Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms

Abstract

Nonmonotone, possibly multivalued stress-strain or reaction-displacement laws give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying (super)potentials the problems generally have nonunique solutions (stable or unstable). In this paper we propose two methods for the solution of the hemivariational inequality problem. The first method is based on the decomposition of the nonconvex superpotential into convex constituents. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. Both methods are based on the solution of convex subproblems and constitute an effective, reliable and versatile family of numerical algorithms for large scale hemivariational inequalities. Finally, the two methods are applied to solve the same problem and the obtained results are compared.