  This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving
large-scale sparse polynomial optimization problems. Its novelty is to exploit
simultaneously correlative sparsity and term sparsity by combining advantages
of two existing frameworks for sparse polynomial optimization. The former is
due to Waki et al. while the latter was initially proposed by Wang et al. and
later exploited in the TSSOS hierarchy. In doing so we obtain CS-TSSOS -- a
two-level hierarchy of semidefinite programming relaxations with (i), the
crucial property to involve blocks of SDP matrices and (ii), the guarantee of
convergence to the global optimum under certain conditions. We demonstrate its
efficiency and scalability on several large-scale instances of the celebrated
Max-Cut problem and the important industrial optimal power flow problem,
involving up to six thousand variables and tens of thousands of constraints.
