2025/01/07 by Guo, Feng, Wang, Jie
#11E25 #12D15 #13J30 #14P10 #15A54 #90C23 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2501.03470
This paper investigates Positivstellensätze for polynomial matrices subject to universally quantified polynomial matrix inequality constraints. We first establish a matrix-valued Positivstellensatz under the Archimedean condition, incorporating universal quantifiers. For scalar-valued polynomial objectives, we further develop a sparse Positivstellensatz that leverages correlative sparsity patterns within these quantified constraints. Moving beyond the Archimedean framework, we then derive a series of generalized Positivstellensätze under analogous settings. These results collectively unify and extend foundational theorems in three distinct contexts: classical polynomial Positivstellensätze, their universally quantified counterparts, and matrix polynomial formulations. Applications of the established Positivstellensätze to robust polynomial matrix optimization are also discussed.