Adaptive stability in combinatorial optimization problems

We consider a general approach to the construction of necessary, sufficient, and necessary and sufficient conditions that allow to “adapt” a known optimal solution of an abstract combinatorial problem with a certain structure to a change in the initial data set for a fixed cost function “easily” from the combinatorial point of view. We call this approach adaptive stability. Apparently, it is the first time that the approach is described for an abstract problem in a rigorous mathematical formalization.