I have a question regarding complementary slackness, the answer should be true of false.
The complementary slackness conditions connect pairs of optimal basic feasible solution of primal and dual linear programs. They correspond to connection between non zero variables in the solution to one linear program and constraints that are satisfied with equality in the other linear program. If there is optimal solution that is not basic feasible solution, complementary slackness condition may not be held.
In my opinion, the answer is false, but I am not completely sure. Why it mentions the optimal solution which is not basic feasible solution? I thought it's impossible, optimal solution has to be one of the BFSs. The following remark made the issue completely ambiguous Deterministic Operations Research: Models and Methods in Linear Optimization. So, on every iteration the complementary slackness conditions are held? But this is not what the complementary slackness theorem was talking about.
If you have good understanding of the topic, please point out what exactly I missed. Thanks!