Ma Yumin, Cai Xingju, Zhang Haiping, Wang Maoran
This paper studies a class of nonconvex and nonsmooth two-block optimization problems, where the objective function consists of two nonconvex and nonsmooth separable functions and a smooth coupling function. To address such problems, we propose an improved algorithm based on the inexact inetial proximal gradient method and Nesterov’s acceleration method, namely the proximal alternating linearized minimization algorithm with two distinct extrapolation parameters. Building upon the traditional proximal alternating linearized minimization framework, the algorithm introduces distinct extrapolation parameter sequences to achieve dual extrapolation for one of the variables. Specifically, during each iteration, the algorithm constructs more tractable subproblems by linearizing the coupling function and adding proximal terms based on two different extrapolated points. Theoretically, under certain assumptions, we prove that any limit point of the bounded sequence generated by the algorithm is a critical point of the objective function. Furthermore, when the objective function satisfies the Kurdyka-Lojasiewicz property, we establish the global convergence of the algorithm. Notably, the proposed algorithm allows extrapolation parameters to take negative values, providing new possibilities for enhancing performance. To validate the effectiveness of the proposed algorithm, we apply it to solve the sparse principal component analysis problem. Extensive numerical experiments demonstrate that the proposed algorithm consistently outperforms existing methods in terms of both convergence speed and computational efficiency. Notably, when one of the extrapolation parameters is negative, the algorithm achieves further performance improvements, demonstrating its enhanced flexibility in parameter selection and underscoring its potential for broader applications.
