  The paper addresses the tolerance approach to the sensitivity analysis of
optimal solutions to the nonlinear optimization problem of the form
  $$\mbox{$\bigoplus\limits_{y\in S}C(y)\to\min$\quad over\quad
$S\in\mathcal{S}$,}$$ where $\mathcal{S}$ is a collection of nonempty subsets
of a finite set $X$ such that the union of $\mathcal{S}$ is $X$ and the
intersection of $\mathcal{S}$ is empty, $C$ is a cost (or weight) function from
$X$ into $\mathbb{R}^+=[0,\infty)$ or $(0,\infty)$, and $\oplus$ is a
continuous, associative, commutative, nondecreasing and unbounded binary
operation of generalized addition on $\mathbb{R}^+$, called an A-operation. We
evaluate and present sharp estimates for upper and lower bounds of costs of
elements from $X$, for which an optimal solution to the above problem remains
stable. These bounds present new results in the sensitivity analysis as well as
extend most known results in a unified way. We define an invariant of the
optimization problem---the tolerance function, which is independent of optimal
solutions, and establish its basic properties, among which we mention a
characterization of the set of all optimal solutions, the uniqueness of optimal
solutions and extremal values of the tolerance function on an optimal solution.
