J Optim Theory Appl
Second-Order Optimality Conditions and Improved
Convergence Results for Regularization Methods for
Cardinality-Constrained Optimization Problems
· Alexandra Schwartz
Received: 5 September 2017 / Accepted: 22 May 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract We consider nonlinear optimization problems with cardinality constraints.
Based on a continuous reformulation, we introduce second-order necessary and suf-
ﬁcient optimality conditions. Under such a second-order condition, we can guarantee
local uniqueness of Mordukhovich stationary points. Finally, we use this observation to
provide extended local convergence theory for a Scholtes-type regularization method,
which guarantees the existence and convergence of iterates under suitable assumptions.
This convergence theory can also be applied to other regularization schemes.
Keywords Cardinality constraints · Strong stationarity · Mordukhovich stationarity ·
Second-order optimality conditions · Regularization method · Scholtes regularization
Mathematics Subject Classiﬁcation 90C27 · 90C30 · 90C33 · 90C46 · 65K05
In this article, we consider cardinality-constrained optimization problems, i.e., ﬁnite-
dimensional optimization problems, which in addition to standard constraints have a
bound on the maximum number of nonzero components of the optimization variable.
These problems have various applications such as compressed sensing, subset selection
in regression, support vector machines, cash management in automatic teller machines,
Communicated by Nikolai Osmolovskii.
Graduate School Computational Engineering, Technische Universität Darmstadt, Darmstadt,