On Efficient Semidefinite Relaxations for Quadratically Constrained Quadratic Programming

Semidefinite Programming;Quadratically Constrained Quadratic Programming;Quadratic Matrix Programming;Quadratic Assignment Problem;Combinatorics and Optimization

Ding, Yichuan

University of Waterloo

关键词: Semidefinite Programming;    Quadratically Constrained Quadratic Programming;    Quadratic Matrix Programming;    Quadratic Assignment Problem;    Combinatorics and Optimization;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/3044/1/yichuanthesis.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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.