A sensitivity measure of the Pareto set in a vector l ∞ - extreme combinatorial problem

Abstract

We consider a vector minimization problem on system of subsets of finite set with Chebyshev norm in a space of perturbing parameters. The behavior of the Pareto set as a function of parameters of partial criteria of the kind MINMAX of absolute value is investigated. 1. Base definitions and lemma The traditional [1 – 11] statement of vector (n-criteria) trajectorial problem is following. A system of nonempty subsets T ⊆ 2E\∅, | T |> 1 of the set E = {e1, e2, ..., em} is given. A vector criterion f(t, A) = (f1(t, A1), f2(t, A2), ..., fn(t, An)) → min t∈T is defined, where n ≥ 1, m ≥ 2, Ai is the row of a matrix A = [aij ]n×m ∈ R. The elements of set T are called trajectories. We consider the case, where partial criteria are given by fi(t, Ai) = max j∈N(t) | aij |, i ∈ Nn, where Nn = {1, 2, ..., n}, N(t) = {j ∈ Nm : ej ∈ t}. By that, the value fi(t, Ai) is Chebyshev norm l∞ of vector, formed by those elements of matrix A, which correspond to the trajectory t. We define the Pareto set (the set of efficient trajectories) by traditional way [12,13]: P(A) = {t ∈ T : π(t, A) = ∅}, c ©2001 by V.A. Emelichev, A.M. Leonovich

Cite this paper

@inproceedings{Emelichev2008ASM, title={A sensitivity measure of the Pareto set in a vector l ∞ - extreme combinatorial problem}, author={Vladimir A. Emelichev and Andrey M. Leonovich}, year={2008} }