Numerical methods for complementarity problems, and optimal control problems under complementarity constraints

Abstract

Complementarity problems occur in many scientific fields: economics, physics, transport, game theory, and mathematics. In this thesis, we offer several theoretical, algorithmic, and numerical contributions to solve the complementarity problems and optimal control problems under complementarity constraints. We are particularly interested in the regularization methods for the numerical resolution of these types of problems, we have proposed new regularization techniques. Indeed, In the first part, we focused on optimal control problems under complementarity constraints. We studied optimal control problems governed by semilinear elliptical variational inequalities involving constraints on the state. We presented a new regularisation schema for the complementarity constraint. We proved that Lagrange multipliers exist. Then, in the second part, we have studied linear complementarity problems (LCPs) and nonlinear complementarity problems (NCPs) by proposing new methods of regularisation to solve these kind of problems. The idea of these methods takes inspiration from interior point methods. Throughout this manuscript, we have focused on the theoretical properties of algorithms and their digital applications.