Suppose we have a primal problem $P$ which is stated as a maximization problem $\max c^{T} x$.
The dual problem is defined (Introduction to Linear Optimization by Dimitris Bertsimas) only for a primal minimization problem.
Then what is the dual problem of $P$ ?
Is it implicit, that the dual problem of $P$ is the dual problem of $P$ stated as the minimization problem $\min -c^T x$ ?
Surely these two primal problems are equivalent in the sense that their optimal solution $ \bar x$ are equal (if it exist). However, the objective values are the same only if we ignore the sign !