TY - JOUR A1 - Hintermüller, Michael A1 - Surowiec, Thomas T1 - A Bundle-Free Implicit Programming Approach for a Class of MPECs in Function Space JF - Mathematical Programming N2 - Using a standard first-order optimality condition for nonsmooth optimization prob- lems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments. Y1 - 2017 U6 - https://doi.org/10.1007/s10107-016-0983-9 VL - 160 IS - 1-2 SP - 271 EP - 305 ER - TY - JOUR A1 - Hintermüller, Michael A1 - Surowiec, Thomas T1 - On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind JF - Set-Valued and Variational Analysis N2 - The directional differentiability of the solution mapping for a class of variational inequali- ties of the second kind inspired by applications in fluid mechanics and moving free boundary problems is investigated. The result is particularly relevant for the model predictive control or optimal control of such variational inequalities in that it can be used to derive stationarity conditions and efficient numerical methods. Y1 - 2017 U6 - https://doi.org/10.1007/s11228-017-0408-9 VL - 26 SP - 631 EP - 642 ER - TY - INPR A1 - Gahururu, Deborah A1 - Hintermüller, Michael A1 - Surowiec, Thomas T1 - Risk-Neutral PDE-Constrained Generalized Nash Equilibrium Problems N2 - A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player’s actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples. Y1 - 2021 ER - TY - INPR A1 - Geiersbach, Caroline A1 - Hintermüller, Michael T1 - Optimality conditions and Moreau–Yosida regularization for almost sure state constraints N2 - We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau–Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity. Y1 - 2021 ER - TY - INPR A1 - Grimm, Veronika A1 - Hintermüller, Michael A1 - Huber, Olivier A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - A PDE-Constrained Generalized Nash Equilibrium Approach for Modeling Gas Markets with Transport N2 - We investigate a class of generalized Nash equilibrium problems (GNEPs) in which the objectives of the individuals are interdependent and the shared constraint consists of a system of partial differential equations. This setup is motivated by the modeling of strategic interactions of competing firms, which explicitly take into account the dynamics of transporting a commodity, such as natural gas, through a network. We establish the existence of a variational equilibrium of the GNEP. In the case of symmetric firms, we identify an equivalent optimization problem. We use this model to numerically explore the impact of linepacking, that is the use of the network as a temporary storage device. In particular, we study the firms' decisions under various linepacking abilities and analyze which market participants benefit from it. Y1 - ER - TY - INPR A1 - Bongarti, Marcelo A1 - Hintermüller, T1 - Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network N2 - The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. KW - optimal boundary control KW - gas dynamics KW - gas networks KW - isothermal Euler equation KW - compressible fluid dynamics Y1 - 2023 ER -