论文标题
获得或失去观点
Gaining or losing perspective
论文作者
论文摘要
We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[l,u]$, where $z$ is a binary indicatorof $x\in[l,u]$ ($u> \ell > 0$), and $y$ "captures" $f(x)$, which is assumed to be convex on its domain $[l,u]$,但是否则$ y = 0 $当$ x = 0 $。当活动具有运营范围时，我们为执行每项活动支付固定成本，并且活动级别的成本为凸面。 我们使用数量作为比较凸体的量度，研究了该模型的各种连续放松，其中之一是凸起船体，它通过“透视重新印象” $ y \ y \ geq zf（x/z）$实现。我们将其与各种较弱的放松相提并论，研究它们何时可以被视为可行的替代方案。在重要的特殊情况下，当$ f（x）：= x^p $，对于$ p> 1 $时，使用不平等的$ yz^q \ geq x^p $的放松，对于[0，p-1] $，对于$ q \ in [0，p-1] $，是较高维的幂cone，是可代表的，因此在理论上是可拖延的。一个众所周知的具体应用（$ f（x）：= x^2 $）是均值优化（以马克维茨的风格），我们进行了一些实验，以说明我们在该应用程序上的理论。
We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[l,u]$, where $z$ is a binary indicatorof $x\in[l,u]$ ($u> \ell > 0$), and $y$ "captures" $f(x)$, which is assumed to be convex on its domain $[l,u]$, but otherwise $y=0$ when $x=0$. This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the "perspective reformulation" inequality $y \geq zf(x/z)$. We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $f(x) := x^p$, for $p>1$, relaxations utilizing the inequality $yz^q \geq x^p$, for $q \in [0,p-1]$, are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $f(x) := x^2$) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.
