In this talk, an analytical approach for solving nonsmooth convex optimization and variational inequality problems is presented. The proposed iteration is a natural modification of the classical Extragradient algorithm in which it finds the solution of the sum of two monotone operators by evaluating the smooth operator twice per iteration. One of the main advantages of the proposed scheme is to avoid evaluating an extra-gradient step per iteration, this modification is suggested when the composite objective function is a sum of three convex functions, one of them is nonsmooth. The convergence of the generated sequence is established by forcing Fejer monotonicity. Moreover, convergence and complexity analyses for the functional value sequence are derived for the two proposed algorithms, called the Half-Extragradient method and its accelerated version. We provide a sublinear complexity of O(1/k) for the Half-Extragradient and O(1/k2) for its associate accelerated version. Furthermore, two variants of the methods are represented for inclusion problems for cocoarcive and Lipschitz operators based on the cutting-plane (localizations) methods by finding a suitable hyperplane that separates the solution from the current iterate.