An accelerated inexact dampened augmented Lagrangian method for linearly-constrained nonconvex composite optimization problems

Abstract

This paper proposes and analyzes an accelerated inexact dampened augmented Lagrangian (AIDAL) method for solving linearly-constrained nonconvex composite optimization problems. Each iteration of the AIDAL method consists of: (i) inexactly solving a dampened proximal augmented Lagrangian (AL) subproblem by calling an accelerated composite gradient (ACG) subroutine; (ii) applying a dampened and under-relaxed Lagrange multiplier update; and (iii) using a novel test to check whether the penalty parameter of the AL function should be increased. Under several mild assumptions involving the dampening factor and the under-relaxation constant, it is shown that the AIDAL method generates an approximate stationary point of the constrained problem in $$\mathcal{O}(\varepsilon ^{-5/2}\log \varepsilon ^{-1})$$ iterations of the ACG subroutine, for a given tolerance $$\varepsilon >0$$ . Numerical experiments are also given to show the computational efficiency of the proposed method.