Two important topics in the study of Quadratically Constrained Quadratic Programming (QCQP) are how to exactly solve a QCQP with few constraints in polynomial time and how to find an inexpensive and strong relaxation bound for a QCQP with many constraints. In this thesis, we first review some important results on QCQP, like the S-Procedure, and the strength of Lagrangian Relaxation and the semidefinite relaxation. Then we focus on two special classes of QCQP, whose objective and constraint functions take the form trace(X^TQX + 2C^T X) + β, and trace(X^TQX + XPX^T + 2C^T X)+ β respectively, where X is an n by r real matrix. For each class of problems, we proposed different semidefinite relaxation formulations and compared their strength. The theoretical results obtained in this thesis have found interesting applications, e.g., solving the Quadratic Assignment Problem.

Two important topics in the study of Quadratically Constrained Quadratic Programming (QCQP) are how to exactly solve a QCQP with few constraints in polynomial time and how to find an inexpensive and strong relaxation bound for a QCQP with many constraints. In this thesis, we first review some important results on QCQP, like the S-Procedure, and the strength of Lagrangian Relaxation and the semidefinite relaxation. Then we focus on two special classes of QCQP, whose objective and constraint functions take the form trace(X^TQX + 2C^T X) + β, and trace(X^TQX + XPX^T + 2C^T X)+ β respectively, where X is an n by r real matrix. For each class of problems, we proposed different semidefinite relaxation formulations and compared their strength. The theoretical results obtained in this thesis have found interesting applications, e.g., solving the Quadratic Assignment Problem.