We introduce a family of symmetric convex bodies called generalized ellipsoids of degree $d$ (GE-$d$s), with ellipsoids corresponding to the case of $d=0$. Generalized ellipsoids (GEs) retain many geometric, algebraic, and algorithmic properties of ellipsoids. We show that the conditions that the parameters of a GE must satisfy can be checked in strongly polynomial time, and that one can search for GEs of a given degree by solving a semidefinite program whose size grows only linearly with dimension. We give an example of a GE which does not have a second-order cone representation, but show that every GE has a semidefinite representation whose size depends linearly on both its dimension and degree. In terms of expressiveness, we prove that for any integer $m\geq 2$, every symmetric full-dimensional polytope with $2m$ facets and every intersection of $m$ co-centered ellipsoids can be represented exactly as a GE-$d$ with $d \leq 2m-3$. Using this result, we show that every symmetric convex body can be approximated arbitrarily well by a GE-$d$ and we quantify the quality of the approximation as a function of the degree $d$. Finally, we present applications of GEs to several areas, such as time-varying portfolio optimization, stability analysis of switched linear systems, robust-to-dynamics optimization, and robust polynomial regression.

翻译：本文引入了一族称为d次广义椭球体（GE-$d$）的对称凸体，其中椭球体对应于$d=0$的情形。广义椭球体（GEs）保留了椭球体的许多几何、代数与算法性质。我们证明，广义椭球体参数所需满足的条件可在强多项式时间内验证，并且可以通过求解一个规模仅随维度线性增长的半定规划来搜索给定次数的广义椭球体。我们给出了一个不具有二阶锥表示的广义椭球体实例，但证明了每个广义椭球体均存在一个规模随其维度和次数线性增长的半定表示。在表达能力方面，我们证明对于任意整数$m\geq 2$，每个具有$2m$个面的对称满维多面体以及任意$m$个共心椭球体的交集均可精确表示为$d \leq 2m-3$的广义椭球体。基于此结果，我们证明了任意对称凸体均可被广义椭球体任意逼近，并量化了逼近质量与次数$d$的函数关系。最后，我们展示了广义椭球体在时变投资组合优化、切换线性系统稳定性分析、动态鲁棒优化以及鲁棒多项式回归等多个领域的应用。