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A sensitivity measure of the Pareto set in a

vector

l

-extreme combinatorial problem

V.A. Emelichev, A.M. Leonovich

Abstract

We consider a vector minimization problem on system of sub-sets 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 subsetsT ⊆2E\∅, |T |>1 of the setE ={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 ∈Rnm.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,

whereNn={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

matrixA, which correspond to the trajectoryt.

We define the Pareto set (the set of efficient trajectories) by tradi-tional way [12,13]:

Pn(A) ={t∈T :π(t, A) =∅},

c

(2)

where

π(t, A) ={t′∈T :q(t, t′, A)≥0(n), q(t, t′, A)6= 0(n)},

q(t, t′, A) = (q1, q2, ..., qn),

qi =qi(t, t′, Ai) =fi(t, Ai)−fi(t′, Ai), i∈Nn, 0(n)= (0,0, ...,0)∈Rn.

It is natural to call the problem of finding the setPn(A) the vector

l∞-extreme trajectorial problem. If E and T are fixed, we denote the

problem byZn(A).Let us assign the norml∞ for any natural number k∈N in the spaceRk:

kzk= max{|zi |:i∈Nk}, z = (z1, z2, ..., zk)∈Rk.

Under the norm of a matrix we understand the norm of the vector, formed by all its elements. For any numberε >0,let us define the set of perturbing matrices

B(ε) ={B ∈Rnm:kB k< ε}.

By analogy with [1,2,9,14 – 19], we call the problem Zn(A) stable (on vector criterion), if

∃ε >0 ∀B ∈ B(ε) (Pn(A)⊇Pn(A+B)).

It is easy to see that the property of stability of the problemZn(A) is a discrete analogue of upper semicontinuity property by Hauzdorf in a pointA∈Rnm of the optimal mapping (see., for example, [15])

Pn:Rnm→2E.

This point-set (many-valued) mapping assign the Pareto set to any set of parameters (any matrix A).

As usual [1,2,16,20], we say that the value

ρn1(A) = (

sup Ω1(A), if Ω1(A)6=∅,

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where Ω1(A) = {ε > 0 : ∀B ∈ B(ε) (Pn(A) ⊇ Pn(A+B))}, is the

stability radius of the problem Zn(A).

In other words, the stability radius is the limit level (in Chebyshev norm) of independent perturbations of matrixA elements, where new efficient trajectories do not appear.

It is natural to say that ρn1(A) = ∞ in the case Ω1(A) = R+.

Evidently, the problemZn(A) is stable and its stability radius is infinite when equalityPn(A) =T holds.

The problemZn(A),is called nontrivial, ifPn(A) =T\Pn(A)6=∅.

As in [2,3,5,6,16,17,8,21], we call the problemZn(A) quasistable, if the formula

∃ε >0∀B ∈ B(ε) (Pn(A)⊆Pn(A+B)) is valid.

Note, that the property of quasistability of the problem Zn(A) is a discrete analogue of lower semicontinuity property (by Hauzdorf) of the many-valued mapping, that assign the Pareto set Pn(A) to any matrixA∈Rnm.

The value

ρn2(A) = (

sup Ω2(A), if Ω2(A)6=∅,

0 otherwise,

where Ω2(A) ={ε >0 : ∀B∈ B(ε) (P(A)⊆P(A+B))},is called the

quasistability radius of the problemZn(A), n≥1.

By that, the quasistability radius defines the limit of independent perturbations, that retain all the efficient trajectories of initial problem and allow the appearance of new trajectories.

Let t={ej1, ej2, ..., ejs} ∈T, s=|t|, j1 < j2 < ... < js.If we put

t[Ai] = (aij1, aij2, ..., aijs)

for any indexi∈Nn,then we obtain

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Consequently, the next inequalities are correct:

fi(t, Ai)−fi(t, Bi)≤fi(t, Ai+Bi)≤fi(t, Ai) +fi(t, Bi). (2) Really, the right inequality is the axiom (triangle inequality)

ka+bk≤kak+kbk, a, b∈Rn,

true for any norm. Settinga=a′−b′, b=b′,we get

ka′ k − kb′ k≤ka′−b′ k, a′, b′ ∈Rn,

which proves the left inequality of (2).

Lemma 1 If trajectoriestandt′ and a matrix B ∈Rnm are such that the inequality

qi(t, t′, Ai)>2kB k (3)

holds for some indexi∈Nn,then the inequality

qi(t, t′, Ai+Bi)>0

is true.

Really, taking into account (1) and (2) combining them with (3), we easily obtain

qi(t, t′, Ai+Bi) =fi(t, Ai+Bi)−f(t′, Ai+Bi)≥

≥fi(t, Ai)−f(t, Bi)−(fi(t′, Ai) +fi(t′, Bi))≥

≥qi(t, t′, Ai)−2kB´ik≥qi(t, t′, Ai)−2kB k>0.

2.

Stability

Denote

ϕn1(A) = 1

2t∈minPn(A)

max t′T\{t}iminNn)

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Theorem 1 For the stability radiusρn

1(A)of vector nontriviall∞

-extreme problem Zn(A), n≥1,the estimates

ϕn1(A)≤ρn1(A)≤ 1

2 kAk (4)

are valid.

Proof. It is clear thatϕ:=ϕn1(A)≥0.We prove the left inequality of (4) at first. If ϕ= 0,then inequalityρn1(A)≥ϕis evident.

Letϕ >0, B∈ B(ϕ).Then, by definition of the numberϕ, for any trajectory t∈Pn(A) (the existence of such a trajectory is guaranteed by nontriviality of our problem), there exists a trajectoryt 6=t′,such that the inequalities

qi(t, t′, Ai)≥2ϕ >2kB k are valid for any indexi∈Nn.

Therefore using the lemma we obtain, that inequality

qi(t, t′, Ai+Bi)>0

holds for any index i ∈ Nn, i.e. t′ ∈ π(t, A+B). Consequently t ∈

Pn(A+B).Thus we have

∀B ∈ B(ϕ) (Pn(A)⊇Pn(A+B)).

Hence, ρn1(A)≥ϕn1(A).

Now we prove that the right inequality of (4) is valid. If we take the matrixB∗= [bij]n×m with elements

bij = ( 1

2 kAk −aij, if aij ≥0,

−12 kAk −aij, if aij <0, as perturbing, then it is easy to see, that

∀i∈Nn ∀t∈T (fi(t, Ai+Bi∗) = 1

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Consequently, taking into account the nontriviality of our problem, we obtain

Pn(A+B∗) =T 6⊆Pn(A).

This implies that the upper estimate

ρn1(A)≤ 1

2 kAk is valid.

Theorem 1 has been proved.

Corollary 1 Stability radius of any nontrivial problemZn(A), n≥

1, is finite.

The following two examples show, that lower and upper bounds of stability radius, stated by theorem 1, are accessible.

Example 1. Letn=m= 2,

A=

"

1 −2

−2 −3 #

, T ={t1, t2}, t1={e1}, t2 ={e2}.

Then f(t1, A) = (1,2), f(t2, A) = (2,3), P2(A) = {t1}, ϕ21(A) = 12,

kAk= 3.If 12 < β < ε, thenP2(A+B∗) ={t2},where

B∗ = "

β β

−β β

#

.

Therefore we have

∀ε > 1

2 ∃B

∈ B(ε) (P2(A+B)6⊆P2(A)).

Consequentlyρ21(A)≤ 12.

Hence, taking into account theorem 1 we getρ21(A) =ϕ21(A) = 12.

Example 2. Letn=m= 2,

A=

"

2 0 0

0 −2 0 #

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Then we obtain f(t1, A) = (2,2), f(t2, A) = (0,2), P2(A) = {t2}, ϕ21(A) = 0,kAk= 2.

It is easy to see that

q1(t1, t2, A1+B1)>0, q2(t1, t2, A2+B2) = 0

for any matrix B ∈ B(1), i.e. ρ21(A) ≥1. Taking into account (4), we have ρ21(A) = 1 = 12 kAk.

Let us also show, that the value of stability radius can differ from the upper and lower bounds from (4).

Example 3. Letn=m= 2,

A=

"

−2 0 1 0 −2 0

#

, T ={t1, t2}, t1 ={e1, e2}, t2={e2, e3}.

In this case,f(t1, A) = (2,2), f(t2, A) = (1,2), P2(A) ={t2}, ϕ21(A) = 0, k A k= 2 and, by theorem 1, we get 0 ≤ ρ21(A) ≤ 1. Let us show, that the stability radius of the problem is equal to 12.

For any ε > 12, there exists a matrix B∗ ∈ B(ε), such that t2 6∈

P2(A+B∗).For example

B∗ = "

β 0 β

0 0 0 #

,

where 12 < β < ε. Hence,ρ21(A)≤ 12.

On the other hand, for any matrixB ∈ B(12) we have

q1(t1, t2, A1+B1)>0, q2(t1, t2, A2, B2) = 0.

Hence,{t2} ⊆P2(A+B) ∀B ∈ B(12),i.e. ρ21(A)≥ 12.

Thus,ρ21(A) = 12.

Let us assign for our problem Zn(A) the traditional Slater set (the set of weakly efficient trajectories) [12,13]:

S1n(A) ={t∈T :σ1(t, A)6=∅},

where

σ1(t, A) ={t′ ∈T : qi(t, t′, Ai)>0, i∈Nn}. It is obvious, that Pn(A)⊆S1n(A).

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Corollary 2 If Pn(A) = Sn

1(A), then nontrivial problem Zn(A),

n≥1, is stable.

Let us illustrate that the equalityPn(A) =Sn(A) is not a necessary condition for stability of a nontrivial problemZn(A).

Example 4. Let n = 2, m = 3, T = {t1, t2, t3}, t1 = {e1}, t2 =

{e2}, t3 ={e3},

A=

"

−1 2 −3

−2 2 4 #

.

Then f(t1, A) = (1,2), f(t2, A) = (2,2), f(t3, A) = (3,4), P2(A) =

{t1}, S12(A) = {t1, t2}, i.e. P2(A) 6= S12(A). But nontrivial problem

Z2(A) is stable, because ρ21(A)≥ϕ21(A) = 12.

3.

Quasistability

Suppose

ϕn2(A) = 1 2t′minPn

(A)t∈minT\{t′}maxi∈Nn

qi(t, t′, Ai). It is easy to see, that ϕn2(A)≥0 for any matrix A∈Rnm.

Theorem 2 For the quasistability radius ρn2(A) of vector l∞

-extreme problem Zn(A), n≥1,the estimate

ρn2(A)≥ϕn2(A) (5)

is valid.

Proof. Let ϕ := ϕn2(A) > 0 (inequality (5) is trivial in the other case). Then, by definition of the number ϕ, for any trajec-tory t′ ∈ Pn(A) and any trajectory t 6= t′ there exists a number

p∈arg max{qi(t, t′, Ai) :i∈Nn},such that

∀B∈ B(ϕ) (qp(t, t′, Ap)≥2ϕ >2kB k). Thus, by the lemma, we get

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Thereforet′∈P(A+B) for any matrixB ∈ B(ϕ).Thereby we have

∀B ∈ B(ϕ) (Pn(A)⊆Pn(A+B)).

Hence, ρn2(A)≥ϕn2(A).

Theorem 2 has been proved.

Next example shows that the lower bound of quasistability radius, stated by theorem 2, is accessible.

Example 5. Letn=m= 2, T ={t1, t2}, t1={e1}, t2={e2},

A=

"

−1 2 3 −4

#

.

Then we easily obtain

f(t1, A) = (1,3), f(t2, A) = (2,4), P2(A) ={t1}, ϕ22(A) =

1 2. If the perturbing matrix is of the kind

B∗ = "

−β −β

β β

#

,

where 12 < β < ε, then P2(A+B∗) = {t2}, i.e. P2(A) 6⊆P2(A+B∗)

for some perturbing matrix B∗ ∈ B(ε).Hence,ρ22(A) =ϕ22(A) = 12.

Let us show that the quasistability radius can be greater then lower bound, stated by theorem 2.

Example 6. Letn= 2, m= 3,

A=

"

0 1 −3 0 −1 3

#

,

T = {t1, t2, t3}, t1 = {e1, e2}, t2 = {e2}, t3 = {e3}. In this case we have k A k= 3, f(t1, A) = f(t2, A) = (1,1), f(t3, A) = (3,3),

P2(A) ={t1, t2}.It is easy to see that, for any matrix B ∈ B(12),

qi(t1, t2, Ai+Bi) = 0, qi(t1, t3, Ai+Bi)<0, qi(t2, t3, Ai+Bi)<0,

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Hence, {t1, t2} ⊆ P2(A+B).Thus the problem Z2(A) is quasistable, i.e. ρ22(A) > 0. On the other hand q(t1, t2, A) = (0,0), and therefore

ϕ22(A) = 0.

For the problem Zn(A) we assign also the Smale set (or the set of strictly efficient trajectories) [12,13]:

S2n(A) ={t∈T :σ2(t, A) =∅},

where

σ2(t, A) ={t′ ∈T :f(t, A)≥f(t, A)}.

It is clear, that S2n(A) ⊆Pn(A), and the Pareto set is always not empty, but the Smale set can be empty.

From theorem 2 we obtain

Corollary 3 If equality Sn

2(A) = Pn(A) holds, then the problem

Zn(A), n≥1,is stable.

Let us show, that the inequality Pn(A) =S2n(A) is not a necessary condition of quasistability of the problem Zn(A).

Example 7. Let n= 2, m= 4, T ={t1, t2, t3, t4}, ti ={ei}, i∈

N4,

A=

"

−1 −2 2 2

−2 −1 −1 3 #

.

Then f(t1, A) = (1,2), f(t2, A) = (2,1), f(t3, A) = (2,1), f(t4, A) =

(2,3), P2(A) = {t1, t2, t3}, S22(A) = {t1}, i.e. P2(A) 6= S22(A). But,

ρ22(A)≥ϕ22(A) = 12 >0,i.e. the problemZ2(A) is quasistable.

Remark 1. It is easy to see, that all essentials, illustrated by examples 1–7, can be shown for any number of criteria (n >2).

Theorem 3 For stability and quasistability radii of the vectorl∞

-extreme problem Zn(A), n≥1,the next estimate is true:

ρns(A)≥ 1

2imin∈Nn

min

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Proof. Let us consider the nontrivial case, i.e. where the right part of (6) is a positive number. Then, evidently, in any row of the matrix

A,absolute values of elements are different in pairs. Therefore, taking into account partial the criteria definition, for any trajectories t 6= t′

and indexi∈Nn we obtain

qi(t, t′, Ai) = 0 =⇒ ∀B ∈ B(ϕi) (qi(t, t′, Ai+Bi) = 0),

qi(t, t′, Ai)>0 =⇒ ∀B ∈ B(ϕi) (qi(t, t′, Ai+Bi)>0), where

ϕi = 1

21≤minj<k≤m||aij | − |aik||. Thereby, for any matrixB ∈ B(ϕi)

sign qi(t, t′, Ai+Bi) =const. Consequently, setting

ϕ= min{ϕi :i∈Nn}, by definition of the Pareto set we get

∀B ∈ B(ϕ) (Pn(A) =Pn(A+B)).

Hence, inequalities (6) are valid. Theorem 3 has been proved.

Examples 1 and 5 indicate that lower bound, stated by theorem 3, is accessible.

Theorem 3 implies

Corollary 4 If, in any row of a matrixA, absolute values of

ele-ments are different in pairs, then the problem Zn(A), n ≥1,is stable

and quasistable.

Remark 2. Evidently, that relations (1) and (2) are true for any norm lk, k ∈ N (not only for l∞). Therefore it is easy to see, that

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with any index k ∈ N, i.e. for any n−criteria problem with partial criteria

kt[Ai]kk→min

t∈T, i∈Nn.

The norms must be the same, i.e. in the space of matrices Rnm it is also needed to use the normlk.In partial case, wherek= 1,the partial criteria take the form

X

j∈N(t)

|aij |→min

t∈T, i∈Nn,

and in the space of perturbing parameters Rnm we have to assign the norm l1:

kAk1=

n X

i=1

m X

j=1

|aij |.

References

[1] Bakurova A.V., Emelichev V.A., Perepelitsa V.A.On the stability

of multicriteria problems on sistem of subsets, Dokl. Akad. Nauk

Belarusi, 1995, vol. 39, No. 2, pp. 33–35 (Russian)

[2] Emelichev V.A., Kravtsov M.K.On stability in traectorial problem

of vector optimization, Kibernetika i Sistemny Analis, 1995, No.

4, pp. 137–143. (Russian)

[3] Emelichev V.A., Kravtsov M.K., Podkopaev D.P.On

quasistabil-ity radius of multicriteria traectorial problem, Dokl. Akad. Nauk

Belarusi, 1996, vol. 40, No. 1, pp. 9–12. (Russian)

[4] Emelichev V.A., Girlich E., Podkopaev D.P. Several types of

sta-bility of efficient solutions of vector traectorial discrete problem,

I, II, Izv. Akad. Nauk Resp. Moldova, Matem., 1996, No. 3, pp. 5-16; 1997, No. 2, pp. 9–25. (Russian)

[5] Emalichev V.A., Podkopaev D.P.On stability conditions of Pareto

set of vector problem on a sistem of subsets, Vestnik Belorus.

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[6] Emelichev V.A., Kravtsov M.K., Podkopaev D.P.On the stability

of trajectory problems of vector optimization, Mathematical Notes,

1998, vol. 63, No. 1, pp. 19–24.

[7] Avsyukevich O.V., Emelichev V.A., Podkopaev D.P. On stability

radius or multicriteria traectorial botleneck problem, Izv. National

Akad. Nauk Belarusi, Ser. phis.-mat. nauk, 1998, No. 1, pp. 112– 116. (Russian)

[8] Emelichev V.A., Krichko V.N. On stability of Smale optimal, Pareto optimal and Slater optimal solutions in vector traectorial

optamization prblem, Izv. Akad. Nauk Resp. Moldova, Matem.,

1998, No. 3, pp. 81–86. (Russian)

[9] Emelichev V.A., Podkopaev D.P. Condition of stability, pseudo-stability and quasi-pseudo-stability of the Pareto set in a vector

traec-torial problem, Revue d’analyse numerique et de theorie de

l’approximation, 1998, vol. 27, No. 1, pp. 91–97.

[10] Emalichev V.A., Nikulin Yu.V.Conditions of strong quasistability

of linear traectorial problem, Izv. Akad. Nauk Belarusi, Ser.

phis.-mat. nauk, 2000, No. 2, pp. 38-41. (Russian)

[11] Emelichev V.A., Nikulin Yu.V.On strong stability radius of vector

traectorial problem, Vestnik Belorus. Universiteta, Ser. 1, 2000,

No. 1, pp. 47–50. (Russian)

[12] Podinovsky V.V., Nogin V.D. Pareto optimal solutions in

multi-criteria problems, Moskow: Nauka, 1982, p. 256. (Russian)

[13] Dubov Yu.A., Travkin S.I., Yakimets V.N.Multicriteria models of

forming and choosing variants of sistem, Moskow: Nauka, 1986,

p. 296. (Russian)

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[15] Sergienco T.I., Kozeratskaya L.N., Lebedeva T.T.Investigation of

stability and parametric analysis of discrete optimization problem,

Kiev, Naukova dumka, 1995, p.170. (Russian)

[16] Emelichev V.A., Podkopaev D.P. On a quantitative measure of

stability for a vector problem in integer programming, Comput.

Math. and Math. Phys., 1998, vol. 38, No. 11, pp. 1727–1731. [17] Berdysheva R., Emelichev V., Girlich E. Stability, pseudostability

and quasistability of vector traectorial lexicographic optimization

problem, Comp. Science Journal of Moldova, 1998, vol. 6, No. 1,

pp. 33–56.

[18] Emelichev V.A., Nikulin Yu.V. On two tipes of stability of

vec-tor linear quadric boolean programming problem, Diskrete analis i

issled. operaciy, Ser. 2, 2000, vol. 6, No. 2, pp. 23–31. (Russian) [19] Emelichev V.A., Yanushkevich O.A.Stability and regularization of

vector lexicographic problem of diskrete programming, Kibernetika

i Sistemny Analis, 2000, No. 2, pp. 54–62. (Russian)

[20] Leontiev V.K. Stability in linear diskrete problem, Probl. Cyber-net., 1979, vol. 35, pp. 169–184. (Russian)

[21] Sotskov Yu.N., Tanaev V.S., Werner F. Stability radius of an

op-timal schedule: a survey and recent developments, Industrial

Ap-plications of Discrete Optimization, Kluwer, 1998, vol. 16, pp. 72–108.

[22] Emelichev V.A., Nikulin Yu.V. On the stability of efficient

solu-tion in a vector quadratic boolean programming problem, Buletinul

Acad. de st. a Republicii Moldova, Matematica, 2000, No. 1, pp. 33–40.

V.A. Emelichev, A.M. Leonovich, Received May 15, 2001 Belorussian State University

ave. F.Skoriny, 4, Minsk 220050 Belarus

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