  This work is a follow-up and a complement to arXiv:1912.08899 [math.OC] for
solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy
that we propose is a new sparse moment-SOS framework based on term-sparsity and
chordal extension. By exploiting term-sparsity of the input polynomials we
obtain a two-level hierarchy of semidefinite programming relaxations. The
novelty and distinguishing feature of such relaxations is to obtain quasi
block-diagonal matrices obtained in an iterative procedure that performs
chordal extension of certain adjacency graphs. The graphs are related to the
terms arising in the original data and not to the links between variables.
Various numerical examples demonstrate the efficiency and the scalability of
this new hierarchy for both unconstrained and constrained POPs. The two
hierarchies are complementary. While the former TSSOS arXiv:1912.08899
[math.OC] has a theoretical convergence guarantee, the chordal-TSSOS has
superior performance but lacks this theoretical guarantee.
