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The paper addresses the tolerance approach to the sensitivity analysis ofoptimal 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 subsetsof a finite set $X$ such that the union of $\mathcal{S}$ is $X$ and theintersection 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 acontinuous, associative, commutative, nondecreasing and unbounded binaryoperation of generalized addition on $\mathbb{R}^+$, called an A-operation. Weevaluate and present sharp estimates for upper and lower bounds of costs ofelements from $X$, for which an optimal solution to the above problem remainsstable. These bounds present new results in the sensitivity analysis as well asextend most known results in a unified way. We define an invariant of theoptimization problem---the tolerance function, which is independent of optimalsolutions, and establish its basic properties, among which we mention acharacterization of the set of all optimal solutions, the uniqueness of optimalsolutions and extremal values of the tolerance function on an optimal solution.

Concepts of Stability in Discrete Optimization Involving Generalized Addition Operations