We propose an accelerated algorithm with a Frank-Wolfe method as an oracle for solving strongly monotone variational inequality problems. While standard solution approaches, such as projected gradient descent (aka value iteration), involve projecting onto the desired set at each iteration, a distinctive feature of our proposed method is the use of a linear minimization oracle in each iteration. This difference potentially reduces the projection cost, a factor that can become significant for certain sets or in high-dimensional problems. We validate the performance of the proposed algorithm on the traffic assignment problem, motivated by the fact that the projection complexity per iteration increases exponentially with respect to the number of links.

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.

Renewable Energy Communities (RECs) enable local energy sharing, reduce grid dependency, and support the energy transition. This work proposes an embedded-oriented Energy Community Management framework that maximizes shared energy while minimizing individual costs, increasing economic benefits. The architecture uses bilevel programming, decoupled via a reformulation of the objective and subproblems with KKT conditions. Optimization employs a modified ADMM algorithm with Nesterov acceleration for faster convergence. Implemented on low-power microcontrollers (ODROID-N2L and H3+), the framework demonstrates real-time feasibility and highlights the potential of lightweight, decentralized REC management.

We consider a class of Wasserstein distributionally robust Nash equilibrium problems, where agents construct heterogeneous data-driven Wasserstein ambiguity sets using private samples and radii, in line with their individual risk-averse behaviour. By leveraging relevant properties of this class of games, we show that equilibria of the original seemingly infinite-dimensional problem can be obtained as a solution to a finite-dimensional Nash equilibrium problem. We then reformulate the problem as a finite-dimensional variational inequality and establish the connection between the corresponding solution sets. Our reformulation has scalable behaviour with respect to the data size and maintains a fixed number of constraints, independently of the number of samples. To compute a solution, we leverage two algorithms, based on the golden ratio algorithm. The efficiency of both algorithmic schemes is corroborated through extensive simulation studies on an illustrative example and a stochastic portfolio allocation game, where behavioural coupling among investors is modeled.

In this paper, we present monviso (monotone variational inequalities solver), a novel open-source Python package for solving monotone variational inequalities. We detail the package’s structure and baseline functionality, discussing a simple example that illustrates the essential methods and parameters. Moreover, we characterize how the proximal operator, which is the foundation of many iterative schemes, is handled through cvxpy, an open-source Python library for convex optimization. We list the available algorithms and describe the basic implementation of any general iterative method to enable users to build additional and (possibly new) algorithms. Finally, we illustrate several examples of possible use cases for monviso, showcasing the different applications the package can support across various fields, including control, optimization, dynamic game theory, and machine learning.

We study the data-driven finite-horizon linear quadratic regularization (LQR) problem reformulated as a semidefinite program (SDP). Our contribution is to propose two novel accelerated first-order methods for solving the resulting SDP. Our methods enjoy adaptive stepsize and adaptive smoothing parameters that speed up convergence and in turn, enhance scalability. Finally, we compare our accelerated first-order methods and show their benefits via numerical simulations on a benchmark LQR example.