Project 12: Numerical Algorithms for Solving Nonsmooth Optimization Problems and Applications to Image Reconstructions

Project abstract: Many optimization algorithms rely on the gradients or the Hessians of the objective functions, while solving optimization problems with nondifferentiable/nonsmooth objective functions is required in many recent applications.  The main goal of this project is to study mathematical foundation and develop numerical algorithms for solving optimization problems without requiring the differentiability of the data. Our first approach is to study generalized differentiation for nondifferentiable functions and develop numerical algorithms based on generalized derivatives of the objective functions. The second approach involves building smoothing techniques to approximate nondifferentiable functions by differentiable functions and then apply gradient-based optimization methods to these approximations. We will apply our methods to the image recovery problem aiming at building numerical algorithms accompanied with MATLAB codes for reconstructing images with missing or distorted pixels. We will also explore further applications to other areas such as medical imaging, missing data recovery, and building movie recommendation systems.

Keywords:

• Convex Optimization
• Generalized Differentiation
• Smoothing Techniques
• Image Recovery

Faculty Mentor: Mau Nam Nguyen http://web.pdx.edu/~mnn3/

https://sites.google.com/pdx.edu/convex-analysis-optimization

Lab or team: NA

Department: Mathematics and Statistics

Community partner(s): NA

Desired skills (but not required): Programming with MATLAB

Tools to be used: A laptop/desktop with MATLAB software installed

Involves teamwork: No