4,502 research outputs found
A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization
We propose a new first-order primal-dual optimization framework for a convex
optimization template with broad applications. Our optimization algorithms
feature optimal convergence guarantees under a variety of common structure
assumptions on the problem template. Our analysis relies on a novel combination
of three classic ideas applied to the primal-dual gap function: smoothing,
acceleration, and homotopy. The algorithms due to the new approach achieve the
best known convergence rate results, in particular when the template consists
of only non-smooth functions. We also outline a restart strategy for the
acceleration to significantly enhance the practical performance. We demonstrate
relations with the augmented Lagrangian method and show how to exploit the
strongly convex objectives with rigorous convergence rate guarantees. We
provide numerical evidence with two examples and illustrate that the new
methods can outperform the state-of-the-art, including Chambolle-Pock, and the
alternating direction method-of-multipliers algorithms.Comment: 35 pages, accepted for publication on SIAM J. Optimization. Tech.
Report, Oct. 2015 (last update Sept. 2016
Generalized Forward-Backward Splitting
This paper introduces the generalized forward-backward splitting algorithm
for minimizing convex functions of the form , where
has a Lipschitz-continuous gradient and the 's are simple in the sense
that their Moreau proximity operators are easy to compute. While the
forward-backward algorithm cannot deal with more than non-smooth
function, our method generalizes it to the case of arbitrary . Our method
makes an explicit use of the regularity of in the forward step, and the
proximity operators of the 's are applied in parallel in the backward
step. This allows the generalized forward backward to efficiently address an
important class of convex problems. We prove its convergence in infinite
dimension, and its robustness to errors on the computation of the proximity
operators and of the gradient of . Examples on inverse problems in imaging
demonstrate the advantage of the proposed methods in comparison to other
splitting algorithms.Comment: 24 pages, 4 figure
Local Linear Convergence Analysis of Primal-Dual Splitting Methods
In this paper, we study the local linear convergence properties of a
versatile class of Primal-Dual splitting methods for minimizing composite
non-smooth convex optimization problems. Under the assumption that the
non-smooth components of the problem are partly smooth relative to smooth
manifolds, we present a unified local convergence analysis framework for these
methods. More precisely, in our framework we first show that (i) the sequences
generated by Primal-Dual splitting methods identify a pair of primal and dual
smooth manifolds in a finite number of iterations, and then (ii) enter a local
linear convergence regime, which is characterized based on the structure of the
underlying active smooth manifolds. We also show how our results for
Primal-Dual splitting can be specialized to cover existing ones on
Forward-Backward splitting and Douglas-Rachford splitting/ADMM (alternating
direction methods of multipliers). Moreover, based on these obtained local
convergence analysis result, several practical acceleration techniques are
discussed. To exemplify the usefulness of the obtained result, we consider
several concrete numerical experiments arising from fields including
signal/image processing, inverse problems and machine learning, etc. The
demonstration not only verifies the local linear convergence behaviour of
Primal-Dual splitting methods, but also the insights on how to accelerate them
in practice
On the initial estimate of interface forces in FETI methods
The Balanced Domain Decomposition (BDD) method and the Finite Element Tearing
and Interconnecting (FETI) method are two commonly used non-overlapping domain
decomposition methods. Due to strong theoretical and numerical similarities,
these two methods are generally considered as being equivalently efficient.
However, for some particular cases, such as for structures with strong
heterogeneities, FETI requires a large number of iterations to compute the
solution compared to BDD. In this paper, the origin of the bad efficiency of
FETI in these particular cases is traced back to poor initial estimates of the
interface stresses. To improve the estimation of interface forces a novel
strategy for splitting interface forces between neighboring substructures is
proposed. The additional computational cost incurred is not significant. This
yields a new initialization for the FETI method and restores numerical
efficiency which makes FETI comparable to BDD even for problems where FETI was
performing poorly. Various simple test problems are presented to discuss the
efficiency of the proposed strategy and to illustrate the so-obtained numerical
equivalence between the BDD and FETI solvers
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