Taming Nonconvexity with Epigraph Lifting

We consider inverse problems regularized by a trained neural network with convex activations and unrestricted weights. The resulting optimization problem is generally nonconvex, and gradient-based methods may converge to poor local minima. We introduce a penalized epigraph lifting of the network and show that the lifted problem admits a difference-of-convex (DC) decomposition. For Lipschitz continuous activations, we establish finite exactness: the original objective is recovered exactly at a finite and explicitly computable penalty threshold barρ. Building on this structure, we propose H-DC (Homotopic DC), a homotopic scheme that solves a sequence of DC subproblems with increasing ρ, warm-starting each level from the previous solution. Numerical experiments on synthetic problems, a manifold denoiser regularizer, and image inpainting with a learned Mumford--Shah regularizer show that H-DC consistently escapes poor local minima and reaches solutions with lower objective values than gradient-based methods from the same initialization.