Some Global Relaxation Methods for Quadratic and Semidefinite Programming

Dissertation Defense
Tuesday, May 9, 2023 - 11:00am for 1.5 hours (actually 80 minutes)
Skiles 005 and ONLINE
Shengding Sun – Georgia Tech – ssun313@gatech.edu
Shengding Sun

Zoom link:

Quadratic programming and semidefinite programming are vital tools in discrete and continuous optimization, with broad applications. A major challenge is to develop methodologies and algorithms to solve instances with special structures. For this purpose, we study some global relaxation techniques to quadratic and semidefinite programming, and prove theoretical properties about their qualities. In the first half we study the negative eigenvalues of $k$-locally positive semidefinite matrices, which are closely related to the sparse relaxation of semidefinite programming. In the second half we study aggregations of quadratic inequalities, a tool that can be leveraged to obtain tighter relaxation to quadratic programming than the standard Shor relaxation. In particular, our results on finiteness of aggregations can potentially lead to efficient algorithms for certain classes of quadratic programming instances with two constraints.