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📄 Abstract
Abstract: Tracking the solution of time-varying variational inequalities is an
important problem with applications in game theory, optimization, and machine
learning. Existing work considers time-varying games or time-varying
optimization problems. For strongly convex optimization problems or strongly
monotone games, these results provide tracking guarantees under the assumption
that the variation of the time-varying problem is restrained, that is, problems
with a sublinear solution path. In this work we extend existing results in two
ways: In our first result, we provide tracking bounds for (1) variational
inequalities with a sublinear solution path but not necessarily monotone
functions, and (2) for periodic time-varying variational inequalities that do
not necessarily have a sublinear solution path-length. Our second main
contribution is an extensive study of the convergence behavior and trajectory
of discrete dynamical systems of periodic time-varying VI. We show that these
systems can exhibit provably chaotic behavior or can converge to the solution.
Finally, we illustrate our theoretical results with experiments.
Key Contributions
This paper extends existing results on tracking time-varying variational inequalities by providing tracking bounds for problems with sublinear solution paths but non-monotone functions, and for periodic time-varying variational inequalities without sublinear path length. It also offers an extensive study of the convergence behavior and trajectory of discrete dynamical systems for periodic time-varying VIs, which is crucial for understanding the stability and dynamics of these systems.
Business Value
Improved theoretical understanding of dynamic optimization and game theory problems can lead to more robust and efficient algorithms for real-world applications in finance, economics, and control systems.