Year: 2024
Type: preprint
Abstract: We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set $\mathcal{X}$ defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on $\mathcal{X}$, based on Jackson's theorem and a variant of the Łojasiewicz inequality.
Source: arXiv (Cornell University)
Authors: Tran, Hoang Anh, Toh, Kim-Chuan
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4399836756