VI-constrained hemivariational inequalities: distributed algorithms and power control in ad-hoc networks

We consider centralized and distributed algorithms for the numerical solution of a hemivariational inequality (HVI) where the feasible set is given by the intersection of a closed convex set with the solution set of a lower-level monotone variational inequality (VI). The algorithms consist of a main loop wherein a sequence of one-level, strongly monotone HVIs are solved that involve the penalization of the non-VI constraint and a combination of proximal and Tikhonov regularization to handle the lower-level VI constraints. Minimization problems, possibly with nonconvex objective functions, over implicitly defined VI constraints are discussed in detail. The methods developed in the paper are then used to successfully solve a new power control problem in ad-hoc networks. © 2013 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.