Answer by Kevin Dalmeijer for Variable bounds in column generation
When pricing $x_j$, there are two possibilities: $x_j$ is not in the restricted master problem / $x_j$ has not been added before. In this case, $x_j=0$ in the current solution. The constraint $x_j \le...
View ArticleAnswer by Qian Zhang for Variable bounds in column generation
Another understanding can be the following. Denote $X = \{ x \in \Bbb R^n \mid 0 \leq x_j \leq 1, \forall j = 1,\dots, n\}$. When solving the pricing problem successfully to optimal, all variable bound...
View ArticleAnswer by Sune for Variable bounds in column generation
Assuming that the $a_{ij}$'s are either zero or one, and the $c_j$'s are positive, you do not need the upper bound on the variables. To see this, if $x_j=1$ for some $j$, then column $j$ covers all...
View ArticleVariable bounds in column generation
Consider the set covering problem \[ \begin{align} \min&\ \sum_{j=1}^nc_jx_j\\ s.t.:&\ \sum_{j=1}^na_{ij}x_j\geq 1,\quad \forall i=1,\dots,m\\ &\ 0\leq x_j \leq 1 \end{align} \] which, in...
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