Convex Analysis for LQG Systems with Applications to Major Minor LQG Mean-Field Game Systems (1810.07551v4)

Abstract: We develop a convex analysis approach for solving LQG optimal control problems and apply it to major-minor (MM) LQG mean-field game (MFG) systems. The approach retrieves the best response strategies for the major agent and all minor agents that attain an $ε$-Nash equilibrium. An important and distinctive advantage to this approach is that unlike the classical approach in the literature, we are able to avoid imposing assumptions on the evolution of the mean-field. In particular, this provides a tool for dealing with complex and non-standard systems.

• The paper presents a convex analysis approach to LQG control, overcoming classical mean-field assumptions by deriving ε-Nash equilibrium strategies.
• It formulates novel state feedback laws using Riccati equations, applicable to both finite and infinite horizon LQG systems.
• The methodology is extended to major-minor mean-field games, establishing mean-field consistency through fixed-point equations and robust control laws.

Convex Analysis for LQG Systems with Applications to Major Minor LQG Mean-Field Game Systems

Introduction

The paper "Convex Analysis for LQG Systems with Applications to Major Minor LQG Mean-Field Game Systems" investigates the use of convex analysis in solving Linear Quadratic Gaussian (LQG) optimal control problems and applies this methodology to major-minor LQG mean-field game (MFG) systems. By utilizing a convex analysis approach, the paper addresses the limitations of classical methods, particularly the need to impose assumptions on the mean-field evolution.

Convex Analysis Approach

Convex analysis serves as a mathematical foundation for optimization problems, leveraging concepts such as the Gâteaux derivative. This approach is applied to derive the optimal control laws that attain an $\epsilon$-Nash equilibrium in the context of stochastic systems. The analysis circumvents the traditional requirement of assuming mean-field dynamics, which is particularly advantageous for complex and non-standard systems where direct assumptions on control actions are challenging.

Single-Agent LQG Systems

For single-agent LQG systems, the paper outlines an optimization framework using convex analysis to derive the optimal control actions. The unique feature of this approach is the use of a state feedback form, delivering the Riccati and offset equations essential for computing control laws. The approach is extended to cover both finite and infinite horizons, accommodating scenarios where the discount factor is absent.

Major Minor LQG Mean-Field Game Systems

The primary application of the convex analysis framework is in major-minor LQG MFG systems. These systems involve a major agent whose influence remains significant as the population of minor agents increases indefinitely. In such a setup, the paper lays out a methodological framework for deriving best response strategies without pre-assumed mean-field formulations.

The approach includes:

• Extending the state space for major-minor systems to incorporate the mean-field.
• Deriving control laws for both major and minor agents, ensuring the attainment of an $\epsilon$-Nash equilibrium.
• Establishing mean-field consistency through fixed-point equations, a novel aspect that removes the need for a priori control action assumptions.

Implementation Considerations

Implementing the proposed methodology requires:

• Calculating the Riccati matrix and solving the associated differential equations to obtain the control laws.
• Using deterministic representations of the agent interactions as part of the state feedback laws.
• Considering the computational complexity of extending traditional LQG systems to accommodate major-minor dynamics.

Conclusion

Employing convex analysis for LQG systems, particularly in major-minor MFG contexts, offers a robust framework for securing optimal control strategies without relying on imposed assumptions about mean-field dynamics. The methodological advancements not only enhance theoretical understanding but also hold practical significance in systems characterized by complex stochastic interactions among numerous agents.