Gradient-based Optimization and Control of Systems with Two Decision-makers and Delayed Information

In today’s world systems are complex in nature, often involving two or more decision-makers (DMs) each attempting to optimize the performance of the system using an objective function that reflects its own preferences and is usually different from the other. Such systems are very common in power, microeconomics, and other systems. The focus of this research is how such systems are optimized and how the results of optimization are implemented in practice. Without loss of generality, we consider systems with two DMs, each attempting to optimize the system’s by minimizing its own objective function. The solution of these types of problems is known as the Nash equilibrium. One of the main drawbacks of the Nash equilibrium is that neither DM can minimize its own function without having some knowledge of the decision variables or decision process of the other DM. What information do the DMs agree to exchange or share will determine the process by which the Nash equilibrium is reached. We examine this issue in detail. We propose an iterative process for each DM to adjust its decisions based on the direction of the gradient of its own objective function. We explore two possible alternatives in which each DM will decide to share current or past information with the other DM at every iteration in the process. These two alternatives generate two different scenarios that will describe the evolution of the process by which the Nash equilibrium may or may not be reached. We investigate analytically the conditions in each scenario on the parameters in the objective functions that will guarantee convergence of the process to the Nash equilibrium. We investigate the implementation of the gradient-based algorithms to two important real-world applications: the duopoly problem in microeconomics and the microgrid price control in an energy management system.