# Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems

Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods... 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, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Optimization Theory and Applications Springer Journals

# Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems

, Volume OnlineFirst – Jun 4, 2018
28 pages

/lp/springer_journal/second-order-optimality-conditions-and-improved-convergence-results-DLUi7C0jOl
Publisher
Springer Journals
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory
ISSN
0022-3239
eISSN
1573-2878
D.O.I.
10.1007/s10957-018-1320-7
Publisher site
See Article on Publisher Site

### Abstract

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

Journal of Optimization Theory and ApplicationsSpringer Journals

Published: Jun 4, 2018

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