Bridging Identification and Second-Order Acceleration: A Fast Alternating Minimization Framework for Composite Optimization

We consider a class of composite optimization problems involving a smooth function and a proper, lower semicontinuous regularizer, which may be nonconvex and nonsmooth. We propose a novel alternating minimization framework that integrates proximal-gradient steps with cubic-regularized Newton updates restricted to a dynamically identified low-dimensional subspace. Under the Kurdyka--Łojasiewicz (KL) property, we establish global convergence of the proposed method to a stationary point. Moreover, by incorporating an adaptive thresholding strategy guided by the KL exponent, we prove a finite identification property without imposing any nondegeneracy assumptions. We further develop a local convergence analysis and show that the proposed method attains a worst-case iteration complexity of $\mathcal{O}(\varepsilon^{-3/2})$ for achieving approximate second-order stationarity. Numerical experiments on both synthetic and real datasets demonstrate the efficiency and effectiveness of the proposed framework.