Lesson 1

Let \(x_{i}\in\mathbb{Z}_{\ge 0}\) be the number of shares of stock \(i\). Let \(y_{i}\) be a 0/1-variable that records whether stock \(i\) is used at all. The connection between them is \[0\le x_{i}\le m_{i} y_{i},\qquad y_{i}\le x_{i}\quad\text{when }m_{i}>0.\] The second inequality is optional if the objective never rewards setting \(y_{i}=1\) for no reason, but writing it makes the meaning exact.

The budget and sector caps are \[\sum_{i} p_{i} x_{i}\le B,\qquad \sum_{i\in S_j}p_{i} x_{i}\le C_{j}.\] A conflict between \(u\) and \(v\) is \[y_{u}+y_{v}\le 1.\] A dependency \(u\to v\) means that selecting stock \(u\) requires selecting stock \(v\) as well; it is written as \[y_{u}\le y_{v}.\]