摘要翻译：
如果一个集合可以表示为用某种线性矩阵不等式(LMI)表示的高维集合的投影，则称为半定可表示或半定规划(SDP)可表示。本文讨论了形式为S_D(f)={x\in d:f(x)>=0}的凸集的半定可表示性条件。这里d={x\在r^n中:g_1(x)>=0,...,g_m(x)>=0}是由一些“好的”凹多项式g_i(x)定义的凸区域（它们满足一定的凹性证明）,而f(x)是多项式或有理函数。当f(x)在\mc{D}上是凹的时，我们证明了S_D(f)在某些条件下有一些显式的半定表示，称为预序凹性或q-模凹性，这些条件是基于一阶凹性准则的实证主义Stellensatz证明的。当f(x)是在S_D(f)边界上有奇点的多项式或有理函数时，引入透视变换，在一定条件下找到S_D(f)的半定显式表示。在n=2的特殊情况下，如果f(x)在一个奇点上的Laurent展开只有两个连续齐次部分，我们证明了S_D(f)总是允许显式可构造半定表示。
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英文标题：
《First Order Conditions for Semidefinite Representations of Convex Sets
  Defined by Rational or Singular Polynomials》
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作者：
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 is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form S_D(f) = {x \in D: f(x) >= 0}. Here D={x\in R^n: g_1(x) >= 0, ..., g_m(x) >= 0} is a convex domain defined by some "nice" concave polynomials g_i(x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over \mc{D}, we prove that S_D(f) has some explicit semidefinite representations under certain conditions called preordering concavity or q-module concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria. When f(x) is a polynomial or rational function having singularities on the boundary of S_D(f), a perspective transformation is introduced to find some explicit semidefinite representations for S_D(f) under certain conditions. In the particular case n=2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that S_D(f) always admits an explicitly constructible semidefinite representation.
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PDF链接：
https://arxiv.org/pdf/0806.4721