A HYBRID ADAPTIVE RESTART FRAMEWORK FOR ACCELERATED INEXACT PROXIMAL GRADIENT METHODS IN NON-CONVEX OPTIMIZATION

Abstract

Accelerated inexact proximal-gradient methods are effective for non-convex composite optimization, but combining adaptive step sizes with acceleration can lead to numerical instability. We propose a Hybrid Adaptive Accelerated Inexact Proximal Gradient (HA-AIPG) framework designed to balance exploratory steps with convergence safeguards. At each iteration, an adaptive spectral trial step is evaluated alongside a conservative, fixed-step reference proximal point. A relaxed combination of both updates is accepted only if it satisfies a sufficient objective decrease condition relative to the current iterate and reference update; otherwise, the algorithm defaults to the reference step and resets momentum. This structure allows trial steps to adapt to local curvature during transient phases while ensuring that global descent guarantees are preserved by the reference point. Under summable inexactness conditions, every limit point of the sequence is proved to be stationary. Under additional Kurdyka–Łojasiewicz (KL) assumptions, full sequence convergence and local rate bounds are established. Numerical experiments illustrate the practical performance of HA-AIPG, demonstrating competitive objective reduction across non-convex benchmark problems.