Set variational inequalities

摘要

In this paper, we formulate variational inequality problems defined by point to set mappings (SVI). Constructive and nonconstructive proofs for the existence of the solutions of the SVI are discussed. Moreover, we show how the well known Circumscribed Ellipsoid Method can be used to find approximate solutions to the SVI defined by monotone mappings. Thereby, we would like to mention that at each iteration we do not need to find subgradients to construct planes which do not cut off the solution. We study also the connection between the SVI and saddle point problems. Finally, a method for solving nonmonotone variational inequality problems will be considered.