Stability of a vector integer quadratic programming problem with respect to vector criterion and constraints

Abstract

The paper analyzes five types of stability against perturbations of initial data in criterion functions and constraints of vector integer quadratic optimization problems. Necessary and sufficient conditions are proved for all the types of stability. The relationship among stability with respect to changes of vector criterion coefficients, stability with respect to changes of initial data in constraints, and stability with respect to vector criterion and constraints is established.Keywords: multicriterial discrete optimization, quadratic functions of partial criteria, stability with respect to vector criterion and constraints, perturbations of initial data.The present paper continues the studies [1, 2] concerned with the five well-known types of stability of vector discrete-optimization problems. It pertains to the series of studies on correctness, stability of discrete optimization problems conducted under the guidance of the Academician I. V. Sergienko at the V. M. Glushkov Institute of Cybernetics since the 80s of the last century. The paper [1] presents the results from analysis of stability against perturbations of initial data in quadratic criterion functions of vector problems of discrete optimization on a finite feasible set. The paper [2] compares the same five types of stability against perturbations of initial data in constraints for vector optimization problems on integer points of a polyhedron. Here, we will consider necessary and sufficient conditions of stability against variations in initial data occurring in both the vector criteria and constraints of vector integer quadratic optimization problems.

On the regularization of vector integer quadratic programming problems

“…We call the problem Z A b m ( , ) stable (T 3 -stable with respect to a vector criterion in the terminology of [2,[4][5][6]) if there is a number e > 0 such that we have…”

“…For any vector l Î > R m and any number t that satisfies the inequalities 0 < < t g, (6) the ( ( , , ), ( , , )) C A d b l t l t -operator regularizes a vector nontrivial IQP problem Z A b m ( , ), m ³ 2 .…”

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

“…In many publications of works performed by a collective of mathematicians under the direction of academician I. V. Sergienko and published in the journal "Cybernetics and Systems Analysis" [1][2][3][4][5][6][7][8], the questions of correctness, decidability, stability, regularization, and parametric analysis of discrete optimization problems are investigated. In particular, based on the use of properties of point-set mappings, a comparative analysis of five types of stability with respect to functionals and constraints of vector integer problems with linear and quadratic partial criterions is made in [6][7][8] (see also [9,10] in which these and many other results are systematized).…”

“…In particular, based on the use of properties of point-set mappings, a comparative analysis of five types of stability with respect to functionals and constraints of vector integer problems with linear and quadratic partial criterions is made in [6][7][8] (see also [9,10] in which these and many other results are systematized). The mentioned works, as well as other achievements of the collective, substantially stimulated investigations of similar problems in Russia (for example, [11][12][13][14]*), Belarus (for example, [15][16][17][18][19][20][21][22][23][24][25][26][27]*), Poland [28,29], and other countries [30].…”

Stability criterion for a lexicographic Boolean optimization problem

“…As usually [1-3, 6-10], a problem Z A b m ( , ) is said to be stable (T 3 -stable as termed in [2,[14][15][16]) if…”