  The convergence analysis of optimization algorithms using continuous-time
dynamical systems has received much attention in recent years. In this paper,
we investigate applications of these systems to analyze the convergence of
linearized proximal ADMM algorithms for nonconvex composite optimization, whose
objective function is the sum of a continuously differentiable function and a
composition of a possibly nonconvex function with a linear operator. We first
derive a first-order differential inclusion for the linearized proximal ADMM
algorithm, LP-ADMM. Both the global convergence and the convergence rates of
the generated trajectory are established with the use of Kurdyka-\L{}ojasiewicz
(KL) property. Then, a stochastic variant, LP-SADMM, is delved into an
investigation for finite-sum nonconvex composite problems. Under mild
conditions, we obtain the stochastic differential equation corresponding to
LP-SADMM, and demonstrate the almost sure global convergence of the generated
trajectory by leveraging the KL property. Based on the almost sure convergence
of trajectory, we construct a stochastic process that converges almost surely
to an approximate critical point of objective function, and derive the expected
convergence rates associated with this stochastic process. Moreover, we propose
an accelerated LP-SADMM that incorporates Nesterov's acceleration technique.
The continuous-time dynamical system of this algorithm is modeled as a
second-order stochastic differential equation. Within the context of KL
property, we explore the related almost sure convergence and expected
convergence rates.
