# American Institute of Mathematical Sciences

January  2012, 8(1): 229-242. doi: 10.3934/jimo.2012.8.229

## 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

Received  July 2011 Revised  September 2011 Published  November 2011

This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality. Results show that the triality theory holds strongly in the tri-duality form for our problem if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Some numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the local minimum and local maximum.
Citation: David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229
##### References:
 [1] D. Y. Gao, Post-buckling 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. Google Scholar [2] D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications," Nonconvex Optimization and its Applications, 39, Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar [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), 467-493. doi: 10.1080/02331930310001611501.  Google Scholar [4] D. Y. Gao, Canonical duality theory: Theory, method, and applications in global optimization, Comput. Chem., 33 (2009), 1964-1972. doi: 10.1016/j.compchemeng.2009.06.009.  Google Scholar [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), 497-522. doi: 10.1093/qjmam/hbn014.  Google Scholar [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), 257-326.  Google Scholar [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), 3660-3673. doi: 10.1016/j.ijsolstr.2007.08.027.  Google Scholar [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, Cambridge, 1985.  Google Scholar [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), 111-127.  Google Scholar [11] T. W. B. Kibble, Phase transitions and topological defects in the early universe, Aust. J. Phys., 50 (1997), 697-722. doi: 10.1071/P96076.  Google Scholar [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), 197-244. doi: 10.1007/BF01011513.  Google Scholar [13] M. D. Voisei and C. Zălinescu, Some remarks concerning Gao-Strang's complementary gap function, Applicable Analysis, 90 2010, 1111-1121. doi: 10.1080/00036811.2010.483427.  Google Scholar [14] M. D. Voisei and C. Zălinescu, Counterexamples to some triality and tri-duality results, J. Glob. Optim., 49 (2011), 173-183. doi: 10.1007/s10898-010-9592-y.  Google Scholar

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##### References:
 [1] D. Y. Gao, Post-buckling 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. Google Scholar [2] D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications," Nonconvex Optimization and its Applications, 39, Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar [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), 467-493. doi: 10.1080/02331930310001611501.  Google Scholar [4] D. Y. Gao, Canonical duality theory: Theory, method, and applications in global optimization, Comput. Chem., 33 (2009), 1964-1972. doi: 10.1016/j.compchemeng.2009.06.009.  Google Scholar [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), 497-522. doi: 10.1093/qjmam/hbn014.  Google Scholar [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), 257-326.  Google Scholar [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), 3660-3673. doi: 10.1016/j.ijsolstr.2007.08.027.  Google Scholar [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, Cambridge, 1985.  Google Scholar [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), 111-127.  Google Scholar [11] T. W. B. Kibble, Phase transitions and topological defects in the early universe, Aust. J. Phys., 50 (1997), 697-722. doi: 10.1071/P96076.  Google Scholar [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), 197-244. doi: 10.1007/BF01011513.  Google Scholar [13] M. D. Voisei and C. Zălinescu, Some remarks concerning Gao-Strang's complementary gap function, Applicable Analysis, 90 2010, 1111-1121. doi: 10.1080/00036811.2010.483427.  Google Scholar [14] M. D. Voisei and C. Zălinescu, Counterexamples to some triality and tri-duality results, J. Glob. Optim., 49 (2011), 173-183. doi: 10.1007/s10898-010-9592-y.  Google Scholar
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