Weijun Xie - Assistant Professor
Industrial and Systems Engineering, Georgia Tech
Abstract:
The low-rank constrained optimization arises in various machine learning and optimization problems. It minimizes a linear objective function subject to multiple linear inequalities and a low-rank domain set. Although the low-rank constrained optimization is generally NP-hard, a viable approach is to convexify the domain set (i.e., replace the domain with its convex hull), known as “partial convexification.” Partial convexification often leads to a tractable convex relaxation, but its solution quality lacks theoretical guarantees. To fill this gap, we establish the necessary and sufficient conditions under which partial convexification matches the original low-rank constrained optimization. Besides, we also derive an upper bound on the minimum rank among all the optimal solutions of partial convexification and prove its tightness. An effective column generation algorithm and a rank-reduction algorithm are developed to efficiently solve the partial convexification. This combination ensures that the output solution satisfies the theoretical guarantees. Finally, numerical experiments validate the strength of the partial convexification and the effectiveness of the proposed algorithms.
Bio:
Dr. Weijun Xie is the Coca-Cola Foundation Early Career Professor and Assistant Professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech. Before joining ISyE, he was an Assistant Professor at the Grado Department of Industrial and Systems Engineering, Virginia Tech, from August 2017 to July 2022. Dr. Xie obtained his PhD. in Operations Research at the Georgia Institute of Technology in August 2017. His research interests are theory and applications of stochastic, discrete, and convex optimization. His works have received multiple awards, including the 2022 New Investigator Award from the Virginia Space Grant Consortium (NASA), the 2021 NSF CAREER Award, Winner of the 2020 INFORMS Young Researchers Paper Prize. He currently serves as Associate Editor of Mathematical Programming and the Journal of Global Optimization.