Title: From Theory to Practice: Advanced Methods for Solving QCQPs 

Date: Tuesday, April 28th, 2026
Time: 11:00 am – 1:00 pm EST
Location: Groseclose 226 (Georgia Freight Bureau Conference Room) and Zoom

Committee:
Dr. Santanu S. Dey (Advisor), H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Oktay Günlük, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Jean-Philippe P. Richard, Department of Industrial and Systems Engineering, University of Minnesota
Dr. Nick Sahinidis, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Yang Wang, School of Civil and Environmental Engineering, Georgia Institute of Technology

Abstract:
Quadratically constrained programs (QCQPs) model critical real-world problems across engineering, finance, logistics, and data science. For nonconvex QCQPs, finding globally optimal solutions is computationally intractable in general, and even small instances can be challenging for state-of-the-art optimization solvers. This thesis studies the design and analysis of global optimization algorithms and relaxation schemes for QCQPs that are both mathematically rigorous and computationally practical.

Solving QCQPs to global optimality relies on three key components: (i) finding high-quality feasible solutions, (ii) constructing tight relaxations, and (iii) selecting effective branching strategies. These are crucial for the spatial branch-and-bound, which is the dominant approach for solving QCQPs. This thesis contributes to each of these components.

The first part of the thesis addresses the problem of finding high-quality feasible solutions for an energy storage optimization problem whose operational constraints are bilinear. We develop a regularized formulation whose linear programming relaxation yields a feasible solution to the original problem. This approach enables solving previously challenging trilevel optimization problems modeling adversarial attacks on power grids.

The second part of the thesis focuses on building a tighter convex relaxation for bilinear bipartite programs using aggregation techniques to construct second-order cone representable sets. We characterize when aggregation methods yield the exact convex hull and provide illustrative examples. Computational experiments demonstrate that aggregation methods improve bounds over single-row relaxations and commercial solvers, reducing the optimality gap.

The third part of the thesis proposes a novel branching rule, namely extreme strong branching, which jointly optimizes both branching variable and point selection via binary search. This approach leverages objective value improvements to minimize branch-and-bound tree size while producing bound tightening as a byproduct. For certain instances, this technique produces substantially smaller branch-and-bound trees to reach optimality compared to alternative branching rules.

The last part of the thesis studies relaxation schemes in a unified framework, from data-independent to data-dependent approaches. We prove that bounded Lagrangian relaxation provides the strongest bound among data-independent methods under mild conditions. We further analyze data-dependent relaxations and establish a hierarchy of the relative strength of various relaxation methods. 

Collectively, these contributions bridge the gap between theoretical possibility and computational feasibility, expanding the range of QCQP instances that can be addressed effectively in practice.