On distributed convex optimization under inequality and equality constraints via primal-dual subgradient methods

TL;DR

This paper introduces distributed primal-dual subgradient algorithms for multi-agent convex optimization problems with inequality and equality constraints, ensuring convergence to optimal solutions over dynamic networks.

Contribution

The paper develops two novel distributed primal-dual algorithms capable of handling both inequality and equality constraints in multi-agent settings with changing network topologies.

Findings

Algorithms converge to optimal solutions under Slater's condition.

Methods work over networks with time-varying topologies.

Achieves asymptotic agreement among agents on optimal values.

Abstract

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 which are 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 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.