  In this paper, we consider polynomial optimization with correlative sparsity.
We construct correlatively sparse Lagrange multiplier expressions (CS-LMEs) and
propose CS-LME reformulations for polynomial optimization problems using the
Karush-Kuhn-Tucker optimality conditions. Correlatively sparse sum-of-squares
(CS-SOS) relaxations are applied to solve the CS-LME reformulation. We show
that the CS-LME reformulation inherits the original correlative sparsity
pattern, and the CS-SOS relaxation provides sharper lower bounds when applied
to the CS-LME reformulation, compared with when it is applied to the original
problem. Moreover, the convergence of our approach is guaranteed under mild
conditions. In numerical experiments, our new approach usually finds the global
optimal value (up to a negligible error) with a low relaxation order for cases
where directly solving the problem fails to get an accurate approximation.
Also, by properly exploiting the correlative sparsity, our CS-LME approach
requires less computational time than the original LME approach to reach the
same accuracy level.
