Sparse solution to classes of optimization problems has been a major concern for optimization problems arising from many disciplines. For example, in image processing, a sparse representation of the important features is crucial to the effective recovery of the original image, while in portfolio selection, a well-diversified portfolio is critical to hedge the potential risk in the portfolio.
It has been observed that for numerous classes of quadratic optimization problems such as the standard quadratic programming problem (StQP), though theoretically intractable, there always exists sparse optimal solutions for instances from many real-world applications.
Jiming Peng, research assistant professor and instructional assistant professor in the industrial engineering department at the UH Cullen College of Engineering, is working to develop a new theoretical framework to establish the existence of sparse optimal or approximate solutions to classes of intractable and non-convex QPs and derive precise probabilistic characterization of the sparsity of the sparsest optimal or approximate solutions to the underlying QPs. This theoretical insight will be used to design new provably-good efficient algorithms and the new techniques will be applied to portfolio selection and computer vision.
This project began in 2011 while Dr. Peng was a faculty in the industrial and systems engineering department at U of Illinois at Urbana-Champaign and is funded by the National Science Foundation. Other participants include Ph.D. student Rui Yang and colleagues Xin Chen and Shuzhong Zhang.