标题： 矩方法-SOS层次结构在箭头型多项式矩阵不等式中的应用及在结构优化中的应用 显示英文标题

标题： Moment-SOS hierarchies for arrow-type polynomial matrix inequalities with applications to structural optimization

摘要： 箭头分解（AD）技术，最初在[数学规划 190(1-2) (2021)，第105-134页]中引入，在线性矩阵不等式（LMIs）的背景下，如果涉及的矩阵满足适当的假设，则表现出优于经典弦分解的可扩展性。本文的主要目标是将AD方法扩展以解决涉及大规模多项式矩阵不等式（PMIs）的多项式优化问题（POPs），其解决方案框架依赖于矩-平方和（mSOS）层次结构。作为第一步，我们重新审视LMI情况，并弱化了在[数学规划 190(1-2) (2021)，第105-134页]中提出的关键AD定理所需的条件。这种修改使得该方法可以应用于更广泛的问题。接下来，我们提出一种实用的程序，减少额外变量的数量，借鉴结构优化应用中常见的物理解释。对于PMI情况，我们探讨了两种不同的方法将AD技术与mSOS层次结构结合。一种方法是在实施mSOS松弛之前对原始POP应用AD。另一种方法是直接对POP的mSOS松弛应用AD。我们为这两种方法建立了收敛保证，并证明理论性质可以扩展到多项式情况。最后，我们展示了AD应用带来的显著计算优势，特别是在结构优化问题的背景下。 显示英文摘要

摘要： The Arrow Decomposition (AD) technique, initially introduced in [Mathematical Programming 190(1-2) (2021), pp 105-134], demonstrated superior scalability over the classical chordal decomposition in the context of Linear Matrix Inequalities (LMIs) if the matrix in question satisfied suitable assumptions. The primary objective of this paper is to extend the AD method to address Polynomial Optimization Problems (POPs) involving large-scale Polynomial Matrix Inequalities (PMIs), with the solution framework relying on moment-sum of square (mSOS) hierarchies. As a first step, we revisit the LMI case and weaken the conditions necessary for the key AD theorem presented in [Mathematical Programming 190(1-2) (2021), pp 105-134]. This modification allows the method to be applied to a broader range of problems. Next, we propose a practical procedure that reduces the number of additional variables, drawing on physical interpretations often found in structural optimization applications. For the PMI case, we explore two distinct approaches to combine the AD technique with mSOS hierarchies. One approach involves applying AD to the original POP before implementing the mSOS relaxation. The other approach applies AD directly to the mSOS relaxations of the POP. We establish convergence guarantees for both approaches and prove that theoretical properties extend to the polynomial case. Finally, we illustrate the significant computational advantages offered by the application of AD, particularly in the context of structural optimization problems.