发表机构
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出受限乘子正性定理，为带仿射参数的多项式优化建立收敛的矩-SOS 层级，实现全局最优性检测与极小点恢复，并显著提升计算效率。
AI 中文摘要
我们针对在辅助参数中仿射的多项式发展了正性证书，其中平方和乘子仅依赖于主变量。在受限阿基米德序单位条件下，每个严格正的标量多项式都承认这样的证书。我们为带仿射参数的多项式优化问题构造了一个收敛的受限矩-SOS 层级，并建立了用于全局最优性检测和全局极小点恢复的平坦性准则。标量定理可推广到具有任意数量仿射参数的多项式矩阵约束。对于矩阵值目标函数，点态正性在单参数情形下是充分的，但在两个或更多参数情形下可能失效；一致自由正性提供了一个通用的充分替代条件。应用包括鲁棒线性规划、鲁棒李雅普诺夫设计、极小极大多项式优化、输入仿射动力系统的分析与控制。数值实验表明，所提方法显著减小了半定分块尺寸和求解时间，同时在大多数测试案例中，边界质量几乎没有或没有退化。
英文摘要
We develop positivity certificates for polynomials that are affine in auxiliary parameters, with sum-of-squares multipliers depending only on the principal variables. Under a restricted Archimedean order-unit condition, every strictly positive scalar polynomial admits such a certificate. We construct a convergent restricted moment-SOS hierarchy for polynomial optimization problems with affine parameters and establish a flatness criterion for global optimality detection and global minimizer recovery. The scalar theorem extends to polynomial-matrix constraints with any number of affine parameters. For matrix-valued objectives, pointwise positivity is sufficient with one parameter but can fail with two or more parameters; uniform free positivity provides a general sufficient replacement. Applications include robust linear programming, robust Lyapunov design, min-max polynomial optimization, analysis and control of input-affine dynamical systems. Numerical experiments show that the proposed approach substantially reduces semidefinite block sizes and solution times while incurring little or no degradation in bound quality for most test cases.
Comments33 pages, 6 tables
