Implicit augmented Lagrangian and generalized optimization

TL;DR

This paper extends the augmented Lagrangian method to generalized nonlinear programming problems without convexity, introducing a new stationarity concept and proving convergence, with practical benefits demonstrated through numerical examples.

Contribution

It develops a generalized augmented Lagrangian framework for nonconvex problems, introducing a new stationarity concept and convergence analysis for implicit formulations.

Findings

Convergence guarantees for the proposed scheme.

Enhanced modeling versatility without convexity.

Numerical example demonstrating practical benefits.

Abstract

Generalized nonlinear programming is considered without any convexity assumption, capturing a variety of problems that include nonsmooth objectives, combinatorial structures, and set-membership nonlinear constraints. We extend the augmented Lagrangian framework to this broad problem class, preserving an implicit formulation and introducing slack variables merely as a formal device. This, however, gives rise to a generalized augmented Lagrangian function that lacks regularity, due to the marginalization with respect to slack variables. Based on parametric optimization, we develop a tailored stationarity concept to better qualify the iterates, generated as approximate solutions to a sequence of subproblems. Using this variational characterization and the lifted representation, a suitable multiplier update rule is derived, and then asymptotic properties and convergence guarantees are established for a safeguarded augmented Lagrangian scheme. An illustrative numerical example showcases the modelling versatility gained by dropping convexity assumptions and indicates the practical benefits of the advocated implicit approach.