Operator splitting decomposition methods for efficient large-scale optimization in distributed engineering networks

Abstract

Distributed optimization problems on a large scale are concerned with complex interrelations between numerous decision variables, constraints, and interacting subsystems, and require effective mathematical techniques for finding stable equilibrium solutions. In this paper, the author proposes a general operator splitting technique that employs monotone operators and fixed points for distributed optimization in engineering networks. The mathematical method suggested by the author enables the decomposition of a composite operator of an optimization problem into simpler resolvent and forward evaluation operators, allowing for decentralized computation without loss of convergence. The distributed optimization problem is transformed into the monotone inclusion model, and its corresponding operator is analyzed in terms of fixed points. The mathematical technique employed by the author allows one to prove the existence, uniqueness, convergence, and stability properties of solutions for the optimization problem. An iterative splitting algorithm to solve for equilibrium solutions is developed. Numerical simulations for distributed optimization benchmark instances that include 50 – 500 agents show that the method proposed in the paper leads to faster convergence, lower residual errors, and higher stability of solutions in comparison with conventional gradient descent, ADMM, and Douglas–Rachford splitting techniques. It is concluded that the mathematical structure of operator decomposition guarantees convergence properties and scalability of the optimization systems on a large scale.