Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of u ^k+1 = P _Ω[u ^k - α _kd ^k]. Interestingly, many of them can be paired such that ũ ^k = P _Ω[u ^k - β _kF(v ^k)] = P _Ω[ũ ^k - (d ^k ₂ - Gd ^k ₁)], where inf{β _k} > 0 and G is a symmetric positive definite matrix. In other words, this projection equation offers a pair of directions, i.e., d ^k ₁ and d ^k ₂ for each step. In this paper, for various APPAs we present a unified framework involving the above equations. Unified characterization is investigated for the contraction and convergence properties under the framework. This shows some essential views behind various outlooks. To study and pair various APPAs for different types of variational inequalities, we thus construct the above equations in different expressions according to the framework. Based on our constructed frameworks, it is interesting to see that, by choosing one of the directions (d ^k ₁ and d ^k ₂) those studied proximal-like methods always utilize the unit step size namely α _k ≡ 1.