We consider quadratic optimization in variables (x, y) where \(0\le x\le y\), and \(y\in \{ 0,1 \}^n\). Such binary variables are commonly referred to as indicator or switching variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set \(\{ (x,xx^T, yy^T)\,:\,0\le x\le y\in \{ 0,1 \}^n \}\). A representation for the case \(n=1\) is known and has been widely used. We give an exact representation for the case \(n=2\) by starting with a disjunctive representation for the convex hull and then eliminating auxiliary variables and constraints that do not change the projection onto the original variables. An alternative derivation for this representation leads to an appealing conjecture for a simplified representation of the convex hull for \(n=2\) when the product term \(y_1y_2\) is ignored.

我们考虑变量 ( x ,  y ) 中的二次优化，其中\(0\le x\le y\)和\(y\in \{ 0,1 \}^n\)。这种二元变量通常被称为指示变量或开关变量，并且通常出现在应用程序中。解决此类问题的一种方法是基于表示或逼近集合\(\{ (x,xx^T, yy^T)\,:\,0\le x\le y\in \{ 0, 1 \}^n \}\)。情况\(n=1\) 的表示是已知的并且已被广泛使用。我们给出了这种情况的精确表示\(n=2\)通过从凸包的析取表示开始，然后消除不会改变对原始变量的投影的辅助变量和约束。当忽略乘积项\(y_1y_2\)时，此表示的另一种推导导致了对\(n=2\)凸包的简化表示的吸引人的猜想。