Optimality conditions for abs-normal NLPs

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Hegerhorst-Schultchen, Lisa Christine: Optimality conditons for abs-normal NLPs. Hannover : Gottfried Wilhelm Leibniz Universität, Diss., 2020, xi, 130 S. DOI: https://doi.org/10.15488/9867

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Structured nonsmoothness is widely present in practical optimization problems. A particularly attractive class of nonsmooth problems, both from a theoretical and from an algorithmic perspective, are nonsmooth NLPs with equality and inequality constraints in abs-normal form, so-called abs-normal NLPs. In this thesis optimality conditions for this particular class are obtained. To this aim, first the theory for the case of unconstrained optimization problems in abs-normal form of Andreas Griewank and Andrea Walther is extended. In particular, similar necessary and sufficient conditions of first and second order are obtained that are directly based on classical Karush-Kuhn-Tucker (KKT) theory for smooth NLPs. Then, it is shown that the class of abs-normal NLPs is equivalent to the class of Mathematical Programs with Equilibrium Constraints (MPECs). Hence, the regularity assumption LIKQ introduced for the abs-normal NLP turns out to be equivalent to MPEC-LICQ. Moreover, stationarity concepts and optimality conditions under these regularity assumptions of linear independece type are equivalent up to technical assumptions. Next, well established constraint qualifications of Mangasarian Fromovitz, Abadie and Guignard type for MPECs are used to define corresponding concepts for abs-normal NLPs. Then, it is shown that kink qualifications and MPEC constraint qualifications of Mangasarian Fromovitz resp. Abadie type are equivalent. As it remains open if this holds for Guignard type kink and constraint qualifications, branch formulations for abs-normal NLPs and MPECs are introduced. Then, equivalence of Abadie’s and Guignard’s constraint qualifications for all branch problems hold. Throughout a reformulation of inequalities with absolute value slacks is considered. It preserves constraint qualifications of linear independence and Abadie type but not of Mangasarian Fromovitz type. For Guignard type it is still an open question but ACQ and GCQ are preserved passing over to branch problems. Further, M-stationarity and B-stationarity concepts for abs-normal NLPs are introduced and corresponding first order optimality con- ditions are proven using the corresponding concepts for MPECs. Moreover, a reformulation to extend the optimality conditions for abs-normal NLPs to those with additional nonsmooth objective functions is given and the preservation of regularity assumptions is considered. Using this, it is shown that the unconstrained abs-normal NLP always satisfies constraint qualifications of Abadie and thus Guignard type. Hence, in this special case every local minimizer satisfies the M-stationarity and B-stationarity concepts for abs-normal NLPs.
License of this version: Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.
Document Type: doctoralThesis
Publishing status: publishedVersion
Issue Date: 2020
Appears in Collections:Fakultät für Mathematik und Physik

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pos. country downloads
total perc.
1 image of flag of Germany Germany 162 57.86%
2 image of flag of China China 70 25.00%
3 image of flag of United States United States 15 5.36%
4 image of flag of Canada Canada 6 2.14%
5 image of flag of India India 4 1.43%
6 image of flag of Denmark Denmark 4 1.43%
7 image of flag of Austria Austria 3 1.07%
8 image of flag of No geo information available No geo information available 2 0.71%
9 image of flag of Russian Federation Russian Federation 2 0.71%
10 image of flag of United Kingdom United Kingdom 2 0.71%
    other countries 10 3.57%

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