On strong quasistability of a vector problem on substitutions

A type of the stability of the Pareto, Smale, and Slater sets for a problem of minimizing linear forms over an arbitrary set of substitutions of the symmetric group is investigated. This type of stability assumes that at least one substitution preserves corresponding efficiency for ”small” independent perturbations of coefficients of the linear forms. Quantitative bounds of such a type of stability are found. In the paper [1], two types of stability for a vector integer linear programming (ILP) problem are investigated. This problem consists in finding the Pareto set. Note that these types of stability are first introduced for a scalar trajectory problem in [2]. In [1], a formula for the strong quasistability radius is deduced and a necessary and sufficient condition of such a stability for a vector ILP problem is obtained. The aim of this paper is to extend these results to vector combinatorial problems of finding the Pareto, Smale, and Slater sets among substitutions of the symmetric group.