  In this paper, we first introduce a preconditioned primal-dual gradient
algorithm based on conjugate duality theory. This algorithm is designed to
solve 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. In contrast to
existing nonconvex primal-dual algorithms, our proposed algorithm, through the
utilization of conjugate duality, does not require the calculation of proximal
mapping of nonconvex functions. Under mild conditions, we prove that any
cluster point of the generated sequence is a critical point of the composite
optimization problem. In the context of Kurdyka-\L{}ojasiewicz property, we
establish global convergence and convergence rates for the iterates. Secondly,
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 relying on the
Kurdyka-\L{}ojasiewicz inequality. Finally, some preliminary numerical results
are presented to demonstrate the effectiveness of the proposed algorithms.

Optimization and Control