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Convex optimization on mixed domains
On the triality theory for a quartic polynomial optimization problem
1. | School of Sciences, Information Technology and Engineering, University of Ballarat, Victoria 3353, Australia |
2. | School of Science, Information Technology and Engineering, University of Ballarat, Victoria 3353, Australia |
References:
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D. Y. Gao, Post-buckling analysis and anomalous dual variational problems in nonlinear beam theory,, in, (1996). Google Scholar |
[2] |
D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications,", Nonconvex Optimization and its Applications, 39 (2000).
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D. Y. Gao, Perfect duality theory and complete solutions to a class of global optimization problems. Theory, methods and applications of optimization,, Optim., 52 (2003), 467.
doi: 10.1080/02331930310001611501. |
[4] |
D. Y. Gao, Canonical duality theory: Theory, method, and applications in global optimization,, Comput. Chem., 33 (2009), 1964.
doi: 10.1016/j.compchemeng.2009.06.009. |
[5] |
D. Y. Gao and R. W. Ogden, Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation,, Quart. J. Mech. Appl. Math., 61 (2008), 497.
doi: 10.1093/qjmam/hbn014. |
[6] |
D. Y. Gao and H. D. Sherali, Canonical duality theory: Connection between nonconvex mechanics and global optimization,, in, 17 (2009), 257.
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[7] |
D. Y. Gao and H. F. Yu, Multi-scale modelling and canonical dual finite element method in phase transitions of solids,, International Journal of Solids and Structures, 45 (2008), 3660.
doi: 10.1016/j.ijsolstr.2007.08.027. |
[8] |
J. Gallier, The Schur complement and symmetric positive semidefinite (and definite) matrices,, \url{http://www.cis.upenn.edu/~jean/schur-comp.pdf}., (). Google Scholar |
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R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (1985).
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A. Jaffe, Constructive quantum field theory,, in, (2000), 111.
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T. W. B. Kibble, Phase transitions and topological defects in the early universe,, Aust. J. Phys., 50 (1997), 697.
doi: 10.1071/P96076. |
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J. S. Rowlinson, Translation of J. D. van der Waals' "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density",, J. Statist. Phys., 20 (1979), 197.
doi: 10.1007/BF01011513. |
[13] |
M. D. Voisei and C. Zălinescu, Some remarks concerning Gao-Strang's complementary gap function,, Applicable Analysis, 90 (2010), 1111.
doi: 10.1080/00036811.2010.483427. |
[14] |
M. D. Voisei and C. Zălinescu, Counterexamples to some triality and tri-duality results,, J. Glob. Optim., 49 (2011), 173.
doi: 10.1007/s10898-010-9592-y. |
show all references
References:
[1] |
D. Y. Gao, Post-buckling analysis and anomalous dual variational problems in nonlinear beam theory,, in, (1996). Google Scholar |
[2] |
D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications,", Nonconvex Optimization and its Applications, 39 (2000).
|
[3] |
D. Y. Gao, Perfect duality theory and complete solutions to a class of global optimization problems. Theory, methods and applications of optimization,, Optim., 52 (2003), 467.
doi: 10.1080/02331930310001611501. |
[4] |
D. Y. Gao, Canonical duality theory: Theory, method, and applications in global optimization,, Comput. Chem., 33 (2009), 1964.
doi: 10.1016/j.compchemeng.2009.06.009. |
[5] |
D. Y. Gao and R. W. Ogden, Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation,, Quart. J. Mech. Appl. Math., 61 (2008), 497.
doi: 10.1093/qjmam/hbn014. |
[6] |
D. Y. Gao and H. D. Sherali, Canonical duality theory: Connection between nonconvex mechanics and global optimization,, in, 17 (2009), 257.
|
[7] |
D. Y. Gao and H. F. Yu, Multi-scale modelling and canonical dual finite element method in phase transitions of solids,, International Journal of Solids and Structures, 45 (2008), 3660.
doi: 10.1016/j.ijsolstr.2007.08.027. |
[8] |
J. Gallier, The Schur complement and symmetric positive semidefinite (and definite) matrices,, \url{http://www.cis.upenn.edu/~jean/schur-comp.pdf}., (). Google Scholar |
[9] |
R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (1985).
|
[10] |
A. Jaffe, Constructive quantum field theory,, in, (2000), 111.
|
[11] |
T. W. B. Kibble, Phase transitions and topological defects in the early universe,, Aust. J. Phys., 50 (1997), 697.
doi: 10.1071/P96076. |
[12] |
J. S. Rowlinson, Translation of J. D. van der Waals' "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density",, J. Statist. Phys., 20 (1979), 197.
doi: 10.1007/BF01011513. |
[13] |
M. D. Voisei and C. Zălinescu, Some remarks concerning Gao-Strang's complementary gap function,, Applicable Analysis, 90 (2010), 1111.
doi: 10.1080/00036811.2010.483427. |
[14] |
M. D. Voisei and C. Zălinescu, Counterexamples to some triality and tri-duality results,, J. Glob. Optim., 49 (2011), 173.
doi: 10.1007/s10898-010-9592-y. |
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