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Working paper

Concepts of stability in discrete optimization involving generalized addition operations

arxiv.org. math. Cornell University, 2013. No. 1309.4242.
Vyacheslav V. Chistyakov, Pardalos P. M.
The paper addresses the tolerance approach to the sensitivity analysis of optimal solutions to a nonlinear discrete optimization problem, which involves a continuous, associative, commutative, nondecreasing and unbounded binary operation of generalized addition on nonnegative reals, called an A-operation. We evaluate and present sharp estimates for upper and lower bounds of costs of elements from the ground set, 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.