Distributed alternating gradient descent for convex semi-infinite programs over a network

TL;DR

This paper introduces a distributed first-order algorithm for convex semi-infinite programs over dynamic networks, ensuring convergence to an optimal solution through local computations and limited communication.

Contribution

It proposes a novel distributed algorithm combining consensus, gradient descent, and local constraint handling for semi-infinite programs over time-varying networks.

Findings

Algorithm guarantees convergence to an optimizer.

Estimates achieve consensus with vanishing errors.

Simulation confirms theoretical results.

Abstract

This paper presents a first-order distributed algorithm for solving a convex semi-infinite program (SIP) over a time-varying network. In this setting, the objective function associated with the optimization problem is a summation of a set of functions, each held by one node in a network. The semi-infinite constraint, on the other hand, is known to all agents. The nodes collectively aim to solve the problem using local data about the objective and limited communication capabilities depending on the network topology. Our algorithm is built on three key ingredients: consensus step, gradient descent in the local objective, and local gradient descent iterations in the constraint at a node when the estimate violates the semi-infinite constraint. The algorithm is constructed, and its parameters are prescribed in such a way that the iterates held by each agent provably converge to an optimizer. That is, as the algorithm progresses, the estimates achieve consensus, and the constraint violation and the error in the optimal value are bounded above by vanishing terms. Simulation examples illustrate our results.