2017/11/07 by Constantin Christof, Gerd Wachsmuth, Christof, Constantin +1 · 1 citation
Computer Science · Engineering · #Contact Mechanics and Variational Inequalities #Optimization and Variational Analysis #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.1711.02720
This paper is concerned with the differential sensitivity analysis of\nvariational inequalities in Banach spaces whose solution operators satisfy a\ngeneralized Lipschitz condition. We prove a sufficient criterion for the\ndirectional differentiability of the solution map that turns out to be also\nnecessary for elliptic variational inequalities in Hilbert spaces (even in the\npresence of asymmetric bilinear forms, nonlinear operators and nonconvex\nfunctionals). In contrast to classical results, our method of proof does not\nrely on Attouch's theorem on the characterization of Mosco convergence but is\nfully elementary. Moreover, our technique allows us to also study those cases\nwhere the variational inequality at hand is not uniquely solvable and where\ndirectional differentiability can only be obtained w.r.t. the weak or the\nweak-\⋆ topology of the underlying space. As tangible examples, we\nconsider a variational inequality arising in elastoplasticity, the projection\nonto prox-regular sets, and a bang-bang optimal control problem.\n