Implicit augmented Lagrangian and generalized optimization
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909155822927872 |
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| author | De Marchi, Alberto |
| author_facet | De Marchi, Alberto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00363 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Implicit augmented Lagrangian and generalized optimization De Marchi, Alberto Optimization and Control 65K10, 90C26, 65K05 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. |
| title | Implicit augmented Lagrangian and generalized optimization |
| topic | Optimization and Control 65K10, 90C26, 65K05 |
| url | https://arxiv.org/abs/2302.00363 |