Quadratic Programming with Discrete Variables

NIE, TIANTIAN. Quadratic Programming with Discrete Variables. (Under the direction of Dr. Shu-Cherng Fang.) In this dissertation, we study the quadratic programming problem with discrete variables (DQP). DQP is important in theory and practice, but the combination of the quadratic feature of the objective function and the discrete nature of the feasible domain makes it hard to solve. In this thesis, we utilize state-of-the-art continuous optimization techniques to study DQP. A new relaxation of DQP based on the linear conic approach is proposed. Numerical results support the high-quality of lower bounds obtained from the proposed relaxation. We then develop a new linearization method to provide effective 0-1 mixed-integer linear programming (MILP) reformulations of DQP. The new method uses a minimum number of linear inequalities to realize the binary representation of the product of discrete variables. It outperforms other known state-of-the-art linearization methods as supported by intensive computational experiments. We also extend the research to study a special linearly constrained DQP problem, called l1norm constrained convex quadratic programming problem with integer variables (l1-DQP). By exploiting the embedded first-order cone structure, we develop a new type of nonlinear conic cuts induced by the discrete nature of the problem. The proposed conic cuts are shown to work more effectively than other known conic cuts in solving l1-DQP using a branch-and-bound scheme. The research in this dissertation has revealed the potential of utilizing continuous optimization techniques and exploring special discrete structures in developing efficient algorithms for solving DQP. Future research directions are included at the end of this document. Quadratic Programming with Discrete Variables