Fast Algorithms for Optimization, Games and Control Applications

This thesis develops high-performance numerical methods for convex optimization, variational inequalities, and game theory, targeting computational bottlenecks in modern large-scale systems. By leveraging the underlying mathematical structure of these problems, this work bridges the gap between abstract operator theory and real-time control and strategic decision-making applications.
The first core contribution focuses on accelerating first-order methods for smooth and nonsmooth convex optimization. We introduce adaptive step-size rules and coupled smoothing–momentum techniques that achieve optimal convergence rates. These methods are designed to exploit problem structure, ensuring computational efficiency and enabling fast convergence without requiring prior knowledge of global problem parameters.
Extending beyond single-agent optimization, the research adopts the framework of variational inequalities to address complex equilibrium problems. We propose projection-free algorithms and specialized splitting methods for settings in which traditional projection operators are computationally expensive. This unified approach enables efficient computation of equilibria in dynamic games and distributionally robust models, where decision-makers must account for both strategic interactions and data uncertainty.
The practical relevance of these developments is demonstrated through real-world applications and the introduction of an open-source computational toolkit. Collectively, these contributions provide a scalable and robust framework for fast, structure-aware decision-making in complex multi-agent systems.