8 research outputs found
System Modeling and Optimization: 27th IFIP TC 7 Conference, CSMO 2015, Sophia Antipolis, France, June 29 - July 3, 2015, Revised Selected Papers
International audienceBook Front Matter of AICT 49
First-order primal-dual methods for nonsmooth nonconvex optimisation
We provide an overview of primal-dual algorithms for nonsmooth and
non-convex-concave saddle-point problems. This flows around a new analysis of
such methods, using Bregman divergences to formulate simplified conditions for
convergence
Generalized Dynamic Programming Principle and Sparse Mean-Field Control Problems
In this paper we study optimal control problems in Wasserstein spaces, which
are suitable to describe macroscopic dynamics of multi-particle systems. The
dynamics is described by a parametrized continuity equation, in which the
Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize
a cost functional which includes a control norm, thus enforcing a \emph{control
sparsity} constraint. More precisely, we consider a nonlocal restriction on the
total amount of control that can be used depending on the overall state of the
evolving mass. We treat in details two main cases: an instantaneous constraint
on the control applied to the evolving mass and a cumulative constraint, which
depends also on the amount of control used in previous times. For both
constraints, we prove the existence of optimal trajectories for general cost
functions and that the value function is viscosity solution of a suitable
Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract Dynamic
Programming Principle, providing further applications in the Appendix.Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0
license http://creativecommons.org/licenses/by-nc-nd/4.0
Introduction to Nonsmooth Analysis and Optimization
This book aims to give an introduction to generalized derivative concepts
useful in deriving necessary optimality conditions and numerical algorithms for
infinite-dimensional nondifferentiable optimization problems that arise in
inverse problems, imaging, and PDE-constrained optimization. They cover convex
subdifferentials, Fenchel duality, monotone operators and resolvents,
Moreau--Yosida regularization as well as Clarke and (briefly) limiting
subdifferentials. Both first-order (proximal point and splitting) methods and
second-order (semismooth Newton) methods are treated. In addition,
differentiation of set-valued mapping is discussed and used for deriving
second-order optimality conditions for as well as Lipschitz stability
properties of minimizers. The required background from functional analysis and
calculus of variations is also briefly summarized.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0418