Preconditioned primal–dual gradient methods for nonconvex composite and finite-sum optimization

In this paper, we first introduce a preconditioned primal–dual gradient algorithm based on conjugate duality theory. This algorithm is designed to solve a composite optimization problem whose objective function consists of two summands: a continuously differentiable nonconvex function and the composition of a nonsmooth nonconvex function with a linear operator. Under mild conditions, we prove that any cluster point of the generated sequence is a critical point of the composite optimization problem. Under the Kurdyka–Łojasiewicz property, we establish the global convergence and convergence rates for the iterates. Second, for nonconvex finite-sum optimization, we propose a stochastic algorithm that combines the preconditioned primal–dual gradient algorithm with a class of variance-reduced stochastic gradient estimators. Almost sure global convergence and expected convergence rates are derived by relying on the Kurdyka–Łojasiewicz inequality. Finally, preliminary numerical results are presented to demonstrate the effectiveness of the proposed algorithms.