The Isomorphism Problem for Higman-Thompson groups

We prove that the Higman–Thompson groups View the MathML sourceGn,r+ and View the MathML sourceGm,s+ are isomorphic if and only if m=nm=n and gcd(n−1,r)=gcd(n−1,s)gcd(n−1,r)=gcd(n−1,s)

Accelerating Incremental Gradient Optimization with Curvature Information

This paper studies an acceleration technique for incremental aggregated gradient ({\sf IAG}) method through the use of \emph{curvature} information for solving strongly convex finite sum optimization problems. These optimization problems of interest arise in large-scale learning applications. Our technique utilizes a curvature-aided gradient tracking step to produce accurate gradient estimates incrementally using Hessian information. We propose and analyze two methods utilizing the new technique, the curvature-aided IAG ({\sf CIAG}) method and the accelerated CIAG ({\sf A-CIAG}) method, which are analogous to gradient method and Nesterov's accelerated gradient method, respectively. Setting $\kappa$ to be the condition number of the objective function, we prove the $R$ linear convergence rates of $1 - \frac{4c_0 \kappa}{(\kappa+1)^2}$ for the {\sf CIAG} method, and $1 - \sqrt{\frac{c_1}{2\kappa}}$ for the {\sf A-CIAG} method, where $c_0,c_1 \leq 1$ are constants inversely proportional to the distance between the initial point and the optimal solution. When the initial iterate is close to the optimal solution, the $R$ linear convergence rates match with the gradient and accelerated gradient method, albeit {\sf CIAG} and {\sf A-CIAG} operate in an incremental setting with strictly lower computation complexity. Numerical experiments confirm our findings. The source codes used for this paper can be found on \url{http://github.com/hoitowai/ciag/}.Comment: 22 pages, 3 figures, 3 tables. Accepted by Computational Optimization and Applications, to appea

Optimal Pricing to Manage Electric Vehicles in Coupled Power and Transportation Networks

We study the system-level effects of the introduction of large populations of Electric Vehicles on the power and transportation networks. We assume that each EV owner solves a decision problem to pick a cost-minimizing charge and travel plan. This individual decision takes into account traffic congestion in the transportation network, affecting travel times, as well as as congestion in the power grid, resulting in spatial variations in electricity prices for battery charging. We show that this decision problem is equivalent to finding the shortest path on an "extended" transportation graph, with virtual arcs that represent charging options. Using this extended graph, we study the collective effects of a large number of EV owners individually solving this path planning problem. We propose a scheme in which independent power and transportation system operators can collaborate to manage each network towards a socially optimum operating point while keeping the operational data of each system private. We further study the optimal reserve capacity requirements for pricing in the absence of such collaboration. We showcase numerically that a lack of attention to interdependencies between the two infrastructures can have adverse operational effects.Comment: Submitted to IEEE Transactions on Control of Network Systems on June 1st 201

Optimal Electricity Pricing for Societal Infrastructure Systems

We develop a general framework for pricing electricity in order to optimally manage the electricity load of societal infrastructures that interact with power systems through their price-responsive electricity load. In such infrastructure systems, electricity is not the sole resource needed to serve users\u27 needs. Examples include cloud computing infrastructure or electric transportation networks. In these cases, other shared networked resources such as charging stations or communication links and data centers are also required to serve users. Hence, pricing of electricity becomes intertwined to managing other congestible resources not priced by the power system operator, leading to a complex economic dispatch problem. For brevity of notation, our analysis is performed under a static setting. We discuss how the power system operator should model the effects of the mobility of loads and congestion in the infrastructure in the economic dispatch. We numerically study the performance of our algorithms using the example of a simple electric transportation network