Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.6.2024.2.08
Volume 6, Issue 2, 1 August 2024, Pages 291-320

 

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 auxiliary variables merely as a formal device. This, however, gives rise to a generalized augmented Lagrangian function that lacks regularity. 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, asymptotic properties and convergence guarantees are established for a safeguarded augmented Lagrangian scheme. Numerical examples showcase the modelling versatility gained by dropping convexity assumptions and the practical benefits of the advocated implicit approach.

 

How to Cite this Article:
A. De Marchi, Implicit augmented Lagrangian and generalized optimization, J. Appl. Numer. Optim. 6 (2024), 291-320.