In decentralized consensus optimization, a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable, where their information exchange is restricted between the neighbors. To this end, one can first obtain a problem reformulation and then apply the alternating direction method of multipliers (ADMM). The method applies iterative computation at the individual agents and information exchange between the neighbors. This approach has been observed to converge quickly and deemed powerful. This paper establishes its linear convergence rate for the decentralized consensus optimization problem with strongly convex local objective functions.The theoretical convergence rate is explicitly given in terms of the network topology, the properties of local objective functions, and the algorithm parameter. This result is not only a performance guarantee but also a guideline toward accelerating the ADMM convergence.

Consider the consensus problem of minimizing f (x) = n i=1 fi(x), where x ∈ R p and each fi is only known to the individual agent i in a connected network of n agents. To solve this problem and obtain the solution, all the agents collaborate with their neighbors through information exchange. This type of decentralized computation does not need a fusion center, offers better network load balance, and improves data privacy. This paper studies the decentralized gradient descent method [20], in which each agent i updates its local variable x (i) ∈ R n by combining the average of its neighbors' with a local negative-gradient step −α∇fi(x (i) ). The method is described by the iteration (1) depends on the stepsize, function convexity, and network spectrum.

This work develops a distributed optimization strategy with guaranteed exact convergence for a broad class of left-stochastic combination policies. The resulting exact diffusion strategy is shown in Part II [2] to have a wider stability range and superior convergence performance than the EXTRA strategy. The exact diffusion solution is applicable to non-symmetric left-stochastic combination matrices, while many earlier developments on exact consensus implementations are limited to doublystochastic matrices; these latter matrices impose stringent constraints on the network topology. The derivation of the exact diffusion strategy in this work relies on reformulating the aggregate optimization problem as a penalized problem and resorting to a diagonally-weighted incremental construction. Detailed stability and convergence analyses are pursued in Part II [2] and are facilitated by examining the evolution of the error dynamics in a transformed domain. Numerical simulations illustrate the theoretical conclusions.

Phytate (myo-inositol-1,2,3,4,5,6-hexakisphosphate, InsP6) is the most abundant P-containing compound in plants, and an important anti-nutritional factor, due to its ability to complex essential micro-nutrients, e.g. iron and zinc. Analysis of natural variation for InsP6 and Pi accumulation in seeds and leaves for a large number of accessions of Arabidopsis thaliana, using a novel method for InsP6 detection, revealed a wide range of variation in InsP6 and Pi levels, varying from 7.0 mg to 23.1 mg of InsP6 per gram of seed. Quantitative trait locus (QTL) analysis of InsP6 and Pi levels in seeds and leaves, using an existing recombinant inbred line population, was performed in order to identify a gene(s) that is (are) involved in the regulation of InsP6 accumulation. Five genomic regions affecting the quantity of the InsP6 and Pi in seeds and leaves were identified. One of them, located on top of chromosome 3, affects all four traits. This QTL appears as the major locus responsible for the observed variation in InsP6 and Pi contents in the L er/Cvi RIL population; the L er allele decreases the content of both InsP6 and Pi in seeds and in leaves. The InsP6/Pi locus was further fine-mapped to a 99-kb region, containing 13 open reading frames. The maternal inheritance of the QTL and the positive correlation between InsP6 and total Pi levels both in seeds and in leaves indicate that the difference in InsP6 level between L er and Cvi is likely to be caused by a difference in transport rather than by an alteration in the biosynthesis. Therefore, we consider the vacuolar membrane ATPase subunit G, located in the region of interest, as the most likely candidate gene for InsP6/Pi.

Part I of this work [2] developed the exact diffusion algorithm to remove the bias that is characteristic of distributed solutions for deterministic optimization problems. The algorithm was shown to be applicable to a larger set of combination policies than earlier approaches in the literature. In particular, the combination matrices are not required to be doubly stochastic, which impose stringent conditions on the graph topology and communications protocol. In this Part II, we examine the convergence and stability properties of exact diffusion in some detail and establish its linear convergence rate. We also show that it has a wider stability range than the EXTRA consensus solution, meaning that it is stable for a wider range of stepsizes and can, therefore, attain faster convergence rates. Analytical examples and numerical simulations illustrate the theoretical findings.Index Terms-distributed optimization, diffusion, consensus, exact convergence, left-stochastic matrix, doublystochastic matrix, balanced policy, Perron vector.where the {q k } are positive weighting scalars, each J k (w) is convex and differentiable, and the aggregate cost J (w) is strongly-convex. When q 1 · · · = q N , problem (1) reduces to(Problems of the type (1)-(2) find applications in a wide range of areas including including wireless sensor networks [3], distributed adaptation and estimation strategies [4], [5], distributed statistical learning [6], [7] and clustering [8]. Various algorithms have been proposed to solve problem (2) such as [9]-[21]. These algorithms either employ doubly-