Technique of upper relaxation type for the sum of quadratic and convex functionals (Q1901960)

``In the present article, we consider the upper relaxation methods for the problem of minimization of functionals represented in the form of a sum of a quadratic functional and a convex functional \(\varphi\) semicontinuous from below. We investigate both coercive and semi-coercive problems. In the first case, the convergence of iterations to a unique solution of the problem is proved; there is established, in the second case, that the limit points of the iteration sequence are solutions to the problem in question. In the case of uniqueness of the semi-coercive problem, the entire iterational sequence converges to this solution. From the viewpoint of practical realization of the methods in question, their versions of pointwise and block upper relaxation methods under separability condition (correspondingly, ``block'' separability) on the functional \(\varphi\) are of most intereset\dots At the end of the article, we give some examples of grid problems approximating variational inequalities, for which the obtained results on the convergence of upper relaxation methods are applied''.