论文标题
在指示器的凸面二次优化问题上
On the convex hull of convex quadratic optimization problems with indicators
论文作者
论文摘要
我们考虑指标变量的凸二次优化问题，并在指示器上的任意约束。我们表明，在扩展空间中相关的混合刻机集的凸面描述，其中二次数量的附加变量由单个正面半限制约束（明确说明）和线性约束组成。特别是，对这类问题的凸化减少了描述扩展公式中的多面体设置。尽管该多面体集的顶点表示是指数级的，并且一般不容易获得显式线性不等式描述，但我们得出了一种紧凑的混合构成线性公式，其溶液与多面体集合的顶点一致。 We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are ``finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of混合成员非线性集。
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are ``finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.
