The present paper considers leveraging network topology information to improve the convergence rate of ADMM for decentralized optimization, where networked nodes work collaboratively to minimize the objective. Such problems can be solved efficiently using ADMM via decomposing the objective into easier subproblems. Properly exploiting network topology can significantly improve the algorithm performance. Hybrid ADMM explores the direction of exploiting node information by taking into account node centrality but fails to utilize edge information. This paper fills the gap by incorporating both node and edge information and provides a novel convergence rate bound for decentralized ADMM that explicitly depends on network topology. Such a novel bound is attainable for certain class of problems, thus tight. The explicit dependence further suggests possible directions to optimal design of edge weights to achieve the best performance. Numerical experiments show that simple heuristic methods could achieve better performance, and also exhibits robustness to topology changes.
A zoom objective with a varifocal ratio close to 2:1, particularly for a projector, has a stationary positive front component, a single axially shiftable negative lens and a stationary multilens rear component. An object-side positive lens member of the rear component, adjoining the shiftable negative lens, has a focal length smaller than that of the group constituted by the remaining, image-side lens members of this component. The negatively refracting third member of the rear component may be a doublet with a positively refracting cemented surface, convex toward the image side, or an equivalent lens pair separated by a small air gap. An electrically-operated shutter provided with a change-over device or arrangement for effecting daylight and flash photography automatically shifting from a daylight mode to a flash photography mode as a function of the detected brightness level of the field being photographed. One embodiment provides for manually presetting the camera for flash photography which is then automatically carried out with an automatic change-over from a daylight photography mode to a flash photography mode. When carrying out daylight photography the power source energy is used in control and in energizing an electromagnet that controls the exposure termination under control of a delay circuit. When taking flash exposure the delay circuit is not employed and a fixed exposure time is used. The power supply is then used to energize the flashbulb and the accuracy of the shutter timing is unaffected as might be the case if the power source were energizing both the flashbulb and driving or energizing the shutter. The same shutter-operating elements are used in both modes of operation and the change-over arrangement eliminates the use of the delay circuit when the brightness level is below a predetermined level so that flash photography becomes necessary. Recently, several convergence rate results for Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) have been presented in the literature. In this paper, we show global linear convergence rate bounds for Douglas-Rachford splitting and ADMM under strong convexity and smoothness assumptions. We further show that the rate bounds are tight for the class of problems under consideration for all feasible algorithm parameters. For problems that satisfy the assumptions, we show how to select step-size and metric for the algorithm that optimize the derived convergence rate bounds. For problems with a similar structure that do not satisfy the assumptions, we present heuristic step-size and metric selection methods.