Rank-2 Matrix Solution for Semidefinite Relaxations of Arbitrary Polynomial Optimization Problems

This paper is concerned with the study of an arbitrary polynomial optimization via a convex relaxation, namely a semidefinite program (SDP). The existence of a rank-1 matrix solution for the SDP relaxation guarantees the recovery of a global solution of the original problem. The main objective of this work is to show that an arbitrary polynomial optimization has an equivalent formulation in the form of a sparse quadraticallyconstrained quadratic program (QCQP) whose SDP relaxation possesses a matrix solution with rank at most 2. This result offers two new insights into the computational complexity of polynomial optimization and combinatorial optimization as a special case. First, the complexity is only related to finding a rank-1 matrix in a convex set where it is guaranteed that a rank-2 matrix can always be found in polynomial time. Second, the approximation of the rank-2 SDP solution with a rank1 matrix enables the retrieval of an approximate near-global solution for the original polynomial optimization. To derive this result, three graph sparsification techniques are proposed, each of which designs a sparse QCQP problem that is equivalent to the original polynomial optimization.