Boolean problem of sequential minimization of moduli of linear functions and stability theorems

Abstract

A multicriterion Boolean problem of lexicographic optimization with partial criteria represented by moduli (absolute values) of linear functions is considered. Five types of stability of a set of lexicographic optima under "small" variations in the parameters of a vector criterion are investigated.Keywords: multicriteria Boolean problem, absolute values (moduli) of linear functions, lexicographic optima, stability of a problem with respect to a vector criterion.

On five types of stability of the lexicographic variant of the combinatorial bottleneck problem

“…Let us give the definitions of the five most widespread types of stability (see, for example, [1][2][3][4][5][6][7][8][13][14][15][16].…”

“…The most part of the results in this direction is connected with obtaining necessary and sufficient conditions for the five known types of stability of the vector integer problems of linear and quadratic programming, which consist of finding the Pareto, Slater, or Smale optimal solutions (see, for example, [1][2][3][4][5][6]) and also with Boolean problems of sequential minimisation of moduli of linear functions [7].…”