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  Semidefinite optimization relaxations are among the widely used approaches to find global optimal or approximate solutions for many nonconvex problems. Here, we consider a specific quadratically constrained quadratic problem with an additional linear constraint. We prove that under certain conditions the semidefinite relaxation approach enables us to find a global optimal solution of the underlying problem in polynomial time .

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In this paper, we present a new primal-dual predictor-corrector interior-point algorithm for linear optimization problems. In each iteration of this algorithm, we use the new wide neighborhood proposed by Darvay and Takács. Our algorithm computes the predictor direction, then the predictor direction is used to obtain the corrector direction. We show that the duality gap reduces in both predictor and corrector steps. Moreover, we conclude that the complexity bound of this algorithm coincides with the best-known complexity bound obtained for small neighborhood algorithms. Eventually, numerical results show the capability and efficiency of the proposed algorithm.