Title:Splitting algorithms for monotone inclusions with minimal dimension
| Speaker: | David Torregrossa Belén |
| Affiliation: | Center for Mathematical Modeling, University of Chile |
| Location: | MC 5501 |
Abstract: Many situations in convex optimization can be modeled as the problem of finding a zero of a monotone operator, which can be regarded as a generalization of the gradient of a differentiable convex function. In order to numerically address this monotone inclusion problem, it is vital to be able to exploit the inherent structure of the operator defining it. The algorithms in the family of the splitting methods achieve this by iteratively solving simpler subtasks that are defined by separately using some parts of the original problem.
In the first part of this talk, we will introduce some of the most relevant monotone inclusion problems and present their applications to optimization. Subsequently, we will draw our attention to a common anomaly that has persisted in the design of methods in this family: the dimension of the underlying space —which we denote as lifting— of the algorithms abnormally increases as the problem size grows. This has direct implications on the computational performance of the method as a result of the increase of memory requirements. In this framework, we characterize the minimal lifting that can be obtained by splitting algorithms adept at solving certain general monotone inclusions. Moreover, we present splitting methods matching these lifting bounds, and thus having minimal lifting.