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

Weiwei Kong () and Renato D. C. Monteiro ()
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Weiwei Kong: Oak Ridge National Laboratory
Renato D. C. Monteiro: Georgia Institute of Technology

Computational Optimization and Applications, 2023, vol. 85, issue 2, No 6, 509-545

Abstract: 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 $$\mathcal{O}(\varepsilon ^{-5/2}\log \varepsilon ^{-1})$$ O ( ε - 5 / 2 log ε - 1 ) iterations of the ACG subroutine, for a given tolerance $$\varepsilon >0$$ ε > 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; Inner accelerated first-order methods; Iteration complexity (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00464-5

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