Accelerated Prox-Level Methods for Unknown Piecewise-Smooth Optimization I: Convex Optimization

Abstract

We introduce a 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 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. For this setting, APEX achieves the best-known oracle-complexity result among existing first-order methods, improving the dependence on the condition number relative to prior bundle-level guarantees. Furthermore, APEX generates a verifiable and accurate termination certificate, enabling a robust, nearly parameter-free implementation. To the best of our knowledge, APEX is the first algorithm to simultaneously achieve the best-known first-order oracle complexity for PWS optimization and provide certificate guarantees.