First order conditions for semidefinite representations of convex sets defined by rational or singular polynomials
First order conditions for semidefinite representations of convex sets defined by rational or singular polynomials
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DOI：
10.1007/s10107-009-0339-9
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发表时间：
2008-06
影响因子：
2.7
通讯作者：
Jiawang Nie
中科院分区：
文献类型：
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作者：
Jiawang Nie
一个集合称为半定可表示或半定规划（SDP）可表示，如果它等于一个由线性矩阵不等式（LMI）定义的高维集合的投影。本文讨论了以下形式的凸集的半定可表示性条件。这里，是一个由一些“漂亮的”凹多项式sgi（x）定义的凸域（它们满足某些证明），并且f（x）是一个多项式或有理函数。当f（x）是凹上的时，我们证明了在一定条件下有一些显式半定表示，称之为预序凹或q-模的凸表示，这些条件是基于一阶凸准则的正半定证明：$$f（u）+ \nabla f（u）^T（x-u）-f（x）\geq 0，\quad \for all\，x，u \in \mathcal {D}.$$当f（x）是在边界上有奇点的多项式或有理函数时，在一定条件下，引入透视变换，得到了f（x）的显式半定表示。在一个特殊的情形中，如果（x）的Laurent展开在一个奇点附近只有两个连续的齐次部分，我们证明了（x）的Laurent展开总是存在一个显式可构造的半定表示.
A set is calledsemidefinite representableorsemidefinite programming(SDP)representableif it equals the projection of a higher dimensional set which is defined by someLinear Matrix Inequality(LMI). This paper discusses the semidefinite representability conditions for convex sets of the form. Here,is a convex domain defined by some “nice” concave polynomialsgi(x) (they satisfy certain concavity certificates), andf(x) is a polynomial or rational function. Whenf(x) is concave over, we prove thathas some explicit semidefinite representations under certain conditions calledpreordering concavityorq-module concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria:$$f(u) + \nabla f(u)^T(x-u) -f(x) \geq 0, \quad \forall \, x, u \in \mathcal {D}.$$Whenf(x) is a polynomial or rational function having singularities on the boundary of, aperspective transformationis introduced to find some explicit semidefinite representations forunder certain conditions. In the special casen= 2, if the Laurent expansion off(x) around one singular point has only two consecutive homogeneous parts, we show thatalways admits an explicitly constructible semidefinite representation.