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A Proximal Augmented Lagrangian Method for Linearly Constrained Nonconvex Composite Optimization Problems

Author

Listed:
  • Jefferson G. Melo

    (Universidade Federal de Goiás)

  • Renato D. C. Monteiro

    (Georgia Institute of Technology)

  • Hairong Wang

    (Georgia Institute of Technology)

Abstract

This paper proposes and establishes the iteration complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. For any given (possibly infeasible) initial point and tolerance $$\rho >0$$ ρ > 0 , it is shown that IPAAL generates an approximate stationary solution in $${{\mathcal {O}}}(\rho ^{-3}\log (\rho ^{-1}))$$ O ( ρ - 3 log ( ρ - 1 ) ) ACG iterations, which can be improved to $${{\mathcal {O}}}(\rho ^{-2.5}\log (\rho ^{-1}))$$ O ( ρ - 2.5 log ( ρ - 1 ) ) if it is further assumed that a certain Slater condition holds.

Suggested Citation

  • Jefferson G. Melo & Renato D. C. Monteiro & Hairong Wang, 2024. "A Proximal Augmented Lagrangian Method for Linearly Constrained Nonconvex Composite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 388-420, July.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-023-02218-z
    DOI: 10.1007/s10957-023-02218-z
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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. repec:inm:orijoo:v:3:y:2021:i:4:p:373-397 is not listed on IDEAS
    3. Qihang Lin & Runchao Ma & Yangyang Xu, 2022. "Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization," Computational Optimization and Applications, Springer, vol. 82(1), pages 175-224, May.
    4. 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.
    5. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
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