  In this paper, we propose a novel primal-dual proximal splitting algorithm
(PD-PSA), named BALPA, for the composite optimization problem with equality
constraints, where the loss function consists of a smooth term and a nonsmooth
term composed with a linear mapping. In BALPA, the dual update is designed as a
proximal point for a time-varying quadratic function, which balances the
implementation of primal and dual update and retains the proximity-induced
feature of classic PD-PSAs. In addition, by this balance, BALPA eliminates the
inefficiency of classic PD-PSAs for composite optimization problems in which
the Euclidean norm of the linear mapping or the equality constraint mapping is
large. Therefore, BALPA not only inherits the advantages of simple structure
and easy implementation of classic PD-PSAs but also ensures a fast convergence
when these norms are large. Moreover, we propose a stochastic version of BALPA
(S-BALPA) and apply the developed BALPA to distributed optimization to devise a
new distributed optimization algorithm. Furthermore, a comprehensive
convergence analysis for BALPA and S-BALPA is conducted, respectively. Finally,
numerical experiments demonstrate the efficiency of the proposed algorithms.
