The inexact power augmented Lagrangian method for constrained nonconvex optimization

• URL: http://arxiv.org/abs/2410.20153v1
• Date: Sat, 26 Oct 2024 11:31:56 GMT
• Title: The inexact power augmented Lagrangian method for constrained nonconvex optimization
• Authors: Alexander Bodard, Konstantinos Oikonomidis, Emanuel Laude, Panagiotis Patrinos,
• Abstract summary: This work introduces an unconventional augmented Lagrangian term, where the augmenting term is a Euclidean norm raised to a power. We show that using lower powers for augmenting term to faster rate, albeit with a slower decrease in residual. Our results further show that using lower powers for augmenting term to faster rate, albeit with a slower decrease in residual.
• Score: 44.516958213972885
• License:
• Abstract: This work introduces an unconventional inexact augmented Lagrangian method, where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems, that involve nonlinear equality constraints over a convex set under a mild regularity condition. First, we conduct a full complexity analysis of the method, leveraging an accelerated first-order algorithm for solving the H\"older-smooth subproblems. Next, we present an inexact proximal point method to tackle these subproblems, demonstrating that it achieves an improved convergence rate. Notably, this rate reduces to the best-known convergence rate for first-order methods when the augmenting term is a squared Euclidean norm. Our worst-case complexity results further show that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease in the dual residual. Numerical experiments support our theoretical findings, illustrating that this trade-off between constraint satisfaction and cost minimization is advantageous for certain practical problems.

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