Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions
<p/> <p>We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed (<in...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2009-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://www.fixedpointtheoryandapplications.com/content/2009/957407 |
| _version_ | 1818838806694985728 |
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| author | Agarwal RaviP Verma RamU |
| author_facet | Agarwal RaviP Verma RamU |
| author_sort | Agarwal RaviP |
| collection | DOAJ |
| description | <p/> <p>We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i3.gif"/></inline-formula>)-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal (<inline-formula> <graphic file="1687-1812-2009-957407-i4.gif"/></inline-formula>)-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the relaxed proximal point algorithm by Eckstein and Bertsekas (1992). Even for the linear convergence analysis for the overrelaxed (or super-relaxed) (<inline-formula> <graphic file="1687-1812-2009-957407-i5.gif"/></inline-formula>)-proximal point algorithm, the fundamental model for Rockafellar's case does the job. Furthermore, we attempt to explore possibilities of generalizing the Yosida regularization/approximation in light of maximal (<inline-formula> <graphic file="1687-1812-2009-957407-i6.gif"/></inline-formula>)-monotonicity, and then applying to first-order evolution equations/inclusions.</p> |
| first_indexed | 2024-12-19T03:44:15Z |
| format | Article |
| id | doaj.art-bdfe618c1d56424b90e5a8498be23060 |
| institution | Directory Open Access Journal |
| issn | 1687-1820 1687-1812 |
| language | English |
| last_indexed | 2024-12-19T03:44:15Z |
| publishDate | 2009-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Fixed Point Theory and Applications |
| spelling | doaj.art-bdfe618c1d56424b90e5a8498be230602022-12-21T20:37:10ZengSpringerOpenFixed Point Theory and Applications1687-18201687-18122009-01-0120091957407Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational InclusionsAgarwal RaviPVerma RamU<p/> <p>We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i3.gif"/></inline-formula>)-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal (<inline-formula> <graphic file="1687-1812-2009-957407-i4.gif"/></inline-formula>)-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the relaxed proximal point algorithm by Eckstein and Bertsekas (1992). Even for the linear convergence analysis for the overrelaxed (or super-relaxed) (<inline-formula> <graphic file="1687-1812-2009-957407-i5.gif"/></inline-formula>)-proximal point algorithm, the fundamental model for Rockafellar's case does the job. Furthermore, we attempt to explore possibilities of generalizing the Yosida regularization/approximation in light of maximal (<inline-formula> <graphic file="1687-1812-2009-957407-i6.gif"/></inline-formula>)-monotonicity, and then applying to first-order evolution equations/inclusions.</p>http://www.fixedpointtheoryandapplications.com/content/2009/957407 |
| spellingShingle | Agarwal RaviP Verma RamU Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions Fixed Point Theory and Applications |
| title | Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions |
| title_full | Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions |
| title_fullStr | Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions |
| title_full_unstemmed | Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions |
| title_short | Super-Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i1.gif"/></inline-formula>)-Proximal Point Algorithms, Relaxed (<inline-formula> <graphic file="1687-1812-2009-957407-i2.gif"/></inline-formula>)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions |
| title_sort | super relaxed inline formula graphic file 1687 1812 2009 957407 i1 gif inline formula proximal point algorithms relaxed inline formula graphic file 1687 1812 2009 957407 i2 gif inline formula proximal point algorithms linear convergence analysis and nonlinear variational inclusions |
| url | http://www.fixedpointtheoryandapplications.com/content/2009/957407 |
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