Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1453 - 1468, Volume 31, Issue 4, December 2011      doi:10.3934/dcds.2011.31.1453

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Radu Strugariu - Technical University Iaşi, Department of Mathematics, 700506–Iaşi, Romania (email)
Mircea D. Voisei - Towson University, Department of Mathematics, 7800 York Rd, Towson, Maryland 21252, United States (email)
Constantin Zălinescu - University “Al.I.Cuza” Iaşi, Faculty of Mathematics, and Institute of Mathematics Octav Mayer, 700506–Iaşi, Romania (email)

Abstract: 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:  Counter-examples, bi-duality, triality, tri-duality.
Mathematics Subject Classification:  Primary: 49N15; Secondary: 90C26.

Received: June 2009;      Revised: July 2010;      Available Online: September 2011.