Date of Award
5-2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
School of Mathematical and Statistical Sciences
Committee Chair/Advisor
Dr. Boshi Yang
Committee Member
Dr. Hao Hu
Committee Member
Dr. Cheng Guo
Committee Member
Dr. Yuyuan Ouyang
Abstract
In general, convex programs have nicer properties than nonconvex programs. Notably, in a convex program, every locally optimal solution is also globally optimal. For this reason, there is interest in finding convex reformulations of nonconvex programs. These reformulation often come in the form of a conic program. For example, nonconvex quadratically-constrained quadratic programs (QCQPs) are often relaxed to semidefinite programs (SDPs) and then tightened with valid inequalities. This dissertation gives a few different problems of interest and shows how conic reformulations can be usefully applied.
In one chapter, we consider two variants of the trust-region subproblem. For each of these variants, we show that their feasible regions can be written as the union of two sub-regions with known lifted convex hulls. We are then able to use this fact to reformulate each variant into a semidefinite program. In the next chapter, we consider the unit commitment problem, which is typically formulated as a mixed-binary program. A convex reformulation of the unit commitment problem is desired as its shadow prices may be used to formulate a pricing scheme. Based on a completely positive programming reformulation, we obtain an SDP relaxation of the unit commitment problem. The last application that we look at focuses on identifying implicit inequalities in a linear system of inequalities. While there are several methods that we examine, one of the most efficient methods works by solving a linear program that has been constructed to have a conic feasible region.
Recommended Citation
Kelly, Sarah, "Applications of Conic Programming Reformulations" (2024). All Dissertations. 3586.
https://tigerprints.clemson.edu/all_dissertations/3586