Convex Algebraic Geometry
Convex Algebraic Geometry
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凸代数几何
DOI：
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
M. Schweighofer
中科院分区：
文献类型：
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作者：
M. Schweighofer
在科学和技术中自然产生的许多几何物体都具有两个理想的性质。它们是凸的和半代数的。凸集具有这样的性质，即人们可以沿着沿着一条直线在凸集的任意两点之间移动而不离开凸集。半代数集合可以通过简单的逻辑运算组合多项式不等式来描述。数学领域主要调查这些对象分别是凸分析和真实的代数几何。在数学上，一个集合的凸性和半代数描述都可以单独利用。然而，目前，这些方法是完全不同的和不相交的巨大局限性。凸性可以导致用于导航几何对象的非常快速的数值算法。然而，为了使这些算法工作，需要额外的结构，例如在集合内部的易于计算的自协调障碍函数[17]。对于半代数集，已知非常通用的符号算法来研究和处理它们[4]。然而，这些算法对于实际目的来说通常不够有效。尽管凸半代数集是一种普遍存在的集合，但其特殊性质的研究却长期被忽视。只有在最近几年才有新的结果和方法出现，导致这些几何对象受到广泛的关注，包括古典代数几何，复杂性理论，控制理论，凸几何，泛函分析，优化理论和真实的代数几何[22，13，15，24，1，27]。从不到十年前开始，已经有越来越多的会议，来自这些领域的人们聚集在一起，凸半代数集作为共同感兴趣的中心工具。组织这次会议背后的动机是认识到，现在是时候出现一个研究领域，凸半代数集是研究的中心对象，而不是支持工具。我们呼吁这方面的凸代数几何和它致力于系统的研究凸半代数集。
Many geometric objects arising naturally in science and technology possess two desirable properties. They are convex and semialgebraic. Convex sets have the property that one can move between any two of its points along a straight line without leaving the set. Semialgebraic sets can be described by combining polynomial inequalities by simple logical operations. The areas of mathematics primarily investigating these objects are Convex Analysis and Real Algebraic Geometry, respectively. Algorithmically, the property of being convex and a semialgebraic description of a set can both be exploited each on its own. However, at the moment, these methods are totally different and disjoint with huge limitations. Convexity can lead to very fast numerical algorithms for navigating a geometric object. However, for these algorithms to work, one needs additional structure such as an easily computable self-concordant barrier function on the interior of the set [17]. For semialgebraic sets, very general symbolic algorithms are known to investigate and handle them [4]. However, these algorithms are often not efficient enough for practical purposes. In spite of their ubiquity, the investigation of the special features of convex semialgebraic sets have been neglected for a long time. Only in recent years have new results and methods come up that have resulted in these geometric objects receiving attention from a wide range of areas including Classical Algebraic Geometry, Complexity Theory, Control Theory, Convex Geometry, Functional Analysis, Optimization Theory and Real Algebraic Geometry [22, 13, 15, 24, 1, 27]. Starting less than a decade ago, there have been more and more meetings where people from some of these areas have come together, with convex semialgebraic sets serving as a central tool of common interest. The motivation behind organizing this meeting was the realization that it is now time for the emergence of an area of research where convex semialgebraic sets are the central objects of study rather than supporting tools. We call this area Convex Algebraic Geometry and it is devoted to the systematic study of convex semialgebraic sets.
三维多面体可以用三个多项式不等式来描述
影响因子：
0.8
作者：
Averkov
通讯作者：
Averkov