text: This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms.
  @cite_37: In this paper we propose a novel smooth vector field whose trajectories globally converge to the saddle point of the Lagrangian associated with a convex and constrained optimization problem. Under suitable assumptions, we prove global convergence of the trajectories for the class of strictly convex problems and we propose a vector field for linear programs.
  @cite_32: Recursive algorithms where random observations enter are studied in a fairly general framework. An important feature is that the observations my depend on previous "outputs" of the algorithm. The considered class of algorithms contains, e.g., stochastic approximation algorithm, recursive identification algorithm, and algorithms for adaptive control of linear systems. It is shown how a deterministic differential equation can be associated with the algorithm. Problems like convergence with probability one, possible convergence points and asymptotic behavior of the algorithm can all be studied in terms of this differential equation. Theorems stating the precise relationships between the differential equation and the algorithm are given as well as examples of applications of the results to problems in identification and adaptive control.
  @cite_0: Abstract We establish a number of properties associated with the dynamical system H = [H,[H, N]] , where H and N are symmetric n by n matrices and [ A , B ] = AB − BA . The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices. We are especially interested in the role of this equation as an analog computer. For example, we show how to map the data associated with a linear programming problem into H (0) and N in such a way as to have H = [H[H, N]] evolve to a solution of the linear programming problem. This result can be applied to find systems which solve a variety of genetic combinatorial optimization problems, and it even provides an algorithm for diagonalizing symmetric matrices.
  @cite_23: In this paper, we propose a novel computation model for solving the distributed optimization problem where the objective function is formed by the sum of convex functions available to individual agent. Our approach differentiates from the existing approach by local convex mixing and gradient searching in that we force the states of the model to the global optimal point by controlling the subgradient of the global optimal function. In this way, the model we proposed does not suffer from the limitation of diminishing step size in gradient searching and allows fast asymptotic convergence. The model also shows robustness to additive noise, which is a main curse for algorithms based on convex mixing or consensus.
  @cite_31: In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.
  @cite_20: 
summary: Dynamical system solutions to distributed optimization problem have been considered for more than fifty years. The Arrow-Hurwicz-Uzawa flow was shown to converge to the set of saddle points for a constrained convex optimization problem @cite_20 . In @cite_0 , a simple and elegant continuous-time protocol was presented to solve linear programming problems. More recently, in @cite_23 , a continuous-time solution having second-order node dynamics was proposed for solving distributed optimization problems for fixed bidirectional graphs. In @cite_37 , a smooth vector field was shown to be able to drive the system trajectory to converge to the saddle point of the Lagrangian of a convex and constrained optimization problem. In @cite_31 , a network of first-order dynamical system was proposed to solve convex intersection computation problems with directed time-varying communication graphs. Besides optimization, a continuous-time interpretation to discrete-time algorithms was discussed for recursive stochastic algorithms in @cite_32 .