Accelerated primal-dual methods for linearly constrained convex optimization problems
This work proposes an accelerated primal-dual dynamical system for affine
constrained convex optimization and presents a class of primal-dual methods
with nonergodic convergence rates. In continuous level, exponential decay of a
novel Lyapunov function is established and in discrete level, implicit,
semi-implicit and explicit numerical discretizations for the continuous model
are considered sequentially and lead to new accelerated primal-dual methods for
solving linearly constrained optimization problems. Special structures of the
subproblems in those schemes are utilized to develop efficient inner solvers.
In addition, nonergodic convergence rates in terms of primal-dual gap, primal
objective residual and feasibility violation are proved via a tailored discrete
Lyapunov function. Moreover, our method has also been applied to decentralized
distributed optimization for fast and efficient solution.