In this talk, we discuss the solution of an Inverse Eigenvalue Complementarity Problem (IEiCP). Some applications of the IEiCP are first described. Two nonlinear formulations, NLP1 and NLP2, of the IEiCP are presented. A necessary and sufficient condition for a stationary point of NLP1 to be a solution of the IEiCP is established. An enumerative algorithm is designed for the solution of the IEiCP by finding a global minimum of NLP2. The use of additional implied constraints for enhancing the efficiency of the algorithm is also discussed. Computational results are provided to highlight the performance of the algorithm in practice.