  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.

Optimization and Control