Abstract
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.
Highlights
Lagrangian relaxation and dual decomposition are extremely effective in solving large-scale convex optimization problems [1,2,3,4,5,6]
The optimization problem is generally divided into two steps, a first step pertaining the calculation of the local subgradients of the Lagrangian dual function, and a second step consisting of the global update of the dual variables by projected subgradient ascent
In one way or another, all these methods use a combination of all the approximate primal solutions that are generated, while the dual decomposition scheme converges to a near-optimal dual solution
Summary
Lagrangian relaxation and dual decomposition are extremely effective in solving large-scale convex optimization problems [1,2,3,4,5,6]. In one way or another, all these methods use a combination of all the approximate primal solutions that are generated, while the dual decomposition scheme converges to a near-optimal dual solution. Having in mind the development of methods to update the dual variables while the optimization problem varies [15,16,17], it is of key importance to employ a constant stepsize In this way, the capability of the subgradient scheme to track the dual optimal solutions does not change over time due to a vanishing stepsize approach. We propose a primal recovery scheme to generate approximate primal solutions from consensus-based dual decomposition This scheme is proven to converge to the optimal primal cost up to a bounded error floor.
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