Convergence Analysis of Approximate Primal Solutions in Dual First-Order Methods

Abstract

Dual first-order methods are powerful techniques for large-scale convex optimization. Although an extensive research effort has been devoted to studying their convergence properties, explicit convergence rates for the primal iterates have essentially only been established under global Lipschitz continuity of the dual gradient. This is a rather restrictive assumption that does not hold for several important classes of problems. In this paper, we demonstrate that primal convergence rate guarantees can also be obtained when the dual gradient is only locally Lipschitz. The class of problems that we analyze admits general convex constraints including nonlinear inequality, linear equality, and set constraints. As an approximate primal solution, we take the minimizer of the Lagrangian computed when evaluating the dual gradient. We first derive error bounds for this approximate primal solution in terms of the errors of the dual variables. Then we establish convergence rates of the dual variables when dual problem...