This talk discusses joint work with Venkat Chandrasekaran, Jose Israel Rodriguez, and Kevin Shu, where we initiate the study of Lagrangian dual sections. This theory gives rise to sufficient conditions for the "hidden convexity" of certain nonconvex optimization problems. Notable examples include spectral inverse problems and certain unbalanced Procrustes problems. As an additional bonus, when the constraint set is a compact Riemannian manifold, the Lagrangian formulation allows us to solve these problems using a numerical continuation algorithm based on Riemannian gradient descent.