SIAM Journal on Optimization, Volume 30, Issue 1, Page 1033-1047, January 2020.
A convex optimization problem in conic form involves minimizing a linear functional over the intersection of a convex cone and an affine subspace. In some cases, it is possible to replace a conic formulation using a certain cone, with a “lifted” conic formulation using another cone that is higher-dimensional, but simpler, in some sense. One situation in which this can be computationally advantageous is when the higher-dimensional cone is a Cartesian product of many “low-complexity” cones, such as second-order cones, or small positive semidefinite cones. This paper studies obstructions to a convex cone having a lifted representation with such a product structure. The main result says that whenever a convex cone has a certain neighborliness property, then it does not have a lifted representation using a finite product of cones, each of which has only short chains of faces. This is a generalization of recent work of Averkov [SIAM J. Appl. Alg. Geom., 3 (2019), pp. 128--151], which considers only lifted representations using products of positive semidefinite cones of bounded size. Among the consequences of the main result is that various cones related to nonnegative polynomials do not have lifted representations using products of “low-complexity” cones, such as smooth cones, the exponential cone, and cones defined by hyperbolic polynomials of low degree.

中文翻译：

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没有长脸链的凸锥体表达能力的限制

SIAM优化杂志，第30卷，第1期，第1033-1047页，2020年1月。
圆锥形的凸优化问题涉及最小化凸锥和仿射子空间相交处的线性泛函。在某些情况下，可以使用某个圆锥体替换圆锥形配方，而使用另一个具有较高尺寸但在某种意义上更简单的圆锥体的“提升”圆锥形配方。一种可能在计算上具有优势的情况是，高维圆锥是许多“低复杂度”圆锥（例如二阶圆锥或小的正半定圆锥）的笛卡尔积。本文研究了具有这种产品结构的凸起圆锥体的障碍物。主要结果表明，只要凸锥具有一定的相邻性，那么使用锥的有限乘积就不会具有提升的表示，每个人只有短短的面孔链。这是Averkov [SIAM J. Appl。海藻 Geom。，3（2019），pp.128--151]，它仅考虑使用正大小为正的半定锥的乘积的提升表示。主要结果的后果之一是，与非负多项式相关的各种圆锥都没有使用“低复杂度”圆锥的乘积来提升表示，例如光滑圆锥，指数圆锥和由低度双曲多项式定义的圆锥。