# Dual problem of a maximization primal problem $P$?

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 !

• The dual problem of P should have dual variables. And where are your constraints ? Commented Nov 14, 2014 at 11:24

So, concerning your question the dual of a $\max$ problem is a $\min$ problem without any need to transform the $\max$ firstly to a $\min$ and then take the dual.
If you insist transforming first to a $\min$ problem and then taking the dual, then it is correct (as you say) that $$\max c^Tx=-\min(-c^T)x$$ so the objective value will be the same but with opposite signs.