J Optim Theory Appl https://doi.org/10.1007/s10957-018-1320-7 Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems 1 1 Max Bucher · 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 1 Introduction 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,
Journal of Optimization Theory and Applications – Springer Journals
Published: Jun 4, 2018
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