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Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments

Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments
Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments
We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization problem with a penalization of the value function constraint.
For treating the latter problem, we develop a framework that does not rely on the direct computation of the lower-level value function or its derivatives. For each penalty parameter, the framework leads to a semismooth system of equations. This allows us to extend the semismooth Newton method to bilevel optimization. Besides global convergence properties of the method, we focus on achieving local superlinear convergence to a solution of the semismooth system. To this end, we formulate an appropriate CD-regularity assumption and derive sufficient conditions so that it is fulfilled. Moreover, we develop conditions to guarantee that a solution of the semismooth system is a local solution of the bilevel optimization problem. Extensive numerical experiments on 124 examples of nonlinear bilevel optimization problems from the literature show that this approach exhibits a remarkable performance, where only a few penalty parameters need to be considered.
Fischer, Andreas
36c6e6ff-f3aa-4b32-8f61-ded55736e925
Zemkoho, Alain
30c79e30-9879-48bd-8d0b-e2fbbc01269e
Zhou, Shenglong
d183edc9-a9f6-4b07-a140-a82213dbd8c3
Fischer, Andreas
36c6e6ff-f3aa-4b32-8f61-ded55736e925
Zemkoho, Alain
30c79e30-9879-48bd-8d0b-e2fbbc01269e
Zhou, Shenglong
d183edc9-a9f6-4b07-a140-a82213dbd8c3

Fischer, Andreas, Zemkoho, Alain and Zhou, Shenglong (2019) Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments. arXiv, (1912.07079).

Record type: Article

Abstract

We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization problem with a penalization of the value function constraint.
For treating the latter problem, we develop a framework that does not rely on the direct computation of the lower-level value function or its derivatives. For each penalty parameter, the framework leads to a semismooth system of equations. This allows us to extend the semismooth Newton method to bilevel optimization. Besides global convergence properties of the method, we focus on achieving local superlinear convergence to a solution of the semismooth system. To this end, we formulate an appropriate CD-regularity assumption and derive sufficient conditions so that it is fulfilled. Moreover, we develop conditions to guarantee that a solution of the semismooth system is a local solution of the bilevel optimization problem. Extensive numerical experiments on 124 examples of nonlinear bilevel optimization problems from the literature show that this approach exhibits a remarkable performance, where only a few penalty parameters need to be considered.

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FischerZemkohoZhouP - Accepted Manuscript
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More information

Accepted/In Press date: 15 December 2019
e-pub ahead of print date: 15 December 2019

Identifiers

Local EPrints ID: 436769
URI: http://eprints.soton.ac.uk/id/eprint/436769
PURE UUID: d5530648-1eed-4402-9051-172bd7dcef80
ORCID for Alain Zemkoho: ORCID iD orcid.org/0000-0003-1265-4178
ORCID for Shenglong Zhou: ORCID iD orcid.org/0000-0003-2843-1614

Catalogue record

Date deposited: 06 Jan 2020 17:31
Last modified: 22 Nov 2021 03:07

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Contributors

Author: Andreas Fischer
Author: Alain Zemkoho ORCID iD
Author: Shenglong Zhou ORCID iD

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