On Distributed Convex Optimization Under Inequality and Equality Constraints

• Published in: IEEE transactions on automatic control Vol. 57; no. 1; pp. 151 - 164
• Main Authors: Minghui Zhu, Martinez, S.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptative systems; Algorithm design and analysis; Algorithms; Applied sciences; Asymptotic properties; Automatic control; Computer science; control theory; systems; Control theory. Systems; Convergence; Convex functions; Cooperative control; distributed optimization; Exact sciences and technology; Heuristic algorithms; Inequalities; Lagrangian functions; multi-agent systems; Network topology; Networks; Optimal control; Optimization; Saddle points; Studies; multi-agent systems; Subgradient optimization; distributed optimization; Cooperative control; Transmission protocol; Duality; Topology; Adaptive control; Convex programming; Saddle point; Penalty function; Multiagent system; Cooperation; Distributed algorithm; Inequality constraint; Equality constraint; Primal dual method; Distributed control; Asymptotic approximation; Objective function; Lagrangian function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2011.2167817

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set. The objective function is defined by a sum of local objective functions, while the global constraint set is produced by the intersection of local constraint sets. In particular, we study two cases: one where the equality constraint is absent, and the other where the local constraint sets are identical. We devise two distributed primal-dual subgradient algorithms based on the characterization of the primal-dual optimal solutions as the saddle points of the Lagrangian and penalty functions. These algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the Slater's condition.