Counterexamples to a triality theorem in “Canonical dual least square method”

In this short note we give two counterexamples to the Triality Theorem concerning the optimization problem presented in Ruan et al., Comput. Optim. Appl. (2008). In order to understand the problem first we introduce the framework of this paper and quote the theorem we have in view. Then, while trying to verify this theorem we encounter several difficulties and note their nature. We end by revealing our counterexamples.