the feasible solutions in networks. To solve this problem, this paper analyzes the properties of the optimal solution of MCLPP problem, and then proposes a design principle for the exact algorithm. Based on the design principle, an exact algorithm called link-disjoint optimal multi-constrained paths algorithm (LIDOMPA) is presented, which is able to find multi-constrained shortest link-disjoint path pair for arbitrary networks. To reduce the complexity, this paper introduces three key concepts: The candidate optimal solution, the constricted constraint vector and the structure-aware non-dominance, which effectively reduce the search space of LIDOMPA without loss of exactness. Extensive experiments show that LIDOMPA outperforms the existing algorithms in terms of the ability of obtaining solutions and achieves acceptable running time overhead.