2 research outputs found
Locating the least 2-norm solution of linear programs via a path-following method
Abstract. A linear program has a unique least 2-norm solution provided that the linear program has a solution. To locate this solution, most of the existing methods were devised to solve certain equivalent perturbed quadratic programs or unconstrained minimization problems. Different from these traditional methods, we provide in this paper a new theory and an effective numerical method to seek the least 2-norm solution of a linear program. The essence of this method is a (interior-point-like) path-following algorithm that traces a newly introduced regularized central path which is fairly different from the central path used in interior-point methods. One distinguishing feature of the method is that it imposes no assumption on the problem. The iterates generated by this algorithm converge to the least 2-norm solution whenever the linear program is solvable; otherwise, the iterates converge to a point which gives a minimal KKT residual when the linear program is unsolvable. Key words. Linear programming, path-following algorithm, regularized central path, least 2-norm solution. AMS subject classifications. 90C05, 90C33, 90C51, 65K05 1. Introduction. Conside
A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization
In this paper we consider a general problem set-up for a wide class of convex
and robust distributed optimization problems in peer-to-peer networks. In this
set-up convex constraint sets are distributed to the network processors who
have to compute the optimizer of a linear cost function subject to the
constraints. We propose a novel fully distributed algorithm, named
cutting-plane consensus, to solve the problem, based on an outer polyhedral
approximation of the constraint sets. Processors running the algorithm compute
and exchange linear approximations of their locally feasible sets.
Independently of the number of processors in the network, each processor stores
only a small number of linear constraints, making the algorithm scalable to
large networks. The cutting-plane consensus algorithm is presented and analyzed
for the general framework. Specifically, we prove that all processors running
the algorithm agree on an optimizer of the global problem, and that the
algorithm is tolerant to node and link failures as long as network connectivity
is preserved. Then, the cutting plane consensus algorithm is specified to three
different classes of distributed optimization problems, namely (i) inequality
constrained problems, (ii) robust optimization problems, and (iii) almost
separable optimization problems with separable objective functions and coupling
constraints. For each one of these problem classes we solve a concrete problem
that can be expressed in that framework and present computational results. That
is, we show how to solve: position estimation in wireless sensor networks, a
distributed robust linear program and, a distributed microgrid control problem.Comment: submitted to IEEE Transactions on Automatic Contro