On the triality theory for a quartic polynomial optimization problem

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

On the triality theory for a quartic polynomial optimization problem

David Yang Gao ^1, and Changzhi Wu ^2,

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

Abstract
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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.

Keywords: global optimization, counter-examples., triality, polynomial optimization, Canonical duality.

Mathematics Subject Classification: 90C26, 90C30, 90C46, 90C4.

Citation: David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial and 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.
[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), 467-493. doi: 10.1080/02331930310001611501.
[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.
[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.
[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.
[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.
[8]	J. Gallier, The Schur complement and symmetric positive semidefinite (and definite) matrices, http://www.cis.upenn.edu/~jean/schur-comp.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), 111-127.
[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.
[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.
[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.
[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.

show all references

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.
[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), 467-493. doi: 10.1080/02331930310001611501.
[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.
[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.
[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.
[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.
[8]	J. Gallier, The Schur complement and symmetric positive semidefinite (and definite) matrices, http://www.cis.upenn.edu/~jean/schur-comp.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), 111-127.
[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.
[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.
[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.
[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.

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