I assume the \(a_i\) and \(b_i\) are variables. First, strict inequalities are anathema in optimization models. You need either to switch to a weak inequality (\(\min\{a_1,\dots,a_m\}\le \max\{b_1,\dots,b_m\}\)) or else specify a minimum acceptable difference (\(\min\{a_1,\dots,a_m\}\le \max\{b_1,\dots,b_m\}-\epsilon\) for some \(\epsilon > 0\)). Second, I don't think there is any way to linearize this without introducing binary variables. What you are saying equates to \(a_i \le b_j\) for some \(i\) and \(j\). You can introduce binary variables \(x_{ij},\,1\le i,j\le m\), constraints \(a_i\le b_j + M(1-x_{ij})\,\forall i,j\) and one more constraint \(\sum_{i=1}^m \sum_{j=1}^m x_{ij} \ge 1\), where \(M\) is some sufficiently large positive constant.
answered
Paul Rubin ♦♦ |