# Penalized Push-Sum Algorithm for Constrained Distributed Optimization with Application to Energy Management in Smart Grid

## TL;DR

This paper introduces a penalty-based push-sum algorithm for distributed convex constrained optimization in multi-agent networks, demonstrating its effectiveness in energy management within smart grids.

## Contribution

It adapts the push-sum protocol to handle convex constraints using penalty functions, with proven convergence and practical application to smart grid energy management.

## Key findings

- Proven convergence of the algorithm under certain assumptions.
- Effective application to distributed energy management in smart grids.
- Advantages over existing distributed optimization methods.

## Abstract

We study distributed convex constrained optimization on a time-varying multi-agent network. Each agent has access to its own local cost function, its local constraints, and its instant number of out-neighbors. The collective goal is to minimize the sum of the cost functions over the set of all constraints. We utilize the push-sum protocol to be able to solve this distributed optimization problem. We adapt the push-sum optimization algorithm, which has been studied in context of unconstrained optimization so far, to convex constrained optimization by introducing an appropriate choice of penalty functions and penalty parameters. Under some additional technical assumptions on the gradients we prove convergence of the distributed penalty-based push-sum algorithm to the optimal value of the global objective function. We apply the proposed penalty-based push-sum algorithm to the problem of distributed energy management in smart grid and discuss the advantages of this novel procedure in comparison with existing ones.