In this letter, we study distributed optimization, where a network of agents, abstracted as a directed graph, collaborates to minimize the average of locally-known convex functions. Most of the existing approaches over directed graphs are based on push-sum (type) techniques, which use an independent algorithm to asymptotically learn either the left or right eigenvector of the underlying weight matrices. This strategy causes additional computation, communication, and nonlinearity in the algorithm. In contrast, we propose a linear algorithm based on an inexact gradient method and a gradient estimation technique. Under the assumptions that each local function is strongly-convex with Lipschitz-continuous gradients, we show that the proposed algorithm geometrically converges to the global minimizer with a sufficiently small step-size. We present simulations to illustrate the theoretical findings.

In this paper, we consider distributed optimization problems where the goal is to minimize a sum of objective functions over a multiagent network. We focus on the case when the interagent communication is described by a strongly connected, directed graph. The proposed algorithm, Accelerated Distributed Directed OPTimization (ADDOPT), achieves the best known convergence rate for this class of problems, O(μ ^k ), 0 <; μ <; 1, given strongly convex, objective functions with globally Lipschitz-continuous gradients, where k is the number of iterations. Moreover, ADD-OPT supports a wider and more realistic range of step sizes in contrast to existing work. In particular, we show that ADD-OPT converges for arbitrarily small (positive) step sizes. Simulations further illustrate our results.

This paper considers a distributed optimization problem over a multi-agent network, in which the objective function is a sum of individual cost functions at the agents. We focus on the case when communication between the agents is described by a directed graph. Existing distributed optimization algorithms for directed graphs require at least the knowledge of the neighbors' out-degree at each agent (due to the requirement of column-stochastic matrices). In contrast, our algorithm requires no such knowledge. Moreover, the proposed algorithm achieves the best known rate of convergence for this class of problems, $O(\mu^k)$ for $0 \lt \mu \lt {1}$, where $k$ is the number of iterations, given that the objective functions are strongly-convex and have Lipschitz-continuous gradients. Numerical experiments are also provided to illustrate the theoretical findings.

Given the increase in flood events in recent years, accurate flood risk assessment is an important component of flood mitigation in urban areas. This research aims to develop updated and accurate flood risk maps in the Don River Watershed within the Great Toronto Area (GTA). The risk maps use geographical information systems (GIS) and multi-criteria analysis along with the application of Analytical Hierarchy Process methods to define and quantify the optimal selection of weights for the criteria that contribute to flood risk. The flood hazard maps were generated for four scenarios, each with different criteria (S1, S2, S3, and S4). The base case scenario (S1) is the most accurate, since it takes into account the floodplain map developed by the Toronto and Region Conservation Authority. It also considers distance to streams (DS), height above nearest drainage (HAND), slope (S), and the Curve Number (CN). S2 only considers DS, HAND, and CN, whereas S3 considers effective precipitation (EP), DS, HAND, and S. Lastly, S4 considers total precipitation (TP), DS, HAND, S, and CN. In addition to the flood hazard, the social and economic vulnerability was included to determine the total flood vulnerability in the watershed under three scenarios; the first one giving a higher importance to the social vulnerability, the second one giving equal importance to both social and economic vulnerability, and the third one giving more importance to the economic vulnerability. The results for each of the four flood scenarios show that the flood risk generated for S2 is the most similar to the base case (S1), followed by S3 and S4. The inclusion of social and economic vulnerability highlights the impacts of floods that are typically ignored in practice. It will allow watershed managers to make more informed decisions for flood mitigation and protection. The most important outcome of this research is that by only using the digital elevation model, the census data, the streams, land use, and soil type layers, it is possible to obtain a reliable flood risk map (S2) using a simplified method as compared to more complex flood risk methods that use hydraulic and hydrological models to generate flood hazard maps (as was the case for S1).

We study distributed optimization to minimize a sum of smooth and strongly-convex functions. Recent work on this problem uses gradient tracking to achieve linear convergence to the exact global minimizer. However, a connection among different approaches has been unclear. In this paper, we first show that many of the existing first-order algorithms are related with a simple state transformation, at the heart of which lies a recently introduced algorithm known as $\mathcal {AB}$ . We then present distributed heavy-ball , denoted as $\mathcal {AB}m$ , that combines $\mathcal {AB}$ with a momentum term and uses nonidentical local step-sizes. By simultaneously implementing both row- and column-stochastic weights, $\mathcal {AB}m$ removes the conservatism in the related work due to doubly stochastic weights or eigenvector estimation. $\mathcal {AB}m$ thus naturally leads to optimization and average consensus over both undirected and directed graphs. We show that $\mathcal {AB}m$ has a global $R$ -linear rate when the largest step-size and momentum parameter are positive and sufficiently small. We numerically show that $\mathcal {AB}m$ achieves acceleration, particularly when the objective functions are ill-conditioned.

In this article, we provide a distributed optimization algorithm, termed as TV-AB, that minimizes a sum of convex functions over time-varying, random directed graphs. Contrary to the existing work, the algorithm we propose does not require eigenvector estimation to estimate the (non-1) Perron eigenvector of a stochastic matrix. Instead, the proposed approach relies on a novel information mixing approach that exploits both rowand column-stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed. We show that TV-AB converges linearly to the optimal solution when the global objective is smooth and strongly convex, and the underlying time-varying graphs exhibit bounded connectivity, i.e., a union of every C consecutive graphs is strongly connected. We derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument. Simulations confirm the findings in this article.