On the T1-stability Radius of a Multicriteria Linear Boolean Problem with Holder Norms in Parameter Spaces.

Discrete optimization models of decision making are widespread in design, control, economics and many other fields of applied research. Many of the decision-making problems can be formulated as multicriteria Boolean problems. Solutions of the problems are
reduced to a choice of variables values from a discrete set that are the best in some sense, which is determined by the physical or economical meaning. Pareto-optimal alternatives are usually regarded as the best solutions. One of the approaches to investigate these problems is the stability analysis carried out under the uncertainty assumption, which means that the given coefficients of the objective vector-function are known up to a certain accuracy degree or may change unpredictably. In these conditions, an investigation of the Pareto set behavior becomes topical since these investigations allow one to specify reliability limits of the decisions made.
In this work, we address an issue of deriving quantitative stability characteristics. Such a characteristic called stability radius is defined as the limit level of perturbations that retains a certain property of a solutions set given in advance. Stability of a multicriteria optimization problem is usually understood as a property of semi-continuity of a multi-valued mapping, which determines the choice function in the Hausdorff sense; i. e., there exists a neighborhood in the space of the initial parameters such that new Pareto optimums are impossible to arise inside it. A weakening of this requirement results in the $T_1$-stability concept, which is treated as an existence of an initial data neighborhood of in a space of problem parameters such that, although new Pareto optimums may arise within it, for each perturbation, there exists at least one efficient solution to the initial problem (not necessarily the same) that retains Pareto optimality.
We study the $T_1$-stability radius of Pareto set to parameter perturbations in the vector criterion on an assumption that arbitrary Hölder norms are given in criteria and solutions spaces of the multicriteria Boolean problem. Specifying problems classes for which bounds estimates have obtained guarantees that the estimates are achievable. We also present an easily computable achievable upper-bound estimate, which is equal to a norm of a vector criterion coefficients matrix. These results naturally lead to the known estimates for the $T_1$-stability radius for the case of Chebyshev norm in the spaces of solutions and criteria.

Keywords: multicriteria Boolean problem, Pareto set, stability radius, $T_1$-stability, Hölder norm.