Extensions to Inequality Constraints, Inexact Solves, Adaptive Parameters, and Globalization

Develop and analyze extensions of the nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that accommodate inequality constraints, inexact linear-system or primal solves, adaptive algorithmic parameters, and globalization mechanisms.

Background

The paper analyzes NR-LALM for smooth nonconvex optimization with nonlinear equality constraints, using a fixed penalty parameter, a linearized primal step, and the classical nonlinear-residual multiplier update. It establishes deterministic and stochastic complexity guarantees under local regularity, and also studies an optional second-order correction.

The conclusion explicitly identifies four directions not resolved by the presented theory: extending the method to inequality constraints, allowing inexact solves, adapting parameters, and developing globalization procedures. These are listed together as remaining open problems rather than merely suggested future work.