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摘要翻译：
如果一个集合$S\子集合Q\re^n$等于一个集合在高维空间中的投影，且该集合可以用某种线性矩阵不等式(LMI)来描述，则称其为{它半正定(SDP)}可表示。对于有界SDP可表示集$w_1,...,w_m$,我们给出了$\cv{\cup_{k=1}^mw_k}$的一个SDP表示的显式构造。这提供了一种从局部SDP表示构建全局SDP表示的技术。(ii)对于紧凸半代数集$S$的SDP可表示性，我们证明了边界$\bds$是正弯曲的充分条件和必要条件:$\bds$在光滑点和非退化角上具有非负曲率。这相当于定义多项式在$\bds$上消失的点上的严格和非严格准凹性。它们之间的间隙是$\bds$，具有正曲率和非负曲率，光滑点和非光滑点。绕过间隙的一个充分条件是当$s$的一些定义多项式是sos凹的。(iii)对于紧致非凸半代数集$T$的凸包的SDP可表示性，我们发现临界对象是$\pt_ct$,$\pt_ct$包含在$\pt\cv{T}$中的$\pt_ct$的最大子集。我们证明了SDP可表示性的充分条件:$\pt_ct$是正弯曲的，必要条件:$\pt_ct$在光滑点和非退化角上具有非负曲率。它们之间的差距类似于案例（二）。本文还讨论了正定拉格朗日-黑森(PDLH)条件。
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英文标题：
《Sufficient and Necessary Conditions for Semidefinite Representability of
Convex Hulls and Sets》
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作者：
J. William Helton and Jiawang Nie
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\bdS$ is positively curved, and necessary condition: $\bdS$ has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on $\bdS$ where they vanish. The gaps between them are $\bdS$ having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ are sos-concave. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\pt_cT$, the maximum subset of $\pt T$ contained in $\pt \cv{T}$. We prove sufficient conditions for SDP representability: $\pt_cT$ is positively curved, and necessary conditions: $\pt_cT$ has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between them are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.
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PDF链接：
https://arxiv.org/pdf/0709.4017
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