2019/06/21 by Constantin Christof, Gerd Wachsmuth, Christof, Constantin +1
Mathematics · Computer Science · #Numerical methods in inverse problems #Optimization and Variational Analysis #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1906.09065
This paper is concerned with second-order optimality conditions for Tikhonov\nregularized optimal control problems governed by the obstacle problem. Using a\nsimple observation that allows to characterize the structure of optimal\ncontrols on the active set, we derive various conditions that guarantee the\nlocal/global optimality of first-order stationary points and/or the\nlocal/global quadratic growth of the reduced objective function. Our analysis\nextends and refines existing results from the literature, and also covers those\nsituations where the problem at hand involves additional box-constraints on the\ncontrol. As a byproduct, our approach shows in particular that Tikhonov\nregularized optimal control problems for the obstacle problem can be\nreformulated as state-constrained optimal control problems for the Poisson\nequation, and that problems involving a subharmonic obstacle and a convex\nobjective function are uniquely solvable. The paper concludes with three\ncounterexamples which illustrate that rather peculiar effects can occur in the\nanalysis of second-order optimality conditions for optimal control problems\ngoverned by the obstacle problem, and that necessary second-order conditions\nfor such problems may be hard to derive.\n