A quadratically constrained quadratic programming (QCQP) problem is to minimize a quadratic objective function subject to quadratic constraints where the objective and constraints are not necessary to be convex. Nonconvex QCQPs are generally NP-hard. Many engineering problems, such as optimal control, especially for polynomial optimal control problems, can be equivalently formulated as QCQPs via the discretization technique. Semidefinite relaxation (SDR) is one of the existing methods to solve nonconvex QCQPs. By replacing the rank-one constraint with semidefinite constraint on the unknown matrix, SDR finds a lower bound of the primal optimal value for the QCQP. In this paper, a novel iterative algorithm combining the alternating projection and rank-one approximation technique is proposed to solve QCQPs. Furthermore, the global convergence proof of the proposed algorithm under certain conditions is provided based on the strong convexity property of the augmented Lagrangian. Finally, simulation results for the mixed-integer boolean quadratic programming problems and path-planning problems of an unmanned aerial vehicle with multiple avoidance zones are presented to verify the effectiveness and improved computational performance comparing to the results from the state-of-art methods.
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