Estimating Sequences with Memory for Minimizing Convex Non-smooth Composite Functions

First-order optimization methods are crucial for solving large-scale data processing problems, particularly those involving convex non-smooth composite objectives. For such problems with convex non-smooth composite objectives, we introduce a new class of generalized composite estimating sequences, devised by exploiting the information embedded in the iterates generated during the minimization process. Building on these sequences, we propose a novel accelerated first-order method tailored for such objective structures. This method features a backtracking line-search strategy and achieves an accelerated convergence rate, regardless of whether the true Lipschitz constant is known. Additionally, it exhibits robustness to imperfect knowledge of the strong convexity parameter, a property of significant practical importance. The method's efficiency and robustness are substantiated by comprehensive numerical evaluations on both synthetic and real-world datasets, demonstrating its effectiveness in data processing applications.