  Nonlinear Convex Cone Programming (NCCP) problems are important and have many
practical applications. In this paper, we introduces a flexible first-order
primal-dual algorithm called the Variant Auxiliary Problem Principle (VAPP) for
solving NCCP problems when the objective function and constraints are smooth
and may be nonsmooth. Each iteration of VAPP generates a nonlinear
approximation to the primal problem of an augmented Lagrangian method. The
approximation incorporates both linearization and a variable distance-like
function, and then the iterations of VAPP provide one decomposition property
for NCCP. Motivated by recent applications in big data analysis, there has been
an explosive growth in interest in the convergence rate analysis of parallel
computing algorithms for large scale optimization problem. This paper proposes
an iteration-based error bound and linear convergence of VAPP. Some verifiable
sufficient conditions of this error bound are also discussed. For the general
convex case (without error bound), we establish $O(1/t)$ convergence rate for
primal suboptimality, feasibility and dual suboptimality. By adaptively setting
in parameters at different iterations, we show an $O(1/t^2)$ rate for the
strongly convex case. We further present Forward-Backward Splitting (FBS)
formulation of VAPP method and establish the connection between VAPP and other
primal-dual splitting methods. Finally, we discuss some issues in the
implementation of VAPP.
