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

Author

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  • Weiwei Kong

    (Oak Ridge National Laboratory)

  • Renato D. C. Monteiro

    (Georgia Institute of Technology)

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.

Suggested Citation

  • Weiwei Kong & Renato D. C. Monteiro, 2023. "An accelerated inexact dampened augmented Lagrangian method for linearly-constrained nonconvex composite optimization problems," Computational Optimization and Applications, Springer, vol. 85(2), pages 509-545, June.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:2:d:10.1007_s10589-023-00464-5
    DOI: 10.1007/s10589-023-00464-5
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    References listed on IDEAS

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    1. Ya-Feng Liu & Xin Liu & Shiqian Ma, 2019. "On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 632-650, May.
    2. Renato D. C. Monteiro & Camilo Ortiz & Benar F. Svaiter, 2016. "An adaptive accelerated first-order method for convex optimization," Computational Optimization and Applications, Springer, vol. 64(1), pages 31-73, May.
    3. Bo Jiang & Tianyi Lin & Shiqian Ma & Shuzhong Zhang, 2019. "Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis," Computational Optimization and Applications, Springer, vol. 72(1), pages 115-157, January.
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