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An Accelerated Inexact Dampened Augmented Lagrangian Method for Linearly-Constrained Nonconvex Composite Optimization Problems

Weiwei Kong(wwkong92***at***gmail.com)
Renato D.C. Monteiro(monteiro***at***isye.gatech.edu)

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 O(p^(-5/2) log p^(-1)) iterations of the ACG subroutine, for a given tolerance p > 0. Numerical experiments are also given to show the computational efficiency of the proposed method.

Keywords: inexact proximal augmented Lagrangian method, linearly constrained smooth nonconvex composite programs, accelerated first-order methods, iteration-complexity

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )


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Entry Submitted: 10/21/2021
Entry Accepted: 10/21/2021
Entry Last Modified: 10/21/2021

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