Different types of stability of vector integer optimization problem: General approach
- PDF / 121,291 Bytes
- 5 Pages / 595.276 x 793.701 pts Page_size
- 66 Downloads / 296 Views
DIFFERENT TYPES OF STABILITY OF VECTOR INTEGER OPTIMIZATION PROBLEM: GENERAL APPROACH
UDC 519.8
T. T. Lebedeva and T. I. Sergienko
The paper relates the stability of a vector (multiobjective) integer optimization problem to the stability of optimal and nonoptimal solutions of this problem. It is shown that the analysis of several types of stability of the problem of searching for Pareto optimal solutions can be reduced to the analysis of two sets consisting of points that stably belong and do not stably belong to the Pareto set. Keywords: multiobjective integer optimization, stability with respect to a vector criterion and constraints, perturbations of initial data.
There are two main approaches to the stability analysis of multiobjective (vector) discrete optimization problems. One of them (so-called theoretical) is oriented (see, for example, [1–4]) to deriving qualitative results, namely, to establish and study the conditions under which the set of optimal solutions (Pareto, Slater or Smale set) possesses some property that definitely characterizes the problem stability against small perturbations of initial data. The other well-known approach is called constructive; it implies (see, for example, [5–7]) deriving and studying quantitative characteristics of the admissible perturbations of the initial data of vector discrete optimization problems, in particular, of the stability radius. The present paper is concerned with the theoretical trend and deals with searching for a unified approach to studying different types of stability of vector integer optimization problems, namely, searching for concepts that can compose a general basis for the description of various types of stability, and using these concepts to formulate necessary and sufficient stability conditions. Let us consider a multiobjective discrete optimization problem Q ( F , X ) : max {F ( x ) | x Î X }, where F ( x ) = ( f1 ( x ), f 2 ( x ), ... , f l ( x )) is a vector criterion, f i , i Î N l , are real-valued functions, f i : R n ® R 1 , N l = {1, ... , l}, l ³ 2 is the number of partial criteria, X Ì Z n , Z n is the set of all integer vectors in R n , and 2 £ | X | < ¥. We will consider the problem Q ( F , X ) as searching for elements of one of the following sets: a set P ( F , X ) of Pareto optimal (efficient) solutions, a set Sl ( F , X ) of Slater optimal (weakly efficient) solutions, and a set Sm ( F , X ) of Smale optimal (strongly efficient) solutions. If we search for elements of a specific set (one of the indicated ones), we will use the notation QP ( F , X ) , QSl ( F , X ) , or QSm ( F , X ) , respectively. According to [8, 9], P ( F , X ) = {x Î X | p ( x, F , X ) = Æ}, Sl ( F , X ) = {x Î X | s ( x, F , X ) = Æ}, Sm ( F , X ) = {x Î X | h ( x, F , X ) = Æ}, where p ( x, F , X ) = { y Î X | F ( y ) ³ F ( x ), F ( y ) ¹ F ( x )}, s ( x, F , X ) = { y Î X | F ( y ) > F ( x )}, and h( x, F , X ) = { y Î X | y ¹ x, F ( y ) ³ F ( x )}. It is clear that Sm( F , X ) Ì P( F , X ) Ì Sl( F , X ) and " x Î X : s( x, F , X ) Ì p ( x, F , X )
Data Loading...
