Activity Identification and Local Linear Convergence of Forward--Backward-type methods

In this paper, we consider a class of Forward--Backward (FB) splitting methods that includes several variants (e.g. inertial schemes, FISTA) for minimizing the sum of two proper convex and lower semi-continuous functions, one of which has a Lipschitz continuous gradient, and the other is partly smooth relatively to a smooth active manifold $\mathcal{M}$. We propose a unified framework, under which we show that, this class of FB-type algorithms (i) correctly identifies the active manifolds in a finite number of iterations (finite activity identification), and (ii) then enters a local linear convergence regime, which we characterize precisely in terms of the structure of the underlying active manifolds. For simpler problems involving polyhedral functions, we show finite termination. We also establish and explain why FISTA (with convergent sequences) locally oscillates and can be slower than FB. These results may have numerous applications including in signal/image processing, sparse recovery and machine learning. Indeed, the obtained results explain the typical behaviour that has been observed numerically for many problems in these fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name only a few.Comment: Full length version of the previous short on

Challenges in Optimal Control of Nonlinear PDE-Systems

The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Operator Splitting Methods for Convex and Nonconvex Optimization

This dissertation focuses on a family of optimization methods called operator splitting methods. They solve complicated problems by decomposing the problem structure into simpler pieces and make progress on each of them separately. Over the past two decades, there has been a resurgence of interests in these methods as the demand for solving structured large-scale problems grew. One of the major challenges for splitting methods is their sensitivity to ill-conditioning, which often makes them struggle to achieve a high order of accuracy. Furthermore, their classical analyses are restricted to the nice settings where solutions do exist, and everything is convex. Much less is known when either of these assumptions breaks down.This work aims to address the issues above. Specifically, we propose a novel acceleration technique called inexact preconditioning, which exploits second-order information at relatively low computation cost. We also show that certain splitting methods still work on problems without solutions, in the sense that their iterates provide information on what goes wrong and how to fix. Finally, for nonconvex problems with saddle points, we show that almost surely, splitting methods will only converge to the local minimums under certain assumptions