Sonja Steffensen

A New Relaxation Scheme for Mathematical Programs with Equilibrium Constraints

We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently nondegenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The relaxation scheme is such that a strongly stationary solution of the MPEC is also a solution of the relaxed problem if the relaxation parameter is chosen sufficiently small. We discuss the properties of the resulting parameterized nonlinear programs and compare stationary points and solutions. We further prove that a limit point of a sequence of stationary points of a sequence of relaxed problems is Clarke-stationary if it satisfies a so-called MPEC-constant rank constraint qualification, and it is Mordukhovich-stationary if it satisfies the MPEC-linear independence constraint qualification and the stationary points satisfy a second order sufficient condition. From this relaxation scheme, a numerical approach is derived that is applied to a comprehensive test set. The numerical results show that the approach combines good efficiency with high robustness.

Implicit-Explicit Runge--Kutta Schemes for Numerical Discretization of Optimal Control Problems

Implicit-explicit (IMEX) Runge--Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and nonstiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX Runge--Kutta methods in the context of optimal control problems. The analysis of the schemes is based on the continuous optimality system. Using suitable transformations of the adjoint equation, order conditions up to order three are proven, and the relation between adjoint schemes obtained through different transformations is investigated as well. Conditions for the IMEX Runge--Kutta methods to be symplectic are also derived. A numerical example illustrating the theoretical properties is presented.

Relaxation approaches to the optimal control of the Euler equations

The treatment of control problems governed by systems of conservation laws poses serious challenges for analysis and numerical simulations. This is due mainly to shock waves that occur in the solution of nonlinear systems of conservation laws. In this article, the problem of the control of Euler flows in gas dynamics is considered. Numerically, two semi-linear approximations of the Euler equations are compared for the purpose of a gradient-based algorithm for optimization. One is the Lattice-Boltzmann method in one spatial dimension and five velocities (D1Q5 model) and the other is the relaxation method. An adjoint method is used. Good results are obtained even in the case where the solution contains discontinuities such as shock waves or contact discontinuities.

Stabilization of Networked Hyperbolic Systems with Boundary Feedback

We summarize recent theoretical results as well as numerical results on the feedback stabilization of first order quasilinear hyperbolic systems (on networks). For the stabilization linear feedback controls are applied at the nodes of the network. This yields the existence and uniqueness of a C 1-solution of the hyperbolic system with small C 1-norm. For this solution an appropriate L 2-Lyapunov function decays exponentially in time. This implies the exponential stability of the system. A numerical discretization of the Lyapunov function is presented and a numerical analysis shows the expected exponential decay for a class of first-order discretization schemes. As an application for the theoretical results the stabilization of the gas flow in fan-shaped pipe networks with compressors is considered.

Relaxation approach for equilibrium problems with equilibrium constraints

We study a generalization of the relaxation scheme for mathematical programs with equilibrium constraints (MPECs) studied in Steffensen and Ulbrich (2010) [31] to equilibrium problems with equilibrium constraints (EPECs). This new class of optimization problems arise, for example, as reformulations of bilevel models used to describe competition in electricity markets. The convergence results of Steffensen and Ulbrich (2010) [31] are extended to parameterized MPECs and then further used to prove the convergence of the associated method for EPECs. Moreover, the proposed relaxation scheme is used to introduce and discuss a new relaxed sequential nonlinear complementarity method to solve EPECs. Both approaches are numerically tested and compared to existing diagonalization and complementarity approaches on a randomly generated test set.

On the relaxation approximation of boundary control of the isothermal Euler equations

We consider the isothermal Euler equations without friction that simulate gas flow through a pipe. We consider the problem of boundary stabilisation of this system locally around a given stationary state. We present a feedback law that is linear in the physical variables and yields exponential decay of the system state. For the numerical solution of hyperbolic systems of conservation laws, the Jin–Xin relaxation scheme can be used. Therefore, we also consider the boundary stabilisation of the relaxation system by the linear Riemann feedback and present numerical examples that show the rapid exponential decay of the stabilised system.

Numerical Methods for the Optimal Control of Scalar Conservation Laws

We are interested in a class of numerical schemes for the optimization of nonlinear hyperbolic partial differential equations. We present continuous and discretized relaxation schemes for scalar, one– conservation laws. We present numerical results on tracking typew problems with nonsmooth desired states and convergence results for higher–order spatial and temporal discretization schemes.

Global Solution of Bilevel Programming Problems

We discuss the global solution of Bilevel Programming Problems using their reformulations as Mathematical Programs with Complementarity Constraints and/or Mixed Integer Nonlinear Programs. We show that under suitable assumptions the Bilevel Program can be reformulated and globally solved via MINLP refomulation. We also briefly discuss some simplifications and suitable additional constraints.