Path-following techniques in PDE-constrained optimization with low multiplier regularity

where ≤ denotes the ordering in L2(ω). By duality theory, the Lagrange multiplier associated with the inequality constraint involving x1 is assumed to exhibit low regularity only, i.e., it does not admit a pointwise interpretation. On the other hand, the multiplier pertinent to φl ≤ x2 ≤ φu is supposed to be regular and the mapping x2-to-adjoint state is assumed to be smoothing. The regularization employed is of a generalized Moreau-Yosida-type (i.e., including a multiplier shift, which may yield feasibility of the regularized solution with respect to the original constraints) and yields regular approximations to low regularity multipliers of the original problem. First the consistency of the regularization is shown, and then regularity properties of the path are discussed. In particular, under a strict complementarity assumption differentiability with respect to the path/regularization parameter is established. This property is useful in devising highly efficient extrapolation schemes within numerical solution algorithms. Further, the path structure allows us to define approximating models, which are used for controlling the path parameter in an iterative process for computing a solution of the original problem. This strategy turns out to be crucial in avoiding potential ill-conditioning due to a rapid increase of the path/regularization parameter. The Moreau–Yosida regularized subproblems of the new path-following technique are solved efficiently by semismooth Newton methods. Due to the regularization the latter method can be analysed successfully in function space. The overall algorithmic concept is provided, and numerical tests (including a comparison with primal-dual path-following interior point methods) for simultaneously state and control constrained optimal control problems show the efficiency of the new concept.