Counter-examples in bi-duality, triality and tri-duality

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December 2011, 31(4): 1453-1468. doi: 10.3934/dcds.2011.31.1453

Counter-examples in bi-duality, triality and tri-duality

Radu Strugariu ^1, , Mircea D. Voisei ^2, and Constantin Zălinescu ^3,

1.	Technical University Iaşi, Department of Mathematics, 700506–Iaşi, Romania
2.	Towson University, Department of Mathematics, 7800 York Rd, Towson, Maryland 21252, United States
3.	University “Al.I.Cuza” Iaşi, Faculty of Mathematics, and Institute of Mathematics Octav Mayer, 700506–Iaşi, Romania

Received June 2009 Revised July 2010 Published September 2011

Abstract
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In this paper, by providing simple counterexamples, several important results in bi-duality, triality and tri-duality, an optimization theory established and presented by D.Y. Gao in his book "Duality Principles in Nonconvex Systems. Theory, Methods and Applications," Kluwer Academic Publishers, Dordrecht, 2000, are proven to be false. Other results concerning this optimization theory from subsequent papers by D.Y. Gao and his collaborators are analyzed, false claims are exposed and when possible corrected, while the possibility or impossibility of obtaining correct various alternatives to the classical minimax relations are discussed.

Keywords: bi-duality, tri-duality., triality, Counter-examples.

Mathematics Subject Classification: Primary: 49N15; Secondary: 90C2.

Citation: Radu Strugariu, Mircea D. Voisei, Constantin Zălinescu. Counter-examples in bi-duality, triality and tri-duality. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1453-1468. doi: 10.3934/dcds.2011.31.1453

References:

[1]	D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory, Methods and Applications," Kluwer Academic Publishers, Dordrecht, 2000.
[2]	D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optim., 17 (2000), 127-160. doi: 10.1023/A:1026537630859.
[3]	D. Y. Gao, Bi-duality in nonconvex optimization, in "Encyclopedia of Optimization" (eds. C. A. Floudas and P. M. Pardalos), 2^nd edition, Springer, New York, (2009), 814-818.
[4]	D. Y. Gao, Mono-duality in convex optimization, in "Encyclopedia of Optimization" (eds. C. A. Floudas and P. M. Pardalos), 2^nd edition, Springer, New York, (2009), 818-822.
[5]	D. Y. Gao and H. D. Sherali, Canonical duality: Connection between nonconvex mechanics and global optimization, in "Advances in Applied Mathematics and Global Optimization," Springer, New York, (2009), 257-326.
[6]	C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002.

show all references

References:

[1]	D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory, Methods and Applications," Kluwer Academic Publishers, Dordrecht, 2000.
[2]	D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optim., 17 (2000), 127-160. doi: 10.1023/A:1026537630859.
[3]	D. Y. Gao, Bi-duality in nonconvex optimization, in "Encyclopedia of Optimization" (eds. C. A. Floudas and P. M. Pardalos), 2^nd edition, Springer, New York, (2009), 814-818.
[4]	D. Y. Gao, Mono-duality in convex optimization, in "Encyclopedia of Optimization" (eds. C. A. Floudas and P. M. Pardalos), 2^nd edition, Springer, New York, (2009), 818-822.
[5]	D. Y. Gao and H. D. Sherali, Canonical duality: Connection between nonconvex mechanics and global optimization, in "Advances in Applied Mathematics and Global Optimization," Springer, New York, (2009), 257-326.
[6]	C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002.

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