# Distributed Continuous-Time and Discrete-Time Optimization With Nonuniform Unbounded Convex Constraint Sets and Nonuniform Stepsizes

## TL;DR

This paper develops distributed optimization algorithms for multi-agent systems with unbounded and nonuniform constraints and stepsizes, ensuring convergence without requiring strong connectivity or bounded gradients.

## Contribution

It introduces novel continuous and discrete-time distributed algorithms accommodating nonuniform, unbounded convex constraints and stepsizes, extending the scope of multi-agent optimization.

## Key findings

- Agents reach consensus while minimizing the objective.
- Algorithms work with unbounded, nonuniform constraints and stepsizes.
- Numerical examples confirm theoretical convergence results.

## Abstract

This paper is devoted to distributed continuous-time and discrete-time optimization problems with nonuniform convex constraint sets and nonuniform stepsizes for general differentiable convex objective functions. The communication graphs are not required to be strongly connected at any time, the gradients of the local objective functions are not required to be bounded when their independent variables tend to infinity, and the constraint sets are not required to be bounded. For continuous-time multi-agent systems, a distributed continuous algorithm is first introduced where the stepsizes and the convex constraint sets are both nonuniform. It is shown that all agents reach a consensus while minimizing the team objective function even when the constraint sets are unbounded. After that, the obtained results are extended to discrete-time multi-agent systems and then the case where each agent remains in a corresponding convex constraint set is studied. To ensure all agents to remain in a bounded region, a switching mechanism is introduced in the algorithms. It is shown that the distributed optimization problems can be solved, even though the discretization of the algorithms might deviate the convergence of the agents from the minimum of the objective functions. Finally, numerical examples are included to show the obtained theoretical results.