470 research outputs found
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
Parallel LQP alternating direction method for solving variational inequality problems with separable structure
In this paper, we propose a logarithmic-quadratic proximal alternating direction method for structured variational inequalities. The predictor is obtained by solving series of related systems of nonlinear equations, and the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along a new descent direction. Global convergence of the new method is proved under certain assumptions. Preliminary numerical experiments are included to verify the theoretical assertions of the proposed method.Qatar University Start-Up Grant: QUSG-CAS-DMSP-13/14-8.Scopu
Applications of a splitting algorithm to decomposition in convex programming and variational inequalities
Cover title.Includes bibliographical references.Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058by Paul Tseng
A Primal-Dual Algorithmic Framework for Constrained Convex Minimization
We present a primal-dual algorithmic framework to obtain approximate
solutions to a prototypical constrained convex optimization problem, and
rigorously characterize how common structural assumptions affect the numerical
efficiency. Our main analysis technique provides a fresh perspective on
Nesterov's excessive gap technique in a structured fashion and unifies it with
smoothing and primal-dual methods. For instance, through the choices of a dual
smoothing strategy and a center point, our framework subsumes decomposition
algorithms, augmented Lagrangian as well as the alternating direction
method-of-multipliers methods as its special cases, and provides optimal
convergence rates on the primal objective residual as well as the primal
feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure
An alternating direction method for solving convex nonlinear semidefinite programming problems
An alternating direction method is proposed for solving convex semidefinite optimization problems. This method only computes several metric projections at each iteration. Convergence analysis is presented and numerical experiments in solving matrix completion problems are reported
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