On stability of the gradient algorithm in convex discrete optimisation problems and related questions

We introduce the notion of steepness of a coordinate-convex function of discrete argument on an ordinal-convex set. In terms of guaranteed estimates it is shown that in problems of optimisation of coordinate-convex functions on an ordinal-convex set the gradient coordinatewise lifting algorithm is stable under small perturbations of the utility function. As corollaries we obtain improved guaranteed estimates for accuracy of the gradient algorithm, and also new sufficient conditions for the values of the utility function of the problem under consideration to coincide in the global and gradient extrema.