Efficient Algorithms for Nonsmooth Optimization in Imaging

Abstract

This joint French-Austrian research project is concerned with the development of efficient algorithms for solving nonsmooth optimization problems arising in imaging. Solving these problems is very challenging, since the objective function is very often non-differentiable and the number of unknowns is usually very large. Hence, standard algorithms such as gradient methods or Newton simply fail. Recently, there has been an increased interest in so-called proximal-like methods. Although developed already in the seventies, they have been revitalized just because of their ability to deal with large nonsmooth problems. Although in principle slower than more sophisticated algorithms such as interior point methods, their simplicity makes them particularly well suited to be implemented on highly parallel architectures, such as GPUs. This can lead to a drastic performance gain - usually a factor of 30 or more. Very recent results indicate that a specific class of proximal-like algorithms - so called first-order primal-dual algorithms - are particularly promising. On the one hand they can deal with a reasonably large class of structure convex problems and on the other hand they seem to be particularly well suited to solve large nonsmooth problems. Despite their good performance, there are still many open problems that need to be addressed in order to further improve these algorithms. Interesting questions include the optimal preconditioning, the optimal acceleration on smoother problems and the application to very large-scale problems. Finding answers to these questions would mean solving larger problems in shorter time, and possibly on more basic hardware (such as cell phones). In turn, it allow researchers to investigate more complex and hence more realistic models.

 

 

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