论文标题
使用切换变量的二次优化：$ n = 2 $的凸壳
Quadratic Optimization with Switching Variables: The Convex Hull for $n = 2$
论文作者
论文摘要
我们考虑变量中的二次优化$（x，y）$其中$ 0 \ le x \ le y $，而$ y \ y \ in \ {0,1 \}^n $。这种二进制$ y $通常称为“指标”或“切换”变量，通常发生在应用中。解决此类问题的一种方法是基于表示或近似集合$ \ {（x，xx^t，yy^t）的凸壳：0 \ le x \ le x \ le y \ in \ in \ {0,1 \}^n \} $。 $ n = 1 $的情况的表示形式已知，并且已被广泛使用。我们通过从凸船体的分析表示开始，然后消除不会将投影变为原始变量的辅助变量和约束来提供确切的$ n = 2 $。当产品项$ y_1y_2 $被忽略时，该表示形式的替代派生导致了3 = 2 $的简化凸船体的吸引人猜想。
We consider quadratic optimization in variables $(x,y)$ where $0\le x\le y$, and $y\in\{0,1\}^n$. Such binary $y$ are commonly refered 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 auxilliary 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.
