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

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
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

References

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