有界次数 SOS 加 SONC 多项式优化层级
A Bounded Degree SOS Plus SONC Hierarchy for Polynomial Optimization
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中文总结 AI 辅助
提出有界次数 SOS+SONC 层级 B-SOS+SONC，结合半定与稀疏结构，证明完备性并给出 SDP-REP 重构，数值上比 B-SOS 下界更紧。
中文摘要 AI 辅助
我们针对约束多项式优化提出了一种有界次数的 SOS+SONC 层级，称为 B-SOS+SONC。从 Lasserre 的有界次数 SOS 框架出发，我们将证书锥从 SOS 扩展至最近引入的 SOS+SONC 锥，从而结合了半定松弛的代数优势与电路多项式所捕获的稀疏结构。我们证明，对于每个固定的证书次数，该层级是完备的，即其最优值单调并收敛至全局最优。此外，我们推导了一个显式的 SDP-REP 重构，使得每个松弛可在半定锥和相对熵锥上的可处理凸优化框架内求解。在优化层级本身之外，我们研究了 SONC 锥的结构性质，并引入了 SONC-凸性的一阶和二阶概念。这为 B-SOS+SONC 层级的一阶精确性提供了新的充分条件。数值实验表明，所提出的层级通常比 B-SOS 松弛产生更紧的下界，同时保持可处理性。
英文摘要
We propose a bounded degree SOS+SONC hierarchy for constrained polynomial optimization, termed B-SOS+SONC. Starting from Lasserre's bounded-degree SOS framework, we enlarge the certificate cone from SOS to the recently introduced SOS+SONC cone, thereby combining the algebraic strength of semidefinite relaxations with the sparse structure captured by circuit polynomials. We show that, for each fixed certificate degree, the resulting hierarchy is complete, that is, its optimal values are monotone and converge to the global optimum. Moreover, we derive an explicit SDP-REP reformulation, so that each relaxation can be solved within a tractable convex optimization framework over semidefinite and relative entropy cones. Beyond the optimization hierarchy itself, we investigate structural properties of the SONC cone and introduce the notions of first-order and second-order SONC-convexity. This leads to a new sufficient condition for first-level exactness of the B-SOS+SONC hierarchy. Numerical experiments illustrate that the proposed hierarchy often yields tighter lower bounds than the B-SOS relaxation while remaining tractable.
发表机构
- University of New South Wales（新南威尔士大学）
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