The $k$-eigenvalue problem plays a central role in the analysis of nuclear systems, as it governs the dynamics of neutron-induced fission chain reactions. The dominant eigenvalue provides key information about the system’s state—whether it is subcritical, critical, or supercritical. The most widely adopted numerical strategy for solving this problem is the inverse power iteration method, along with its variants. Recent work by \cite{KWMF} introduced a low-rank inverse power method that significantly reduces both memory usage and computational cost. Building upon this idea, we propose a rank-adaptive inverse power method that dynamically adjusts the rank throughout the iterative process. By progressively increasing the rank as the algorithm converges, we effectively balance model fidelity and efficiency, further lowering computational demands. We apply this multi-fidelity framework to the optimization of a simplified nuclear reactor model, wherein the system is parameterized and the goal is to identify the parameter configuration that achieves criticality. Numerical experiments demonstrate the robustness and superior performance of the proposed method in comparison with traditional techniques. This is a joint work with L. Einkemmer, J, Kusch, R. McClarren. \begin{thebibliography}{99} \bibitem{CL} Ceruti, G., and Lubich, C. An unconventional robust integrator for dynamical low-rank approximation. BIT Numerical Mathematics (2022) 62:23–44. https://doi.org/10.1007/s10543-021-00905-3. \bibitem{GKS} Guglielmi, N., Kressner, D., and Scalone, C. Computing low-rank rightmost eigenpairs of a class of matrix-valued linear operators. Advances in Computational Mathematics (2021) 47:62. https://doi.org/10.1007/s10444-021-09908-1. \bibitem{KWMF} Kusch, J., Whewell, B., McClarren, R., and Frank, M. A low-rank power iteration scheme for neutron transport criticality problems. Journal of Computational Physics (2022) 470:111587. https://doi.org/10.1016/j.jcp.2022.111587. \bibitem{jomp} Scalone C., Einkemmer L., Kusch J., McClarren R., A multi-fidelity adaptive dynamical low-rank based optimization algorithm for fission criticality problems, to apper in Journal of Scientific Computing. \end{thebibliography}