NSF Award on Polynomial Optimization: Solution Methods and Applications

Recently, a NSF grant was award to Shuzhong Zhang on polynomial optimization. 

The research objective of this project is to develop numerical tools for solving polynomial and tensor optimization models arising from engineering applications such as design of radar waveforms in signal processing, information extraction in magnetic resonance imaging, and gene expression data analysis in bioinformatics. An integrated approach for polynomial and tensor optimization models will be studied and tested in this project. The key components of the integrated approach include relaxation techniques, approximations to optimal solutions, randomized sampling techniques utilizing known structure of the model, and disciplined local improvements of feasible solutions. Numerical tools to be developed under the framework of the integrated approach will be investigated experimentally and validated using the data obtained from practical engineering applications including signal processing and bioinformatics. Performance of the algorithms developed from the project will be measured and compared against currently existing benchmarks. 

If successful, the results of this research will enhance the effectiveness of methods for solving nonlinear optimization models where polynomial functions are found in the objective as well as in the constraints. In particular, such algorithms will lead to substantial practical improvements in radar waveform design, image processing, and data co-clustering analysis. As polynomial functions are known to be pervasive in the context of engineering design and decision making, success of the project will lead to broad societal benefits. Moreover, the proposed research addresses fundamental challenges in optimization theory, because most current research has not gone beyond polynomial models of degree 2; thus, results extending the theory beyond these models would constitute a major advance on the theory side, as well.