SIAM Journal on Optimization, Volume 30, Issue 2, Page 1251-1273, January 2020.
We present a unified geometrical analysis on the completely positive programming (CPP) reformulations of quadratic optimization problems (QOPs) and their extension to polynomial optimization problems (POPs) based on a class of geometrically defined nonconvex conic programs and their convexification. The class of nonconvex conic programs minimize a linear objective function in a vector space $\mathbb{V}$ over the constraint set represented geometrically as the intersection of a nonconvex cone $\mathbb{K} \subset \mathbb{V}$, a face $\mathbb{J}$ of the convex hull of $\mathbb{K}$, and a parallel translation $\mathbb{L}$ of a hyperplane. We show that under moderate assumptions, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of $\mathbb{J}$ and $\mathbb{L}$. The replacement procedure is applied for deriving the CPP reformulations of QOPs and their extension to POPs.

中文翻译：

---

二次多项式最优化问题的凸圆锥重构的几何分析

SIAM优化杂志，第30卷，第2期，第1251-1273页，2020年1月。
我们基于一类几何定义的非凸圆锥程序及其凸性，对二次优化问题（QOP）的完全正编程（CPP）重构及其对多项式优化问题（POP）的扩展提出了统一的几何分析。一类非凸圆锥程序在几何上表示为非凸圆锥$ \ mathbb {K} \ subset \ mathbb {V} $的交集的约束集上，向量空间$ \ mathbb {V} $中的线性目标函数最小化， $ \ mathbb {K} $的凸包的面$ \ mathbb {J} $和超平面的平行平移$ \ mathbb {L} $。我们表明，在中等假设下，通过将约束集替换为$ \ mathbb {J} $和$ \ mathbb {L} $的交集，可以将原始的非凸圆锥程序等效地重新构造为凸圆锥程序。替换程序适用于推导QOP的CPP格式及其对POP的扩展。