PRIMAL-DUAL METHODS FOR CONSTRAINED OPTIMIZATION: ANALYSIS AND APPLICATIONS

Constrained optimization plays a crucial role in numerous scientific and engineering fields, where solutions must satisfy specific constraints while optimizing an objective function. The complexity of these problems has driven the development of efficient algorithms. Primal-dual methods, particularly, have been a powerful class of algorithms capable of tackling constrained optimization problems. This dissertation introduces and analyzes new Lagrangian-based primal-dual methods, exploring their applications in the equilibrium computation of generalized Nash games and non-convex constrained optimization problems. Generalized Nash games, also known as generalized Nash equilibrium problems, expand the concept of classical Nash games by incorporating coupled constraints, which substantially increase their computational complexity. These games are prevalent in a variety of real-world applications, such as electricity markets, economic markets, transportation networks, and various multi-agent systems, where decision-makers are required to engage in strategic actions while also considering coupled constraints. We develop a primal-dual first-order method for efficient computation of generalized Nash equilibrium, providing its theoretical foundations and practical implementation. Non-convex constrained optimization problems often emerge across various application domains, presenting significant theoretical and computational challenges due to the presence of non-convex constraints and objective functions. To address these challenges, we propose and analyze novel Lagrangian-based primal-dual methods designed to manage non-convex constraints and establish their convergence properties. We perform extensive numerical experiments to demonstrate the practicality and versatility of our proposed methods. The results show the efficacy of our methods in tackling the computational challenges associated with generalized Nash games and non-convex constrained optimization.