Let ${\bf c}_{1}$, ${\bf c}_{2}\in \mathbb{R}^{n}$, ${\bf A}\in\mathbb{R}^{m\times n}$ and ${\bf b}\in\mathbb{R}^{m}$.
Show how one can solve the optimization problem:
min $\,$ min$\left({\bf c}^{T}_{1}{\bf x}\,,\, {\bf c}^{T}_{2}{\bf x}\right)$
$\,\,$s.t. $\,\,\,\,$ ${\bf Ax=b}$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ ${\bf x\ge 0}$
If this were a minimizing the maximum problem, I would have defined a free variable to address the maximum and then the objective would just be minimizing the new free variable.
However, I can't do the same for this. So how should I approach this problem?
It's a linear programming problem, so it only has one optimal solution. If $c_1$ and $c_2$ are fixed, then just solve them separately to get the minimum one as your solution.
If $c_1$ and $c_2$ is not fixed, I think your question is
$min: c^Tx\\ s.t. Ax=b\\ x\geq0\\ c\in R^n$
Assume there exists $x$ satisfies $Ax=b, \ x\geq0$, then we can always find a $c$ to make the objective function smaller. Therefore, the minimum of this problem is $-\infty$.