A Majorization-Minimization Approach to Design of Power Transmission Networks

We propose an optimization approach to design cost-effective electrical power transmission networks. That is, we aim to select both the network structure and the line conductances (line sizes) so as to optimize the trade-off between network efficiency (low power dissipation within the transmission network) and the cost to build the network. We begin with a convex optimization method based on the paper ``Minimizing Effective Resistance of a Graph'' [Ghosh, Boyd \& Saberi]. We show that this (DC) resistive network method can be adapted to the context of AC power flow. However, that does not address the combinatorial aspect of selecting network structure. We approach this problem as selecting a subgraph within an over-complete network, posed as minimizing the (convex) network power dissipation plus a non-convex cost on line conductances that encourages sparse networks where many line conductances are set to zero. We develop a heuristic approach to solve this non-convex optimization problem using: (1) a continuation method to interpolate from the smooth, convex problem to the (non-smooth, non-convex) combinatorial problem, (2) the majorization-minimization algorithm to perform the necessary intermediate smooth but non-convex optimization steps. Ultimately, this involves solving a sequence of convex optimization problems in which we iteratively reweight a linear cost on line conductances to fit the actual non-convex cost. Several examples are presented which suggest that the overall method is a good heuristic for network design. We also consider how to obtain sparse networks that are still robust against failures of lines and/or generators.Comment: 8 pages, 3 figures. To appear in Proc. 49th IEEE Conference on Decision and Control (CDC '10

Lagrangian Relaxation for MAP Estimation in Graphical Models

We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a ``duality gap'', and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular tree-reweighted max-product (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we propose a new class of multiscale relaxations that introduce ``summary'' variables. The potential benefits of such generalizations include: reducing or eliminating the ``duality gap'' in hard problems, reducing the number or Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.Comment: 10 pages, presented at 45th Allerton conference on communication, control and computing, to appear in proceeding

Regulated peristalsis into the acidic region of the _Drosophila_ larval midgut is controlled by a novel component of the Autonomic Nervous System

The underlying cellular and molecular mechanisms that regulate and coordinate critical physiological processes such as peristalsis are complex, often cryptic, and involve the integration of multiple tissues and organ systems within the organism. We have identified a completely novel component of the larval autonomic nervous system in the _Drosophila_ larval midgut that is essential for the peristaltic movement of food from the anterior midgut into the acidic region of the midgut. We have named this region the Superior Cupric Autonomic Nervous System or SCANS. Located at the junction of the anterior and the acidic portions of the midgut, the SCANS is characterized by a cluster of a novel neuro-enteroendocrine cells that we call Lettuce Head Cells, a valve, and two anterior muscular tethers to the dorsal gastric caeca. Using cell ablation and ectopic activation via expression of the _Chlamydomonas reinhardtii_ blue-light activated channelrhodopsin, we demonstrate that the SCANS and in particular the Lettuce Head Cells are both necessary and sufficient for peristalsis and perhaps serve a larger role by coordinating digestion throughout the anterior midgut with development and growth

Fixing Convergence of Gaussian Belief Propagation

Gaussian belief propagation (GaBP) is an iterative message-passing algorithm for inference in Gaussian graphical models. It is known that when GaBP converges it converges to the correct MAP estimate of the Gaussian random vector and simple sufficient conditions for its convergence have been established. In this paper we develop a double-loop algorithm for forcing convergence of GaBP. Our method computes the correct MAP estimate even in cases where standard GaBP would not have converged. We further extend this construction to compute least-squares solutions of over-constrained linear systems. We believe that our construction has numerous applications, since the GaBP algorithm is linked to solution of linear systems of equations, which is a fundamental problem in computer science and engineering. As a case study, we discuss the linear detection problem. We show that using our new construction, we are able to force convergence of Montanari's linear detection algorithm, in cases where it would originally fail. As a consequence, we are able to increase significantly the number of users that can transmit concurrently.Comment: In the IEEE International Symposium on Information Theory (ISIT) 2009, Seoul, South Korea, July 200

Estimation in Gaussian Graphical Models Using Tractable Subgraphs: A Walk-Sum Analysis

Graphical models provide a powerful formalism for statistical signal processing. Due to their sophisticated modeling capabilities, they have found applications in a variety of fields such as computer vision, image processing, and distributed sensor networks. In this paper, we present a general class of algorithms for estimation in Gaussian graphical models with arbitrary structure. These algorithms involve a sequence of inference problems on tractable subgraphs over subsets of variables. This framework includes parallel iterations such as embedded trees, serial iterations such as block Gauss-Seidel, and hybrid versions of these iterations. We also discuss a method that uses local memory at each node to overcome temporary communication failures that may arise in distributed sensor network applications. We analyze these algorithms based on the recently developed walk-sum interpretation of Gaussian inference. We describe the walks ldquocomputedrdquo by the algorithms using walk-sum diagrams, and show that for iterations based on a very large and flexible set of sequences of subgraphs, convergence is guaranteed in walk-summable models. Consequently, we are free to choose spanning trees and subsets of variables adaptively at each iteration. This leads to efficient methods for optimizing the next iteration step to achieve maximum reduction in error. Simulation results demonstrate that these nonstationary algorithms provide a significant speedup in convergence over traditional one-tree and two-tree iterations

Maximum Entropy Relaxation for Graphical Model Selection given Inconsistent Statistics

We develop a novel approach to approximate a specified collection of marginal distributions on subsets of variables by a globally consistent distribution on the entire collection of variables. In general, the specified marginal distributions may be inconsistent on overlapping subsets of variables. Our method is based on maximizing entropy over an exponential family of graphical models, subject to divergence constraints on small subsets of variables that enforce closeness to the specified marginals. The resulting optimization problem is convex, and can be solved efficiently using a primal-dual interiorpoint algorithm. Moreover, this framework leads naturally to a solution that is a sparse graphical model