DY Gao solely or together with some of his collaborators applied his Canonical duality theory (CDT) for solving a class of unconstrained optimization problems, getting the so-called triality theorems. Unfortunately, the double-min duality from these results published before 2010 revealed to be false, even if in 2003 DY Gao announced that certain additional conditions are needed for getting it. After 2010 DY Gao together with some of his collaborators published several papers in which they added additional conditions for getting double-min and double-max dualities in the triality theorems. The aim of this paper is to treat rigorously this kind of problems and to discuss several results concerning the triality theory obtained up to now.
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