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

A Correlatively Sparse Lagrange Multiplier Expression Relaxation for Polynomial Optimization