Block Factor-Width-Two Matrices and Their Applications to Semidefinite and Sum-of-Squares Optimization
IEEE Transactions on Automatic Control(2023)
摘要
Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this article, we introduce a new notion of
block factor-width-two matrices
and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the
scaled diagonally dominant sum-of-squares (SDSOS)
polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a tradeoff between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings.
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关键词
Large-scale systems,matrix decomposition,semidefinite optimization,sum-of-squares (SOS) polynomials
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