Projection and Contraction Methods for Solving Symmetric Cone Variational Inequalities

In this paper, we employ projection and contraction methods (PC method) to solve variational inequality defined on a closed and convex symmetric cone (SCVI). By introducing Jordan product operator into a finite-dimensional inner product space V, we get an Euclidean Jordan algebras (V, <; ·, · >;,o). For a given x ∈ V, we have a spectral decomposition of x with respect to a Jordan frame of V. Therefore, we can easily obtain projection of x on the square cone K of V(K is a symmetric cone). We describe some implementation issues of PC-methods for solving SCVI(K, F) with uniformly strong monotone mapping F, while K = R_n⁺, K Λ_n⁺, respectively. Finally, we propose an implementation of PC methods for SCVI(S_n⁺, F) with uniformly strong monotone and Lipschitz continue mapping F : Sⁿ → Sⁿ, and give some numerical results to show validity of the proposed method.