# Distributed optimization with nonconvex velocity constraints, nonuniform position constraints and nonuniform stepsizes

## TL;DR

This paper introduces distributed algorithms for multi-agent systems with complex constraints, ensuring bounded positions and convergence under nonconvex velocity and position constraints with nonuniform stepsizes.

## Contribution

It proposes novel distributed algorithms that handle nonconvex velocity constraints, nonuniform position constraints, and adaptive stepsizes without predesign, ensuring boundedness and convergence.

## Key findings

- Algorithms guarantee agents' positions remain bounded.
- Convergence achieved under jointly strongly connected and balanced communication topologies.
- Numerical examples validate theoretical results.

## Abstract

This note is devoted to the distributed optimization problem of multi-agent systems with nonconvex velocity constraints, nonuniform position constraints and nonuniform stepsizes. Two distributed constrained algorithms with nonconvex velocity constraints and nonuniform stepsizes are proposed in the absence and the presence of nonuniform position constraints by introducing a switching mechanism to guarantee all agents' position states to remain in a bounded region. The algorithm gains need not to be predesigned and can be selected by each agent using its own and neighbours' information. By a model transformation, the original nonlinear time-varying system is converted into a linear time-varying one with a nonlinear error term. Based on the properties of stochastic matrices, it is shown that the optimization problem can be solved as long as the communication topologies are jointly strongly connected and balanced. Numerical examples are given to show the obtained theoretical results.