Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems
We extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an i...
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| Published in | Mathematical programming Vol. 211; no. 1-2; pp. 207 - 243 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Springer Nature B.V
01.05.2025
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| Online Access | Get full text |
| ISSN | 0025-5610 1436-4646 1436-4646 |
| DOI | 10.1007/s10107-024-02082-3 |
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| Abstract | We extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. As consequence, we conclude that the global structure of the Scholtes-type regularization essentially coincides with that of CCOP. |
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| AbstractList | We extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. As consequence, we conclude that the global structure of the Scholtes-type regularization essentially coincides with that of CCOP. |
| Author | Lämmel, Sebastian Shikhman, Vladimir |
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| References | M Červinka (2082_CR11) 2016; 160 C Kanzow (2082_CR3) 2021; 36 H Günzel (2082_CR12) 2008; 57 S Lämmel (2082_CR10) 2023; 8 M Bucher (2082_CR6) 2018; 178 S Scholtes (2082_CR1) 2001; 11 AF Izmailov (2082_CR2) 2009; 142 M Branda (2082_CR7) 2018; 70 HT Jongen (2082_CR8) 2000 S Lämmel (2082_CR9) 2023 HT Jongen (2082_CR13) 2004 2082_CR4 OP Burdakov (2082_CR5) 2016; 26 |
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| SubjectTerms | Constraints Convergence Optimization Regularization Saddle points |
| Title | Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems |
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