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We consider solving high-order semidefinite programming (SDP) relaxations ofnonconvex polynomial optimization problems (POPs) that often admit degeneraterank-one optimal solutions. Instead of solving the SDP alone, we propose a newalgorithmic framework that blends local search using the nonconvex POP intoglobal descent using the convex SDP. In particular, we first design a globallyconvergent inexact projected gradient method (iPGM) for solving the SDP thatserves as the backbone of our framework. We then accelerate iPGM by takinglong, but safeguarded, rank-one steps generated by fast nonlinear programmingalgorithms. We prove that the new framework is still globally convergent forsolving the SDP. To solve the iPGM subproblem of projecting a given point ontothe feasible set of the SDP, we design a two-phase algorithm with phase oneusing a symmetric Gauss-Seidel based accelerated proximal gradient method(sGS-APG) to generate a good initial point, and phase two using a modifiedlimited-memory BFGS (L-BFGS) method to obtain an accurate solution. We analyzethe convergence for both phases and establish a novel global convergence resultfor the modified L-BFGS that does not require the objective function to betwice continuously differentiable. We conduct numerical experiments for solvingsecond-order SDP relaxations arising from a diverse set of POPs. Our frameworkdemonstrates state-of-the-art efficiency, scalability, and robustness insolving degenerate rank-one SDPs to high accuracy, even in the presence ofmillions of equality constraints.

An Inexact Projected Gradient Method with Rounding and Lifting by  Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of  Polynomial Optimization