  The main purpose of this research note is to show that the triality theory
can always be used to identify both global minimizer and the biggest local
maximizer in global optimization. An open problem left on the double-min
duality is solved for a nonconvex optimization problem with double-well
potential in $\real^n$, which leads to a complete set of analytical solutions.
Also a convergency theorem is proved for linear perturbation canonical dual
method, which can be used for solving global optimization problems with
multiple solutions. The methods and results presented in this note pave the way
towards the proof of the triality theory in general cases.
