共正锥上Parrilo平方和分层的连续层的显式分离器
Explicit Separators for Consecutive Levels of Parrilo's Sum-of-Squares Hierarchy over the Copositive Cone
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中文总结 AI 辅助
该研究解决了共正锥上Parrilo平方和分层的连续层差异问题，构造了$n=5$时从第1到4层的显式分离器，证明了连续严格包含关系，并表明该现象可推广到任意大的层次。
中文摘要 AI 辅助
Parrilo锥$\boldsymbol{\tilde{K}_n^r}$构成共正锥$\boldsymbol{\text{COP}_n}$的嵌套半定可表示内近似序列。对于$n=5$，其并集为整个$\text{COP}_5$，但无单一层次能达到，且除经典第一步外，连续层次是否存在差异仍是未解决的问题。据我们所知，此前尚无任何$t\boldsymbol{\bold{\text{≥}}2}$且$n\boldsymbol{\bold{\text{≥}}5}$时，$\boldsymbol{\tilde{K}_n^t\backslash\tilde{K}_n^{t-1}}$中的显式矩阵被发表。我们解决了前三种情况：通过沿正内部方向平移Horn矩阵的对角缩放得到的显式有理矩阵，分别属于$\tilde{K}_5^2\backslash\tilde{K}_5^1$、$\tilde{K}_5^3\backslash\tilde{K}_5^2$和$\tilde{K}_5^4\backslash\tilde{K}_5^3$，给出三个严格包含关系$\tilde{K}_5^1\boldsymbol{\bold{\text{⊂neq}}}\tilde{K}_5^2\boldsymbol{\bold{\text{⊂neq}}}\tilde{K}_5^3\boldsymbol{\bold{\text{⊂neq}}}\tilde{K}_5^4$。每个包含关系都由一个精确有理Gram矩阵和一个精确有理对偶矩泛函证明，并通过整数算术的独立程序重新验证，分离具有鲁棒性：一对固定的证书覆盖了宽度超过$\boldsymbol{3·10^{-3}}$的平移区间，且$\tilde{K}_5^2\backslash\tilde{K}_5^1$具有非空内部。结合Dickinson、Dür、Gijben和Hildebrand的缩放定理，以及Schweighofer和Vargas的完备性定理，进一步表明严格相邻包含关系会在任意大的层次上重复出现。所有分离器均通过一个阈值工具定位：沿内部方向将矩阵$M$映射到$\tilde{K}_5^r$的最小平移$\boldsymbol{\text{eps}_r(M)}$随$r$非递增，且层次间的每一次严格下降都标志着一个分离器窗口。
英文摘要
Parrilo's cones $\Kc{n}{r}$ form a nested sequence of semidefinite-representable inner approximations of the copositive cone $\COP_n$. For $n=5$ their union is all of $\COP_5$, yet no single level attains it, and whether consecutive levels actually differ had remained open beyond the classical first step. No explicit matrix in $\Kc{n}{t}\setminus\Kc{n}{t-1}$ had, to our knowledge, been published for any $t\ge2$ and $n\ge5$. We settle the first three cases. Explicit rational matrices, obtained from diagonal scalings of the Horn matrix shifted along a positive interior direction, lie in $\Kc{5}{2}\setminus\Kc{5}{1}$, in $\Kc{5}{3}\setminus\Kc{5}{2}$, and in $\Kc{5}{4}\setminus\Kc{5}{3}$, giving three consecutive strict inclusions $\Kc{5}{1}\subsetneq\Kc{5}{2}\subsetneq\Kc{5}{3}\subsetneq\Kc{5}{4}$. Each is certified by an exact rational Gram matrix and an exact rational dual moment functional, re-verified by a standalone program in integer arithmetic. The separations are robust. One fixed certificate pair covers an interval of shifts of width exceeding $3\cdot10^{-3}$, and $\Kc{5}{2}\setminus\Kc{5}{1}$ has nonempty interior. Combining a scaling theorem of Dickinson, Dür, Gijben and Hildebrand with the completeness theorem of Schweighofer and Vargas shows further that strict adjacent inclusions recur at arbitrarily large levels. All separators were located by one threshold device: the least shift $\eps_r(M)$ carrying $M$ into $\Kc{5}{r}$ along an interior direction is nonincreasing in $r$, and each strict drop between levels marks a window of separators.
