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_15: Distributed Constraint Satisfaction gives an overview of Constraint Satisfaction Problems (CSPs), adapts related search algorithms and consistency algorithms for applications to multi-agent systems, and consolidates recent research devoted to cooperation in such systems. The techniques introduced are applied to various problems in multi-agent systems. Among the new approaches is a hybrid-type algorithm for weak-commitment search combining backtracking and iterative improvement. Also, an extension of the basic CSP formalization called "Partial CSP" is introduced in order to handle over-constrained CSPs.
  @cite_34: Preface. MATHEMATICAL REVIEW. Methods of Proof and Some Notation. Vector Spaces and Matrices. Transformations. Concepts from Geometry. Elements of Calculus. UNCONSTRAINED OPTIMIZATION. Basics of Set--Constrained and Unconstrained Optimization. One--Dimensional Search Methods. Gradient Methods. Newton's Method. Conjugate Direction Methods. Quasi--Newton Methods. Solving Ax = b. Unconstrained Optimization and Neural Networks. Genetic Algorithms. LINEAR PROGRAMMING. Introduction to Linear Programming. Simplex Method. Duality. Non--Simplex Methods. NONLINEAR CONSTRAINED OPTIMIZATION. Problems with Equality Constraints. Problems with Inequality Constraints. Convex Optimization Problems. Algorithms for Constrained Optimization. References. Index.
  @cite_1: 
summary: Distributed optimization is a classical topic in applied mathematics with several excellent textbooks, e.g., @cite_15 @cite_1 @cite_34 .