Relative-Entropy Geometry and Convergence in Nonconvex Wiretap Optimization

We develop and analyze an entropic descent method for a nonconvex optimization problem on probability simplices arising in classical and quantum secure communication. The optimization variables describe a randomized encoder, allowing its input distribution and stochastic preprocessing to be optimized jointly. An exact relative-entropy identity yields a channel-independent majorization bound and an explicit mirror-descent iteration with a fixed unit step size. We establish quantitative descent, convergence of objective values, and stationarity of accumulation points on their support faces for arbitrary finite classical channels, without assuming degradedness or less noisiness. For strictly positive transition matrices, a square-root parametrization and a Kurdyka--\L{}ojasiewicz argument establish finite length of the transformed trajectory and convergence of the entire sequence, including limits on the simplex boundary. For less-noisy channels, a restricted convex formulation admits global convergence with an explicit \(O(1/t)\) objective-error bound. We extend the relative-entropy construction to noncommuting quantum outputs and derive monotone alternating updates for covariant encoders, including correlated preprocessing over a fixed block. Numerical experiments compare the method with analytic benchmarks and constrained optimization, reveal multiple attracting values, and demonstrate strict improvements from stochastic preprocessing.