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.
Many convex optimization problems can be analyzed from the perspective of @cite_6 @cite_9 . For example, the problem of minimizing @math can be cast as finding the zero of the subdifferntial @math , which happens to be a maximal monotone operator provided @math is convex, closed and proper. However, finding zeros of two operators can be difficult, whereas methods can be employed to split the problem into easier subproblems @cite_38 @cite_16 @cite_9 . One prominent example is @cite_22 @cite_13 .
@cite_38 @cite_22 @cite_9 @cite_6 @cite_16 @cite_13
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Splitting algorithms for the sum of two monotone operators.We study two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators. These algorithms ar... This paper shows, by means of an operator called asplitting operator, that the Douglas--Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas--Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework. For the problem of minimizing a lower semicontinuous proper convex function f on a Hilbert space, the proximal point algorithm in exact form generates a sequence @math by taking @math to be the minimizes of @math , where @math . This algorithm is of interest for several reasons, but especially because of its role in certain computational methods based on duality, such as the Hestenes-Powell method of multipliers in nonlinear programming. It is investigated here in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential @math is replaced by an arbitrary maximal monotone operator T. Convergence is established under several criteria amenable to implementation. The rate of convergence is shown to be “typically” linear with an arbitrarily good modulus if @math stays large enough, in fact superlinear if @math . The case of @math is treated in extra detail. Applicati... 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.