Primal-dual methods are a class of optimization algorithms that involve solving two related optimization problems simultaneously, one in the primal space and the other in the dual space. These methods are commonly used in convex optimization and mathematical programming, and have been shown to be particularly effective in solving large-scale optimization problems. In primal-dual methods, the primal problem is typically formulated as a minimization problem, while the dual problem is formulated as a maximization problem. The two problems are interconnected through a set of constraints and duality relationships. By solving the dual problem alongside the primal problem, primal-dual methods can often achieve faster convergence and better performance than traditional optimization algorithms. These methods are widely used in various applications, including machine learning, computer vision, and signal processing.