Distributed Random Convex Programming via Constraints Consensus

This paper discusses distributed approaches for the solution of random convex programs (RCP). RCPs are convex optimization problems with a (usually large) number N of randomly extracted constraints; they arise in several applicative areas, especially in the context of decision under uncertainty, see [2],[3]. We here consider a setup in which instances of the random constraints (the scenario) are not held by a single centralized processing unit, but are distributed among different nodes of a network. Each node "sees" only a small subset of the constraints, and may communicate with neighbors. The objective is to make all nodes converge to the same solution as the centralized RCP problem. To this end, we develop two distributed algorithms that are variants of the constraints consensus algorithm [4],[5]: the active constraints consensus (ACC) algorithm, and the vertex constraints consensus (VCC) algorithm. We show that the ACC algorithm computes the overall optimal solution in finite time, and with almost surely bounded communication at each iteration. The VCC algorithm is instead tailored for the special case in which the constraint functions are convex also w.r.t. the uncertain parameters, and it computes the solution in a number of iterations bounded by the diameter of the communication graph. We further devise a variant of the VCC algorithm, namely quantized vertex constraints consensus (qVCC), to cope with the case in which communication bandwidth among processors is bounded. We discuss several applications of the proposed distributed techniques, including estimation, classification, and random model predictive control, and we present a numerical analysis of the performance of the proposed methods. As a complementary numerical result, we show that the parallel computation of the scenario solution using ACC algorithm significantly outperforms its centralized equivalent

CHREST+: A simulation of how humans learn to solve problems using diagrams.

This paper describes the underlying principles of a computer model, CHREST+, which learns to solve problems using diagrammatic representations. Although earlier work has determined that experts store domain-specific information within schemata, no substantive model has been proposed for learning such representations. We describe the different strategies used by subjects in constructing a diagrammatic representation of an electric circuit known as an AVOW diagram, and explain how these strategies fit a theory for the learnt representations. Then we describe CHREST+, an extended version of an established model of human perceptual memory. The extension enables the model to relate information learnt about circuits with that about their associated AVOW diagrams, and use this information as a schema to improve its efficiency at problem solving

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex

The asymptotics of some spin foam amplitudes for a quantum 4-simplex is known to display rapid oscillations whose frequency is the Regge action. In this note, we reformulate this result through a difference equation, asymptotically satisfied by these models, and whose semi-classical solutions are precisely the sine and the cosine of the Regge action. This equation is then interpreted as coming from the canonical quantization of a simple constraint in Regge calculus. This suggests to lift and generalize this constraint to the phase space of loop quantum gravity parametrized by twisted geometries. The result is a reformulation of the flat model for topological BF theory from the Hamiltonian perspective. The Wheeler-de-Witt equation in the spin network basis gives difference equations which are exactly recursion relations on the 15j-symbol. Moreover, the semi-classical limit is investigated using coherent states, and produces the expected results. It mimics the classical constraint with quantized areas, and for Regge geometries it reduces to the semi-classical equation which has been introduced in the beginning.Comment: 16 pages, the new title is that of the published version (initial title: A taste of Hamiltonian constraint in spin foam models