927 research outputs found
Primal Recovery from Consensus-Based Dual Decomposition for Distributed Convex Optimization
Dual decomposition has been successfully employed in a variety of distributed
convex optimization problems solved by a network of computing and communicating
nodes. Often, when the cost function is separable but the constraints are
coupled, the dual decomposition scheme involves local parallel subgradient
calculations and a global subgradient update performed by a master node. In
this paper, we propose a consensus-based dual decomposition to remove the need
for such a master node and still enable the computing nodes to generate an
approximate dual solution for the underlying convex optimization problem. In
addition, we provide a primal recovery mechanism to allow the nodes to have
access to approximate near-optimal primal solutions. Our scheme is based on a
constant stepsize choice and the dual and primal objective convergence are
achieved up to a bounded error floor dependent on the stepsize and on the
number of consensus steps among the nodes
A Duality-Based Approach for Distributed Optimization with Coupling Constraints
In this paper we consider a distributed optimization scenario in which a set
of agents has to solve a convex optimization problem with separable cost
function, local constraint sets and a coupling inequality constraint. We
propose a novel distributed algorithm based on a relaxation of the primal
problem and an elegant exploration of duality theory. Despite its complex
derivation based on several duality steps, the distributed algorithm has a very
simple and intuitive structure. That is, each node solves a local version of
the original problem relaxation, and updates suitable dual variables. We prove
the algorithm correctness and show its effectiveness via numerical
computations
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
A Class of Randomized Primal-Dual Algorithms for Distributed Optimization
Based on a preconditioned version of the randomized block-coordinate
forward-backward algorithm recently proposed in [Combettes,Pesquet,2014],
several variants of block-coordinate primal-dual algorithms are designed in
order to solve a wide array of monotone inclusion problems. These methods rely
on a sweep of blocks of variables which are activated at each iteration
according to a random rule, and they allow stochastic errors in the evaluation
of the involved operators. Then, this framework is employed to derive
block-coordinate primal-dual proximal algorithms for solving composite convex
variational problems. The resulting algorithm implementations may be useful for
reducing computational complexity and memory requirements. Furthermore, we show
that the proposed approach can be used to develop novel asynchronous
distributed primal-dual algorithms in a multi-agent context
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