2019/01/05 by Thanh-Hieu Le, Le, Thanh-Hieu, Nhat-Thien Pham +1
Computer Science · Engineering · Mathematics · #68U01 #68U20 #Advanced Optimization Algorithms Research #Computation and Language (cs.CL) #Control Systems and Identification #FOS: Computer and information sciences #FOS: Mathematics #Matrix Theory and Algorithms #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1901.02360
openalex publication_date 2019/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: ℝn; ℝ and its intervals [a,b], [0,∞); and the strips [a,b] × ℝ ⊂ ℝ2. A method for numerically computing such representations is also presented. The methodology is divided into two stages: (S1) diagonalizing the initial polynomial matrix based on the Schmüdgen's procedure \citeSchmudgen09; (S2) for each diagonal element of the resulting matrix, find its low rank sosrf-representation satisfying the Artin's theorem solving the Hilbert's 17th problem. Some numerical tests and illustrations with \textsfOCTAVE are also presented for each type of polynomial matrices.