An accelerated preconditioned primal-dual gradient algorithm for structured nonconvex optimization problems - Long, Xian-Jun; Nie, Jia-Lin; Gou, Zhun; Sun, Xiang-Kai; Li, Gao-Xi (Communications in nonlinear science & numerical simulation, 2026)
•An novel accelerated preconditioned primal-dual gradient algorithm for solving nonconvex optimization problems by the conjugate duality theory of nonconvex functions.•Our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. the computation load may be significantly reduced.•Global convergence under Kurdyka-Lojasiewicz condition.•Numerical results illustrate that the proposed algorithm is quite competitive with some existing algorithms. For a nonconvex problem, the computation of the proximal operator of the nonconvex function is difficult in general. In this paper, based on the conjugate duality theory of nonconvex functions, we present an accelerated preconditioned primal-dual gradient algorithm for a class of nonconvex optimization problems. Compared with the existing algorithms, our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. Hence, the computation load may be significantly reduced. We prove that the sequence generated by the proposed algorithm globally converges to a critical point under Kurdyka-Łojasiewicz framework. Furthermore, we derive the convergence rate of the proposed algorithm. Finally, numerical results on signal recovery, image denoising and sparse principal component analysis illustrate that the proposed algorithm is quite competitive with some existing algorithms.