Semidefinite Approximations of Conical Hulls of Measured Sets

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• 作者：Julián Romero ; Mauricio Velasco
• 关键词：Approximation of convex bodies ; Spectrahedra ; SDr sets
• 刊名：Discrete & Computational Geometry
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：57
• 期：1
• 页码：71-103
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Computational Mathematics and Numerical Analysis;
• 出版者：Springer US
• ISSN：1432-0444
• 卷排序：57

文摘

Let C be a proper convex cone generated by a compact set which supports a measure \(\mu \). A construction due to Barvinok, Veomett and Lasserre produces, using \(\mu \), a sequence \((P_k)_{k\in \mathbb {N}}\) of nested spectrahedral cones which contains the cone \(C^*\) dual to C. We prove convergence results for such sequences of spectrahedra and provide tools for bounding the distance between \(P_k\) and \(C^*\). These tools are especially useful on cones with enough symmetries and allow us to determine bounds for several cones of interest. We compute bounds for semidefinite approximations of cones over traveling salesman polytopes, cones of nonnegative ternary sextics and quaternary quartics and cones non-negative functions on finite abelian groups.