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Linear-quadratic mean-field-type difference games with coupled affine inequality constraints

Partha Sarathi Mohapatra, Puduru Viswanadha Reddy

TL;DR

Addresses discrete-time, finite-horizon linear-quadratic mean-field-type difference games with coupled affine inequality constraints. The authors prove that a mean-field-type generalized Nash equilibrium exists when a multiplier process $\{\mu_k^{i\star}\}$ satisfies implicit complementarity conditions, and they reformulate these conditions as a single large-scale linear complementarity problem (LCP) that can be solved numerically. The equilibrium strategies are given in a linear feedback form $u_k^{i\star}-\mathbb{E}[u_k^{i\star}]=\eta_k^i (x_k^{\star}-\mathbb{E}[x_k^{\star}])$, with $\mathbb{E}[u_k^{i\star}]=\delta_k^i \mathbb{E}[x_k^{\star}] + \bar{\delta}_k^i$, obtained via a five-step direct method and backward recursions for auxiliary coefficients. The method is demonstrated on a microgrid energy-storage problem, showing feasible control actions under coupled constraints and highlighting the practical computability of MFTDG equilibria; the framework extends naturally to matrix-valued dynamics and constraints.

Abstract

In this letter, we study a class of linear-quadratic mean-field-type difference games with coupled affine inequality constraints. We show that the mean-field-type equilibrium can be characterized by the existence of a multiplier process which satisfies some implicit complementarity conditions. Further, we show that the equilibrium strategies can be computed by reformulating these conditions as a single large-scale linear complementarity problem. We illustrate our results with an energy storage problem arising in the management of microgrids.

Linear-quadratic mean-field-type difference games with coupled affine inequality constraints

TL;DR

Addresses discrete-time, finite-horizon linear-quadratic mean-field-type difference games with coupled affine inequality constraints. The authors prove that a mean-field-type generalized Nash equilibrium exists when a multiplier process satisfies implicit complementarity conditions, and they reformulate these conditions as a single large-scale linear complementarity problem (LCP) that can be solved numerically. The equilibrium strategies are given in a linear feedback form , with , obtained via a five-step direct method and backward recursions for auxiliary coefficients. The method is demonstrated on a microgrid energy-storage problem, showing feasible control actions under coupled constraints and highlighting the practical computability of MFTDG equilibria; the framework extends naturally to matrix-valued dynamics and constraints.

Abstract

In this letter, we study a class of linear-quadratic mean-field-type difference games with coupled affine inequality constraints. We show that the mean-field-type equilibrium can be characterized by the existence of a multiplier process which satisfies some implicit complementarity conditions. Further, we show that the equilibrium strategies can be computed by reformulating these conditions as a single large-scale linear complementarity problem. We illustrate our results with an energy storage problem arising in the management of microgrids.
Paper Structure (6 sections, 2 theorems, 46 equations, 2 figures)

This paper contains 6 sections, 2 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Solvability
  5. Numerical Illustration
  6. Conclusion

Key Result

Theorem 1

Let Assumption ass:GameAssumption holds. Assume there exist a multiplier process $\{\mu_k^{i\star}\in \mathbb{R}^{s_i},~i\in \mathsf{N},~ k\in\mathsf{K}_l\}$ satisfying the following complementarity conditions where $\{x_k^{\star}, ~k\in\mathsf{K}_l\}$ evolves as follows with $\mathsf{A}_k:=a_k+\sum_{j\in\mathsf{N}}b^j_k\eta_k^j$, $\bar{\mathsf{A}}_k:=(a_k+\bar{a}_k+\sum_{j\in\mathsf{N}}(b_k^j+\

Figures (2)

  • Figure 1: A microgrid with two generators and one storage unit
  • Figure 2: Panel (a) depicts time varying load and generator outputs, and panel (b) depicts the battery storage level and disturbance signal.

Theorems & Definitions (11)

  • Remark 1
  • Definition 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • ...and 1 more