I asked this question here but I noticed that this notation $x^*$ may actually mean two things: primal optimality and dual optimality. Please, explain this notation particularly here: I understand $\bar 0\in\partial f(x^*)$ so that there is a global minimum but whether this optima is dual or primal, I don't know -- and what the difference is to the notation $\bar 0\in\partial f(x)$?
This definition here appears in topics related to Farkas' lemma, Motzkin and Gordan.
P.s. I am studying nonlinear optimization 2.3139.