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, (1996).   Google Scholar

[2]

D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods and Applications,", Nonconvex Optimization and its Applications, 39 (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.  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.  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.  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, 17 (2009), 257.   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.  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, (1985).   Google Scholar

[10]

A. Jaffe, Constructive quantum field theory,, in, (2000), 111.   Google Scholar

[11]

T. W. B. Kibble, Phase transitions and topological defects in the early universe,, Aust. J. Phys., 50 (1997), 697.  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.  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.  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.  doi: 10.1007/s10898-010-9592-y.  Google Scholar

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).   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.  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.  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.  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, 17 (2009), 257.   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.  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, (1985).   Google Scholar

[10]

A. Jaffe, Constructive quantum field theory,, in, (2000), 111.   Google Scholar

[11]

T. W. B. Kibble, Phase transitions and topological defects in the early universe,, Aust. J. Phys., 50 (1997), 697.  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.  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.  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.  doi: 10.1007/s10898-010-9592-y.  Google Scholar

[1]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[2]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[3]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[4]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[5]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[6]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[7]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[8]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[9]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[10]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]