  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.
