The inexact power augmented Lagrangian method for constrained nonconvex optimization
This incremental work addresses optimization problems with nonlinear equality constraints for researchers in mathematical optimization, offering a trade-off in convergence rates.
The paper tackles constrained nonconvex optimization by introducing an inexact augmented Lagrangian method with a power-based augmenting term, showing that lower powers lead to faster constraint satisfaction but slower dual residual decrease, as supported by numerical experiments.
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 Hölder-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.