Mordukhovich Derivatives of Metric Projection Operator in Hilbert Spaces
In this paper, we study the generalized differentiability of the metric projection operator in Hilbert spaces. We find exact expressions for Mordukhovich derivatives (which are also called Mordukhovich coderivatives) for the metric projection operator onto closed balls in Hilbert spaces and positive...
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| Veröffentlicht in: | Journal of optimization theory and applications Jg. 203; H. 3; S. 2649 - 2678 |
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| Abstract | In this paper, we study the generalized differentiability of the metric projection operator in Hilbert spaces. We find exact expressions for Mordukhovich derivatives (which are also called Mordukhovich coderivatives) for the metric projection operator onto closed balls in Hilbert spaces and positive cones in Euclidean spaces and in real Hilbert space
l
2
. We investigate the connection between Frèchet derivatives, Gâteaux directional derivatives and the Mordukhovich derivatives of the metric projection in Hilbert spaces. |
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| AbstractList | In this paper, we study the generalized differentiability of the metric projection operator in Hilbert spaces. We find exact expressions for Mordukhovich derivatives (which are also called Mordukhovich coderivatives) for the metric projection operator onto closed balls in Hilbert spaces and positive cones in Euclidean spaces and in real Hilbert space l2. We investigate the connection between Frèchet derivatives, Gâteaux directional derivatives and the Mordukhovich derivatives of the metric projection in Hilbert spaces. In this paper, we study the generalized differentiability of the metric projection operator in Hilbert spaces. We find exact expressions for Mordukhovich derivatives (which are also called Mordukhovich coderivatives) for the metric projection operator onto closed balls in Hilbert spaces and positive cones in Euclidean spaces and in real Hilbert space l 2 . We investigate the connection between Frèchet derivatives, Gâteaux directional derivatives and the Mordukhovich derivatives of the metric projection in Hilbert spaces. |
| Author | Li, Jinlu |
| Author_xml | – sequence: 1 givenname: Jinlu surname: Li fullname: Li, Jinlu email: jli@shawnee.edu organization: Department of Mathematics, Shawnee State University |
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| Cites_doi | 10.1109/CDC.2002.1184423 10.1016/B978-0-12-775850-3.50013-3 10.2969/jmsj/02940615 10.1007/978-3-540-31247-5 10.2140/pjm.1995.170.567 10.1007/s10957-023-02329-7 10.1016/B978-0-12-576350-9.50005-9 10.1090/S0002-9947-1973-0326252-2 10.1186/s13660-016-1163-4 10.1090/S0002-9947-1982-0645326-5 10.1007/BF01584382 10.1007/978-1-4419-9467-7 10.1016/0022-247X(88)90213-2 |
| ContentType | Journal Article |
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| Keywords | 46C05 Strict Fréchet derivative Mordukhovich derivative Fréchet derivative 90C25 Metric projection operator Gâteaux directional derivative 65K10 49M27 47H05 |
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| References | A Shapiro (2530_CR16) 1988; 131 Boris S Mordukhovich (2530_CR11) 2006 A Haraux (2530_CR4) 1977; 29 A Shapiro (2530_CR15) 1990; 86 X Qin (2530_CR13) 2016 HH Bauschke (2530_CR1) 2011 RA Tapia (2530_CR17) 1971 2530_CR10 2530_CR7 2530_CR8 JL Li (2530_CR6) 2024; 200 A Shapiro (2530_CR14) 1987; 99 S Fitzpatrick (2530_CR3) 1982; 270 RB Holmes (2530_CR5) 1973; 184 E Zarantonello (2530_CR18) 1971 VI Berdyshev (2530_CR2) 1985; 150 D Noll (2530_CR12) 1995; 170 K Malanowski (2530_CR9) 1985; 33 |
| References_xml | – ident: 2530_CR10 doi: 10.1109/CDC.2002.1184423 – start-page: 237 volume-title: Contributions to Nonlinear Functional Analysis year: 1971 ident: 2530_CR18 doi: 10.1016/B978-0-12-775850-3.50013-3 – ident: 2530_CR7 – ident: 2530_CR8 – volume: 29 start-page: 615 year: 1977 ident: 2530_CR4 publication-title: J. Math. Soc. Jpn. doi: 10.2969/jmsj/02940615 – volume-title: Variational Analysis and Generalized Differentiation I Basic Theory year: 2006 ident: 2530_CR11 doi: 10.1007/978-3-540-31247-5 – volume: 170 start-page: 567 issue: 2 year: 1995 ident: 2530_CR12 publication-title: Pac. J. Math. doi: 10.2140/pjm.1995.170.567 – volume: 99 start-page: 123 year: 1987 ident: 2530_CR14 publication-title: Proc. Am. Math. Soc – volume: 200 start-page: 923 year: 2024 ident: 2530_CR6 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-023-02329-7 – volume: 150 start-page: 58 year: 1985 ident: 2530_CR2 publication-title: Akad. Nauk. SSSR Ural. Nauchn. Tsentr Sverd. – volume: 86 start-page: 77 year: 1990 ident: 2530_CR15 publication-title: J. Math. Anal. Appl. – start-page: 45 volume-title: Nonlinear Functional Analysis and Applications year: 1971 ident: 2530_CR17 doi: 10.1016/B978-0-12-576350-9.50005-9 – volume: 184 start-page: 87 year: 1973 ident: 2530_CR5 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1973-0326252-2 – year: 2016 ident: 2530_CR13 publication-title: J. Inequal. Appl. doi: 10.1186/s13660-016-1163-4 – volume: 270 start-page: 483 year: 1982 ident: 2530_CR3 publication-title: Trans. Am. Math. Soc doi: 10.1090/S0002-9947-1982-0645326-5 – volume: 33 start-page: 352 year: 1985 ident: 2530_CR9 publication-title: Math. Prog. doi: 10.1007/BF01584382 – volume-title: Convex Analysis and Monotone Operator Theory in Hilbert Spaces year: 2011 ident: 2530_CR1 doi: 10.1007/978-1-4419-9467-7 – volume: 131 start-page: 392 year: 1988 ident: 2530_CR16 publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(88)90213-2 |
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| SubjectTerms | Applications of Mathematics Approximation Banach spaces Calculus of Variations and Optimal Control; Optimization Closed balls Codes Engineering Euclidean geometry Hilbert space Mathematics Mathematics and Statistics Operations Research/Decision Theory Operators (mathematics) Optimization Theory of Computation |
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| Title | Mordukhovich Derivatives of Metric Projection Operator in Hilbert Spaces |
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