Activity propagation in systems of linear inequalities and its relation to block-coordinate descent in linear programs

We study a constraint propagation algorithm to detect infeasibility of a system of linear inequalities over continuous variables, which we call activity propagation. Each iteration of this algorithm chooses a subset of the inequalities and if it infers that some of them are always active (i.e., always hold with equality), it turns them into equalities. We show that this algorithm can be described as chaotic iterations and its fixed points can be characterized by a local consistency, in a similar way to traditional local consistency methods in CSP such as arc consistency. Via complementary slackness, activity propagation can be employed to iteratively improve a dual-feasible solution of large-scale linear programs in a primal-dual loop – a special case of this method is the Virtual Arc Consistency algorithm by Cooper et al. As our second contribution, we show that this method has the same set of fixed points as block-coordinate descent (BCD) applied to the dual linear program. While BCD is popular in large-scale optimization, its fixed points need not be global optima even for convex problems and a succinct characterization of convex problems optimally solvable by BCD remains elusive. Our result may open the way for such a characterization since it allows us to characterize BCD fixed points in terms of local consistencies.