This document discusses alternating minimization for solving optimization problems with variables that have natural partitions. It presents two main results: 1. Alternating minimization converges to a second-order stationary point when the objective function is strongly bi-convex with a full-rank cross Hessian at all strict saddles. 2. Proximal alternating minimization, which adds proximal terms to regularize each subproblem, converges to a second-order stationary point when the objective function is L-Lipschitz bi-smooth and the proximal parameter is larger than L. This provides a partial solution to the lack of global optimality guarantees for alternating minimization by establishing second-order convergence under certain conditions.