extragradientmetoden

The extragradient method is an iterative algorithm used for solving variational inequality problems and monotone operator equations. It is a generalization of the gradient descent method, designed to handle situations where simply following the gradient might not converge to a solution. The core idea of the extragradient method involves taking two gradient steps, or more precisely, two projection steps, within each iteration.

In a typical iteration, the method first computes a tentative point by taking a step in the negative gradient direction. This tentative point is then projected onto the feasible set. Subsequently, a second gradient step is computed based on this projected point, and this second step is also projected onto the feasible set to obtain the next iteration's point. This double step ensures that the algorithm makes progress towards a solution, especially in the presence of constraints or complex operator structures.

The extragradient method has theoretical guarantees of convergence under certain conditions, such as monotonicity and Lipschitz continuity of the underlying operator. It has found applications in various fields, including optimization, machine learning, game theory, and economic modeling, where variational inequalities arise naturally. Its robustness compared to simpler gradient-based methods makes it a valuable tool for solving a broader class of problems.