Taming Nonconvexity with Epigraph Lifting

We consider inverse problems regularized by a trained neural network with \emph{convex activations} and \emph{unrestricted weights}. The resulting optimization problem is generally nonconvex, and gradient-based methods may converge to poor local minima. We introduce a \emph{penalized epigraph lifting} of the network and show that the lifted problem admits a \emph{difference-of-convex (DC) decomposition}. For Lipschitz continuous activations, we establish \emph{finite exactness}: the original objective is recovered exactly at a finite and explicitly computable penalty threshold $\barρ$. Building on this structure, we propose \emph{H-DC} (Homotopic DC), a homotopic scheme that solves a sequ...

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