On the regularization of vector integer quadratic programming problems
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ON THE REGULARIZATION OF VECTOR INTEGER QUADRATIC PROGRAMMING PROBLEMS V. A. Emelicheva† and E. E. Gurevskiia
UDC 519.8
For a vector integer quadratic programming problem, a regularizing operator is proposed that acts on a vector criterion and transforms a possibly unstable initial problem into a series of perturbed stable problems with the same Pareto set. The technique of e-regularization is developed that allows replacing the considered problem by perturbed e-stable problems. Keywords: multicriteriality, integer quadratic programming, Pareto set, stability, stabilization, regularization, e-stabilization, e-regularization. INTRODUCTION As is well known [1–3], scalar (one-criterion) linear discrete optimization problems are always stable. The existence of unstable (or ill-posed in the sense of Hadamard) vector discrete optimization problems [1, 2, 4–6] naturally leads to the necessity of creation of a regularizing operator that is a concrete kind of perturbations of initial data of a discrete problem in order that, as in the case of a linear programming problem [7, 8], to replace a possibly unstable problem by a series of perturbed stable ones. The first result in this direction is obtained in [9] in which, based on the theory of cones of promising directions [1], a regularization is proposed with respect to a vector criterion and constraints of the vector integer linear programming (ILP) problem of searching for the Pareto set on a bounded set of admissible alternatives. In this case, the regularization with respect to a vector criterion is based on a scalar perturbation parameter and implies the replacement of a vector ILP problem by a perturbed problem with the Slater set contained in the Pareto set of the original problem. Later on, this result is generalized in [10, 11] in which not a scalar but a vector perturbation parameter of the initial data of a problem is considered that allows the creation of a regularizing operator that transforms a possibly unstable vector ILP problem into a series of not only stable but also equivalent problems, i.e., problems with the original Pareto set. A discussion and a comparison of the results of [9, 10] are presented in [2, pp.155–163]. In this article, the methodology developed in [10] is applied to the vector integer quadratic programming (IQP) problem consisting of searching for Pareto sets. An expedient of regularization and e-regularization of such problems is proposed. We note that an expedient is described in [12, 13] for regularization and e-regularization of vector integer linear and quadratic programming problems with the lexicographic optimality principle. 1. BASIC DEFINITIONS AND LEMMAS Let us consider a vector (m-criterion) IQP problem of the form Z m ( A , b ) : min{ f ( x, A , b ) : x Î X }, m ³ 1, f ( x, A , b ) = ( f1 ( x, A1 , b1 ), f 2 ( x, A 2 , b2 ), K , f m ( x, A m , bm )) ,
where
f i : R n ´n ´ R n ® R, are quadratic functions
a
Belarusian State University, Minsk, Belarus, †[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2
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