Optimal Methods for Unknown Piecewise Smooth Problems I: Convex Optimization

We introduce an optimal and nearly parameter-free algorithm for minimizing piecewise smooth (PWS) convex functions under the quadratic growth (QG) condition, where the locations and structure of the smooth regions are entirely \textit{unknown}. Our algorithm, \apex{} (Accelerated Prox-Level method for Exploring Piecewise Smoothness), is an accelerated bundle-level method designed to adaptively exploit the underlying PWS structure. APEX enjoys optimal theoretical guarantees, achieving a tight oracle complexity bound that matches the lower bound established in this work for convex PWS optimization. Furthermore, APEX generates a verifiable and accurate termination certificate, enabling a robust, almost parameter-free implementation. To the best of our knowledge, APEX is the first algorithm to simultaneously achieve the optimal convergence rate for PWS optimization and provide certificate guarantees.