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Integer Control Approximations for Graphon Dynamical Systems

  • Martin T. Köhler
  • Artemi Makarow
  • Christian Kirches
    • Categories (Mixed) Integer Nonlinear Programming, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , Short URL: https://optimization-online.org/?p=29601
    \(\) Graphons generalize graphs and define a limit object of a converging graph sequence. The notion of graphons allows for a generic representation of coupled network dynamical systems. We are interested in approximating optimal switching controls for graphon dynamical systems. To this end, we apply a decomposition approach comprised of a relaxation and a reconstruction step. We extend the sum-up rounding algorithm to operate on a finite partition of the continuous vertex set in a graphon setting and restore integer feasibility for a given relaxed control solution. Finally, we derive an \( L^1 \) bound for the state variables for structurally similar graphs and approximated switching controls. We verify our claims by simulating the Lotka-Volterra equations on a graph.

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