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

Research output: Contribution to journalResearch articleContributedpeer-review



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.


Original languageEnglish
Pages (from-to)1770-1804
Number of pages35
JournalOptimization Methods and Software
Issue number5
Publication statusPublished - 2022

External IDs

Mendeley e71bca5b-973e-3a32-946f-c94bfb056a23
WOS 000725984900001
Scopus 85117835335
dblp journals/oms/FischerZZ22


DFG Classification of Subject Areas according to Review Boards

Subject groups, research areas, subject areas according to Destatis


  • 90C26, 90C30, 90C46, 90C53, Bilevel optimization, Newton method, lower-level value function, Lower-level value function

Library keywords