Set Intersection and Consistency in Constraint Networks

In this paper, we show that there is a close relation between consistency in a constraint network and set intersection. A proof schema is provided as a generic way to obtain consistency properties from properties on set intersection. This approach not only simplifies the understanding of and unifies many existing consistency results, but also directs the study of consistency to that of set intersection properties in many situations, as demonstrated by the results on the convexity and tightness of constraints in this paper. Specifically, we identify a new class of tree convex constraints where local consistency ensures global consistency. This generalizes row convex constraints. Various consistency results are also obtained on constraint networks where only some, in contrast to all in the existing work,constraints are tight

Set Intersection and Consistency in Constraint Networks

Abstract In this paper, we show that there is a close relation between consistency in a constraint network and set intersection. A proof schema is provided as a generic way to obtain consistency properties from properties on set intersection. This approach not only simplifies the understanding of and unifies many existing consistency results, but also directs the study of consistency to that of set intersection properties in many situations, as demonstrated by the results on the convexity and tightness of constraints in this paper. Specifically, we identify a new class of tree convex constraints where local consistency ensures global consistency. This generalizes row convex constraints. Various consistency results are also obtained on constraint networks where only some, in contrast to all in the existing work, constraints are tight

Geometry of Power Flows and Optimization in Distribution Networks

We investigate the geometry of injection regions and its relationship to optimization of power flows in tree networks. The injection region is the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region. If the voltage magnitudes are fixed, the injection region of a tree network can be written as a linear transformation of the product of two-bus injection regions, one for each line in the network. Using this decomposition, we show that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull. Moreover, the resulting convexified optimal power flow problem can be efficiently solved via }{ semi-definite programming or second order cone relaxations. These results improve upon earlier works by removing the assumptions on active power lower bounds. It is also shown that our practical angle assumption guarantees two other properties: (i) the uniqueness of the solution of the power flow problem, and (ii) the non-negativity of the locational marginal prices. Partial results are presented for the case when the voltage magnitudes are not fixed but can lie within certain bounds.Comment: To Appear in IEEE Transaction on Power System

Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising--like) models in equilibrium statistical mechanics, improving on the mean--field approximation and the Bethe--Peierls approximation, which can be regarded as the lowest level of the CVM. In recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models. The foundations of the CVM are briefly reviewed, and the relations with similar techniques are discussed. The main properties of the method are considered, with emphasis on its exactness for particular models and on its asymptotic properties. The problem of the minimization of the variational free energy, which arises in the CVM, is also addressed, and recent results about both provably convergent and message-passing algorithms are discussed.Comment: 36 pages, 17 figure

Unveiling The Tree: A Convex Framework for Sparse Problems

This paper presents a general framework for generating greedy algorithms for solving convex constraint satisfaction problems for sparse solutions by mapping the satisfaction problem into one of graph traversal on a rooted tree of unknown topology. For every pre-walk of the tree an initial set of generally dense feasible solutions is processed in such a way that the sparsity of each solution increases with each generation unveiled. The specific computation performed at any particular child node is shown to correspond to an embedding of a polytope into the polytope received from that nodes parent. Several issues related to pre-walk order selection, computational complexity and tractability, and the use of heuristic and/or side information is discussed. An example of a single-path, depth-first algorithm on a tree with randomized vertex reduction and a run-time path selection algorithm is presented in the context of sparse lowpass filter design