🤖 AI Summary
For nonconvex optimization problems with nonlinear equality constraints, this paper proposes an inexact augmented Lagrangian method employing a norm penalty with exponent strictly between 1 and 2. The method constructs Hölder-smooth subproblems under convex feasibility and weak regularity assumptions, leveraging the first use of a non-integer-power Euclidean norm as the augmentation term. We establish, for the first time, accelerated first-order algorithm complexity bounds for such subproblems. Theoretically, we reveal an intrinsic trade-off: constraint violation converges faster as the exponent decreases, while dual residual decay remains controllably degraded. Numerical experiments demonstrate that the proposed method achieves superior constraint satisfaction accuracy and iteration efficiency compared to the standard squared-augmented Lagrangian method.
📝 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.