Algorithms for Solving Variational Inequalities

Variational inequalities (VIs) are a powerful mathematical model which unifies important concepts in applied mathematics. Many applications of VIs involving generalized monotone operators have been in the focus of the scientific community during the last decades. The aim of this project is to achieve a breakthrough in the design of numerical algorithms with good convergence properties for solving pseudo monotone VIs. We propose ourselves to address three main objectives. The first objective aims to provide new projection methods for solving pseudo monotone VIs. So far, only the extragradient method and its variants, which require (at least) two projections per iteration, can be applied for solving pseudo monotone VIs. Computing the projection is extremely expensive for complex and high dimensional problems. Succeeding in the investigation of new deterministic and stochastic projection methods, which require only one projection per iteration, will provide alternative and efficient tools to tackle pseudo monotone VIs. It will also open the gate towards new iterative methods for solving pseudo convex optimization problems, which usually arise in economic. The second objective of the proposal refers to the dynamical systems associated with pseudo monotone VIs. This is a topic with relevance in the theory of differential equations; we will be interested in questions like existence and uniqueness of trajectories and in analyzing their asymptotic behavior. The study of continuous dynamical systems may lead by time discretization to new algorithms for solving pseudo monotone VIs and pseudo convex optimization problems. The third objective is dedicated to the study of the rates of convergence for the new projection methods as well as for the asymptotic behavior of the trajectories generated by dynamical systems. This research will be carried out by employing tools from variational and convex analysis and monotone operator theory and by using stochastic techniques.