A sharp Lagrange multiplier theorem for nonlinear programs
For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity—a not very restrictive generalization of convexity which arises natur...
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| Published in | Journal of global optimization Vol. 65; no. 3; pp. 513 - 530 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.07.2016
Springer Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0925-5001 1573-2916 |
| DOI | 10.1007/s10898-015-0379-z |
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| Abstract | For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity—a not very restrictive generalization of convexity which arises naturally in minimax theory—of a finite family of suitable functions. Even if we dispense with the Slater condition, it is proven that the infsup-convexity is nothing more than an equivalent reformulation of the Fritz John conditions for the nonlinear optimization problem under consideration. |
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| AbstractList | For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity-a not very restrictive generalization of convexity which arises naturally in minimax theory-of a finite family of suitable functions. Even if we dispense with the Slater condition, it is proven that the infsup-convexity is nothing more than an equivalent reformulation of the Fritz John conditions for the nonlinear optimization problem under consideration. |
| Audience | Academic |
| Author | Galán, M. Ruiz |
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| CitedBy_id | crossref_primary_10_1016_j_cnsns_2020_105339 crossref_primary_10_1002_mma_5566 crossref_primary_10_1007_s10957_016_0959_1 crossref_primary_10_1007_s11590_017_1124_y crossref_primary_10_1016_j_jmaa_2017_06_007 |
| Cites_doi | 10.1007/BF00934081 10.1080/02331930600819969 10.1155/S1110757X04304018 10.2307/1911819 10.1023/A:1021781308794 10.1007/s11117-008-2220-0 10.2140/pjm.1972.40.709 10.1016/0022-247X(67)90163-1 10.1080/0233193021000031615 10.1007/978-3-0348-0439-4 10.1007/s11117-013-0238-4 10.1007/BF02190126 10.1137/120901805 10.1007/s10957-013-0456-8 10.1080/02331930903552473 10.1007/PL00000466 10.1007/s10957-014-0546-2 10.1016/j.na.2010.09.066 10.1073/pnas.39.1.42 10.1016/j.hm.2003.07.001 10.4153/CMB-2012-028-5 10.1137/100817061 10.1137/1.9781611971088 10.1007/BFb0093633 10.1525/9780520411586-036 10.4064/sm-13-2-137-179 10.1155/2014/453912 |
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| DOI | 10.1007/s10898-015-0379-z |
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| Keywords | Fritz John conditions 90C30 90C46 Lagrange multipliers Karush–Kuhn–Tucker conditions 46A22 Nonlinear programming 26B25 Infsup-convexity Separation theorem |
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| Snippet | For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points... |
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| SubjectTerms | Computer Science Convex analysis Equivalence Euclidean space Inequalities Lagrange multiplier Lagrange multipliers Mathematical analysis Mathematical functions Mathematics Mathematics and Statistics Minimax technique Nonlinear programming Nonlinearity Operations Research/Decision Theory Optimization Real Functions Saddle points Studies Theorems |
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| Title | A sharp Lagrange multiplier theorem for nonlinear programs |
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