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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:
[1] 
D. Y. Gao, Postbuckling analysis and anomalous dual variational problems in nonlinear beam theory, in "Proc. of the Fifth Pan American Congress of Applied Mechanics" (eds. L. A. Godoy and L. E. Suarez), Vol. 4, Applied Mechanics in Americans, The University of Iowa, Iowa City, 1996. 
[2] 
D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications," Nonconvex Optimization and its Applications, 39, Kluwer Academic Publishers, Dordrecht, 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), 467493. doi: 10.1080/02331930310001611501. 
[4] 
D. Y. Gao, Canonical duality theory: Theory, method, and applications in global optimization, Comput. Chem., 33 (2009), 19641972. doi: 10.1016/j.compchemeng.2009.06.009. 
[5] 
D. Y. Gao and R. W. Ogden, Multiple solutions to nonconvex variational problems with implications for phase transitions and numerical computation, Quart. J. Mech. Appl. Math., 61 (2008), 497522. doi: 10.1093/qjmam/hbn014. 
[6] 
D. Y. Gao and H. D. Sherali, Canonical duality theory: Connection between nonconvex mechanics and global optimization, in "Advances in Applied Mathematics and Global Optimization," Adv. Mech. Math., 17, Springer, New York, (2009), 257326. 
[7] 
D. Y. Gao and H. F. Yu, Multiscale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 45 (2008), 36603673. 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/schurcomp.pdf}., (). 
[9] 
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. 
[10] 
A. Jaffe, Constructive quantum field theory, in "Mathematical Physics 2000" (eds. A. Fokas, A. Grigoryan, T. Kibble and B. Zegarlinski), Imperial College Press, London, (2000), 111127. 
[11] 
T. W. B. Kibble, Phase transitions and topological defects in the early universe, Aust. J. Phys., 50 (1997), 697722. 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), 197244. doi: 10.1007/BF01011513. 
[13] 
M. D. Voisei and C. Zălinescu, Some remarks concerning GaoStrang's complementary gap function, Applicable Analysis, 90 2010, 11111121. doi: 10.1080/00036811.2010.483427. 
[14] 
M. D. Voisei and C. Zălinescu, Counterexamples to some triality and triduality results, J. Glob. Optim., 49 (2011), 173183. doi: 10.1007/s108980109592y. 
show all references
References:
[1] 
D. Y. Gao, Postbuckling analysis and anomalous dual variational problems in nonlinear beam theory, in "Proc. of the Fifth Pan American Congress of Applied Mechanics" (eds. L. A. Godoy and L. E. Suarez), Vol. 4, Applied Mechanics in Americans, The University of Iowa, Iowa City, 1996. 
[2] 
D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications," Nonconvex Optimization and its Applications, 39, Kluwer Academic Publishers, Dordrecht, 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), 467493. doi: 10.1080/02331930310001611501. 
[4] 
D. Y. Gao, Canonical duality theory: Theory, method, and applications in global optimization, Comput. Chem., 33 (2009), 19641972. doi: 10.1016/j.compchemeng.2009.06.009. 
[5] 
D. Y. Gao and R. W. Ogden, Multiple solutions to nonconvex variational problems with implications for phase transitions and numerical computation, Quart. J. Mech. Appl. Math., 61 (2008), 497522. doi: 10.1093/qjmam/hbn014. 
[6] 
D. Y. Gao and H. D. Sherali, Canonical duality theory: Connection between nonconvex mechanics and global optimization, in "Advances in Applied Mathematics and Global Optimization," Adv. Mech. Math., 17, Springer, New York, (2009), 257326. 
[7] 
D. Y. Gao and H. F. Yu, Multiscale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 45 (2008), 36603673. 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/schurcomp.pdf}., (). 
[9] 
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. 
[10] 
A. Jaffe, Constructive quantum field theory, in "Mathematical Physics 2000" (eds. A. Fokas, A. Grigoryan, T. Kibble and B. Zegarlinski), Imperial College Press, London, (2000), 111127. 
[11] 
T. W. B. Kibble, Phase transitions and topological defects in the early universe, Aust. J. Phys., 50 (1997), 697722. 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), 197244. doi: 10.1007/BF01011513. 
[13] 
M. D. Voisei and C. Zălinescu, Some remarks concerning GaoStrang's complementary gap function, Applicable Analysis, 90 2010, 11111121. doi: 10.1080/00036811.2010.483427. 
[14] 
M. D. Voisei and C. Zălinescu, Counterexamples to some triality and triduality results, J. Glob. Optim., 49 (2011), 173183. doi: 10.1007/s108980109592y. 
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