Determining the components of a radioactive source/shield system using the system's radiation signature, a type of inverse transport problem, is one of great importance in homeland security, material safeguards, and waste management. Here, the Levenberg-Marquardt (or simply "Marquardt") method, a standard gradient-based optimization technique, is applied to the inverse transport problems of interface location identification, shield material identification, source composition identification, and material mass density identification (both separately and combined) in multilayered radioactive source/shield systems. One-dimensional spherical problems using leakage measurements of neutron-induced gamma-ray lines and two-dimensional cylindrical problems using flux measurements of uncollided passive gamma-ray lines are considered. Gradients are calculated using an adjoint-based differentiation technique that is more efficient than difference formulas. The Marquardt method is iterative and directly estimates unknown interface locations, source isotope weight fractions, and material mass densities, while the unknown shield material is identified by estimating its macroscopic gamma-ray cross sections. Numerical test cases illustrate the utility of the Marquardt method using both simulated data that are perfectly consistent with the optimization process and realistic data simulated by Monte Carlo.