arXiv subjects

Steven-Marian Stengl

Publications and source records attributed to Steven-Marian Stengl.

2 recordsLinked to original sources

arXiv2021

Path-Following Methods for Generalized Nash Equilibrium Problems

Building upon the results in [Hintermüller et al., SIAM J. Optim, '15], generalized Nash equilibrium problems are considered, in which the feasible set of each player is influenced by the decisions of their competitors. This is realized via the existence of one (or more) state constraint(s) establishing a link between the players. Special emphasis is put on the situation of a state encoded in a possibly non-linear operator equation. First order optimality conditions under a constraint qualification are derived. Aiming at a practically meaningful method, an approximation scheme using a penalization technique leading to a sequence of (Nash) equilibrium problems without dependence of the constraint set on the other players' strategies is established. An associated path-following strategy related to a value function is then proposed. This happens at first on the most abstract level and is subsequently established to a narrower framework geared to the presence of partial differential equations in the constraint. Our findings are illustrated with examples having distributed and boundary controls - both involving semi-linear elliptic PDEs.

math.OC↗

arXiv2021

Combined Regularization and Discretization of Equilibrium Problems and Primal-Dual Gap Estimators

The present work aims at the application of finite element discretizations to a class of equilibrium problems involving moving constraints. Therefore, a Moreau--Yosida based regularization technique, controlled by a parameter, is discussed and, using a generalized $Γ$-convergence concept, a priori convergence results are derived. The latter technique is applied to the discretization of the regularized problems and is used to prove the convergence to the orginal equilibrium problem, when both -- regularization and discretization -- are imposed simultaneously. In addition, a primal-dual gap technique is used for the derivation of error estimators suitable for adaptive mesh refinement. A strategy for balancing between a refinement of the mesh and an update of the regularization parameter is established, too. The theoretical findings are illustrated for the obstacle problem as well as numerical experiments are performed for two quasi-variational inequalities with application to thermoforming and biomedicine, respectively.

math.NA↗