Motivated by an obstacle problem for a membrane
subject to cohesion forces, constrained minimization problems involving
a non-convex and non-differentiable objective functional
representing the total potential energy are considered. The associated
first order optimality system leads to a hemi-variational inequality,
which can also be interpreted as a special complementarity
problem in function space. Besides an analytical investigation
of first-order optimality, a primal-dual active set solver is introduced.
It is associated to a limit case of a semi-smooth Newton
method for a regularized version of the underlying problem class.
For the numerical algorithms studied in this paper, global as well as
local convergence properties are derived and verified numerically.
This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.