We derive a-priori estimates on the length of the primal-dual path that results from a Moreau-Yosida approximation of the feasible set for state constrained optimal control problems. These bounds depend on the regularity of the state and the dimension of the problem. Comparison with numerical results indicates that these bounds are sharp and are attained for the case of a single active point.
Interior Point Methods in Function Space for State Constraints - Inexact Newton and Adaptivity
(2009)
We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton path-following method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an a-posteriori error estimator for a problem suited norm.
We consider first order optimality conditions for state constrained optimal control problems. In particular we study the case where the state equation has not enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measures and additional regularity properties.
This paper is concerned with the state-constrained optimal control of the
two-dimensional thermistor problem, a quasi-linear coupled system
of a parabolic and elliptic PDE with mixed boundary conditions.
This system models the heating of a conducting material by means of direct current.
Existence, uniqueness and continuity for the state system are derived by employing
maximal elliptic and parabolic regularity. By similar arguments the
linearized state system is discussed, while the adjoint system involving measures
is investigated using a duality argument. These results allow to derive
first-order necessary conditions for the optimal control problem.
We propose and analyse an interior point path-following method in function space
for state constrained optimal control. Our emphasis is on proving convergence in
function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.
A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.
We consider a control- and state-constrained optimal control problem
governed by a semilinear
elliptic equation with nonlocal interface conditions.
These conditions occur during the
modeling of diffuse-gray conductive-radiative heat transfer.
The nonlocal radiation interface condition and the pointwise state-constraints
represent the particular features of this problem. To deal with the
state-constraints, continuity of the state is shown which allows to
derive first-order necessary conditions. Afterwards, we establish second-order
sufficient conditions that account for strongly active sets and
ensure local optimality in an $L^2$-neighborhood.
A Lavrentiev type regularization technique for
solving elliptic boundary control problems with pointwise state
constraints is considered. The main concept behind this
regularization is to look for controls in the range of the adjoint
control-to-state mapping. After investigating the analysis of the
method, a semismooth Newton method based on the optimality
conditions is presented. The theoretical results are confirmed by
numerical tests. Moreover, they are validated by comparing the
regularization technique with standard numerical codes based on the
discretize-then-optimize concept.