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Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone

A systematic way for generating penalty and barrier methods is proposed in order to solve variational inequalities with a maximal monotone map and over a feasible set which is the intersection of three kinds of inequalities. The first type of inequalities consists of a finite number of convex inequalities, the second concerns an affine map taking its values in the cone of symmetric semidefinite positive matrices, while the third concerns an affine map taking its values in the Lorentz cone. Convergence is proved under essentially the assumptions that Slater's condition holds and that the set of solutions of the variational inequality is nonempty and compact.