ABOUT ONE DISCRETE ANALOG OF HAUSDORFF SEMI-CONTINUITY OF SUITABLE MAPPING IN A VECTOR COMBINATORIAL PROBLEM WITH A PARAMETRIC PRINCIPLE OF OPTIMALITY (“FROM SLATER TO LEXICOGRAPHIC”)

. A multicriteria linear combinatorial problem is considered, principle of optimality of which is deﬁned by a partitioning of partial criteria onto groups with Slater preference relation within each group and the lexicographic preference relation between them. Quasistability of the problem is investigated. This type of stability is a discrete analog of Hausdorﬀ lower semicontinuity of the many-valued mapping that deﬁnes the choice function. A formula of quasistabil-ity radius is derived for the case of metric l ∞ . Some conditions of quasistability are stated as corollaries.


INTRODUCTION
Usually stability of an optimization problem is understood as continuous dependence of solution set on parameters of the problem.The most general approaches to stability analysis of optimization problems are based on properties of many-valued mappings that define optimality principles [1].
Mathematical analysis does not present methods sufficient to investigate stability of a discrete optimization problem.It is greatly due to complexity of discrete models, which can behave unpredictably under small variations of initial data.At the same time, if terminology of general topology is not used, then the formulation of a question of stability can be significantly simplified in the case of a space of acnodes.There are different types of stability of discrete optimization problems (see e.g.[2][3][4][5][6]).Stability of a discrete problem in the broad sense means that there exists a neighborhood of the initial data in the space of parameters such that any problem with parameters from this neighborhood possesses some invariance with respect to the initial problem.In particular, upper (lower) semicontinuity of an optimal mapping is equivalent to nonappearance of new (preserving of initial) optimal solutions under "small" perturbations of the mapping parameters.So concepts of stability [2][3][4][5][6][7] and quasistability [5,[8][9][10][11][12] of discrete optimization problems arise.
In this article under parametrization of optimality principle (from Slater to lexicographic) we obtain a generalized formula of quasistability radius of a multicriteria linear combinatorial problem with partitioning of criteria into groups according to their importance.Slater preference relation is defined within each group and the lexicographic preference relation -between them.We consider the case, where metric l ∞ is defined in the space of problem parameters.

DEFINITIONS AND PROPERTIES
We consider typical vector (n-criteria) combinatorial problem.Let a vector criterion A specific peculiarity of vector optimization is the existence of many principles of optimality (or functions of choice) which could be defined with binary relations of preference.
Let us define three binary relations in the space R p of arbitrary dimension p ∈ N according to formulas where k=min{i ∈ N p : y i = y i }, y=(y 1 , y 2 , . . ., y p ), y =(y 1 , y 2 , . . ., y p ).
These relations create the following well-known objects of vector optimization: Slater set ( weakly efficient trajectories ) Smale set ( strictly efficient trajectories ) and set of lexicographically optimal trajectories Here is the negation of the relation .Obviously that Sm n (A) ⊆ Sl n (A).where I r = ∅, r ∈ N s ; p = q ⇒ I p ∩ I q = ∅.For any such partitioning we define the binary relation Ω n I of strict preference in the space R n between different vectors y = (y 1 , y 2 , . . ., y n ) and y = (y 1 , y 2 , . . ., y n ) as follows:

Due to transitivity of relation
where k = min{i ∈ N s : y I i = y I i }; y I k and y I k are the projections of the vectors y and y correspondingly onto the coordinate axes of the space R n with numbers from the subset I k .
The introduced binary relation Ω n I determines ordering of the shaped subsets of criteria such that any previous subset is significantly more important that any consequent subset.This relation generates the set of I-optimal trajectories , is Slater set Sl n (A), and T n (A, I L ), where I L = {{1}, {2}, . . ., {n}} (s = n), is the set of lexicographically optimal trajectories.So under the parametrization of optimality principle we understand assigning a characteristic to binary relation which in special cases induces well-known Slater and lexicographic optimality principles.
It is easy to show that the binary relation Ω n I is antireflexive, asymmetric, transitive, and hence it is a cyclic.And since the set T is finite, the set T n (A, I) is non-empty for any matrix A and any partitioning I of the set N n .
Hereinafter by Z n (A, I) we denote the problem of finding the set T n (A, I).
Clearly, T 1 (A, {1}) is the set of optimal trajectories of the scalar linear trajectories problem Z 1 (A, {1}), where A ∈ R m .Many extreme combinatorial problems on graphs, boolean programming and scheduling problems and others are reduced to Z 1 (A, {1}) (see e.g.[4,6,7]). Denote The following properties follow directly from the above definitions.
Assume the converse, i.e. t ∈ Sm(A) and t ∈ T n (A, I).Then according to property 2.4 there exists a trajectory t = t, such that f (t, A) Ω n I f (t , A). Hence due to property 2.3 we have Taking into account the inclusion t ∈ Sl 1 (A), we obtain i. e. t ∈ Sm 1 (A), which contradicts to the assumption.
Directly from definition of set Sm 1 we obtain Let us consider such variant of the problem Z n (A, I) stability, which is the discrete analog of Hausdorff lower semicontinuity in the point A of the many-valued mapping R n×m → 2 T , that puts set T n (A, I) to any collection of parameters of the problem from the metric space R n×m with metric l ∞ .For a vector problem Z n (A, I), lower semicontinuity means that set T n (A, I) can only expand under "small" perturbations of elements of matrix A.
Definition 2.8.Vector problem Z n (A, I), n ≥ 1, is called quasistabile ( under perturbations of elements of matrix A ), if there exists ε > 0, such that for any perturbing matrix A ∈ Ψ(ε) the inclusion Definition 2.9.Under the quasistability radius of the vector problem Z n (A, I), n ≥ 1, we understand the number

LEMMAS
For any different trajectories t any t we define the numbers: Obviously, that ∆(t, t ) > 0.
Lemma 3.1.If d n (t, t , A) ≥ ϕ > 0, then the following relation holds for any perturbing matrix A ∈ Ψ(ϕ) : Further suppose that the assertion of the lemma is false, i.e. that there exists a matrix

Then by virtue of property 2.3 and linearity of the functions f
i.e. for any index i ∈ I 1 the inequality holds, which contradicts to (1).
Lemma 3.2.Let t ∈ T n (A, I), t ∈ T \ {t}.For any number α > d n (t, t , A) there exists a matrix Then ||A * || = α and equitably equalities  Proof.Before proving of the theorem, let us note that from the non-emptiness of T n (A, I) and T \ {t} and from the definition of d n (t, t , A), the right side of formula (3), further define it by ϕ, is positive and correctly defined.
Without loss of generality assume that ϕ > 0 (otherwise inequality ( 4) is obvious).From the definition of the number ϕ, it follows that for any trajectories t ∈ T n (A, I) and t = t the inequalities Therefore by virtue of property 2.5 we have t ∈ T n (A + A , I).Thus we conclude ∀ A ∈ Ψ(ϕ) (T n (A, I) ⊆ T n (A + A , I)), this formula proves (4).It remains to show that (5) ρ n (A, I) ≤ ϕ.
Let ε > α > ϕ and trajectories t ∈ T n (A, I) and t = t be such that d n (t, t , A) = ϕ.Then according to lemma 3.2 there exists a matrix A * with norm ||A * || = α such that (2) holds, i.e. t ∈ T n (A + A * , I).Hence we have which proves inequality (5), said above and taking into (4) we obtain (3).

COROLLARIES
Series of corollaries follows directly from theorem.
Corollary 5.1.The quasistability radius of the problem Z n (A, I Sl ), n ≥ 1, of finding Slater set Sl n (A) is expressed by the formula The formula of quasistability radius, led in corollary 5.1, easily pass into the formula of quasistability radius of the scalar trajectorial problem with linear criterion [4] Corollary 5.2 ( [22] ).The quasistability radius of the problem Z n (A, I L ), n ≥ 1, of finding the set of lexicographically optimal trajectories L n (A) is expressed by the formula ∆(t, t ) .
Corollary 5.3.For any partitioning I of the set N n , n ≥ 1, into s subsets, s ∈ N n , the following statements are equivalent for a problem Z n (A, I), n ≥ 1: ( Proof.Equivalence of statements (i) and (ii) follows directly from the theorem.
The implication (ii) ⇒ (iii) is proved by contradiction.Suppose that (ii) holds but (iii) does not.
Then (since T n (A, I) = Sm 1 (A) is assumed) there exists a trajectory t ∈ T n (A, I) ⊆ Sl 1 (A), such that t ∈ Sm 1 (A).It follows that there exists a trajectory t ∈ Sl 1 (A) such that t = t, f I 1 (t, A) = f I 1 (t , A), which contradicts to statement (ii).The implication (iii) ⇒ (i) is obvious by virtue of property 2.7.
From corollary 5.3, we easily get the following attendant results.It is easy to understand that for a scalar linear trajectorial problem the coincidence of Slater and Smale sets is equivalent to existence of a unique optimal solution.Therefore partial case of corollary 5.4 is Corollary 5.5 ( [4] ).Singlecriterion (scalar) linear trajectorial problem is quasistable if and only if it has a unique optimal solution.Corollary 5.6 ( [22] ).The problem Z n (A, I L ), n ≥ 1, of finding the set L n (A) of lexicographically optimal trajectories is quasistable if and only if Sl and finiteness of set T, set Sl n (A) = ∅ for any matrix A ∈ R n×m .But Smale set can be empty, because binary relation Sm is not transitivity.Let s ∈ N n , I = {I 1 , I 2 , . . ., I s } be a partition of the set N n into s nonintersecting nonempty subsets, i.e.N n = r∈Ns I r , this implies (2) by virtue of property 2.2.

Theorem 4 . 1 .
For any partitioning I = {{I 1 }, {I 2 }, . . ., {I s }} of the set N n , n ≥ 1, into s subsets, s ∈ N n , the quasistability radius ρ n (A, I) of a problem Z n (A, I) is expressed by the formula (3) ρ n (A, I) = min t∈T n (A,I) min t ∈T \{t} d n (t, t , A).

Corollary 5 . 4 .
The problem Z n (A, I Sl ), n ≥ 1, of finding Slater set Sl n (A) is quasistable if and only if Sl n (A) and Sm n (A) are coinciding.
Property 2.5.A trajectory t ∈ T n (A, I) if and only if for any trajectory t the relation f (t, A) A). Property 2.4.A trajectory t ∈ T n (A, I) if and only if there exists a trajectory t = t, such that f (t, A) Ω n I f (t , A).