On some type of stability for multicriteria integer linear programming problem of finding extremum solutions Текст научной статьи по специальности «Математика»

udk: 519.854

msc2010: 90c09, 90c27, 90c29, 90c31

ON SOME TYPE OF STABILITY FOR MULTICRITERIA INTEGER LINEAR PROGRAMMING PROBLEM OF FINDING EXTREMUM

SOLUTIONS © Vladimir A. Emelichev*, Yury V. Nikulin**

* Belarusian State University NEZAVISIMOSTI 4, Minsk, 220030, Belarus e-mail: vemelichev@gmail.com ** University of Turku Vesilinnantie 5, Turku, 20014, Finland e-mail: yurnik@utu.fi

Abstract. We consider a wide class of linear optimization problems with integer variables. In this paper, the lower and upper attainable bounds on the T2-stability radius of the set of extremum solutions are obtained in the situation where solution space and criterion space are endowed with various Holder's norms. As corollaries, the Testability criterion is formulated, and, furthermore, the Testability radius formula is specified for the case where criterion space is endowed with Chebyshev's norm.

Keywords: multicriteria integer linear programming, set of extremum solutions, stability radius, T2-stability, Holder's norm, Chebyshev's norm.

Introduction

Multicriteria decision making models of discrete optimization are widely used in economics, management, and many other fields of applied mathematics. One of the major troubles in such models is related to inaccuracy of input date due to many factors such as modeling and/or prediction errors, imprecise calculation, and etc. Sometimes an algebraic formulation may also influence the issue [1]. The accumulated experience of dealing with these aspects clearly shows that none of the real life problems can be properly formulated and correctly solved without deep analyzing its stability.

Any complex decision making problem involves multicriteria choice of a subset of the best alternatives among all the feasible solutions satisfying some chosen optimality principle. Post-optimal and parametric analysis of discrete problems target finding an answer to the question how optimal solutions react to small perturbations of input parameters (problem initial data).

There exist two major approaches, qualitative and quantitative, in post-optimal analysis of discrete optimization problems. Different types of stability are typically

analyzed, where stability can be defined as a certain feature of preserving some invariance properties on the set of optimal solutions. In a family of qualitative approaches, we are seeking for necessary and/or sufficient conditions of stability as well as specifying an area of stability, usually termed as stability region [2-8].

A family of quantitative approaches deals with construction and calculation of numerical characteristics (quantifying measures) of stability. The stability radius is a key concept used inhere. It is defined as a radius of extreme stability ball, i. e. a radius of the largest neighborhood inside the initial problem parameter metric space such that any "perturbed"problem with parameters in it has a certain invariance properties on the set of optimal solutions. The coefficients of a scalar or vector criterion (objective function) are usually subject for perturbations. Sometime we can also have uncertainty related to the feasibility constraints, then the coefficients of the constraints are considered as a subject for perturbation. The main research goal is to find analytical expressions or bounds of the stability radius [9-19] as well as algorithms of their calculation for optimization problems [20-22].

This paper belongs to the family of quantitative approaches. It continues a series of publications [11, 23-26] seeking for analytical bounds on stability radius (different types of stability) for multicriteria problem of Integer Linear Programming (ILP) with Pareto optimality principle.

In multicriteria decision making, we may also deal with some choice functions that are different from the well-known Pareto optimality principle. Such functions have a specific merit in many real life applications (see e.g. [27-30]). In this paper, we consider multicriteria problem of ILP with extremum optimality principle, i. e. with the set of all extremum solutions. We study the so-called T2-type of stability [4] that can easily be interpreted in terms of stability kernel existence, i. e. existence of a subset of extremum solutions such that the solutions are stable with respect to small perturbations of initial problem parameters. Thus, the Testability radius is defined as a supreme level of problem parameter perturbations preserving at least one solution within the stability kernel. In this paper, we specify the lower and upper bounds on the Testability radius of multicriteria ILP problem with extremum solutions for the case where criterion space is endowed with various Holder's norms. Attainability of the estimates (both lower and uppers bounds) is shown. As a corollary, we deduce a known before criterion on Testability of multicriteria ILP problem for the case where criterion space is endowed with Chebyshev's norm.

Introduction

We consider an m-criteria problem of ILP problem in the following formulation. Let C = [cij] E Rmxn be a real valued m x n - matrix with corresponding rows Ci E Rn, i E Nm = {1,2,..., m}, m > 1. Let also X C Zn, 1 < |X| < w, be a set of feasible solutions (integer vectors) x = (x\,x2,... ,xn)T, n > 2. We define a vector criterion

Cx = (C1x, C2x,..., Cmx) —> min,

v ' xex

with partial criteria being linear functions.

In this paper, Zm(C), C G Rmxn, is a problem of finding the set of extremum solutions defined in traditional way (see e.g. [27-29]):

Em(C) = {x G X : 3k G Nm Vx' G X (Cfc(x) < Ck(x'))}.

Thus, the choice of extremum solutions can be interpreted as finding best solutions for each of m criteria, and then combining them into one set. In other words, the set of extremum solutions contains all the individual minimizers of each objective. Obviously, E 1(C), C G Rn is the set of optimal solutions for scalar problem Z 1(C) with C G Rmxn. Taking into account that X is finite, the following formulae below are true for any

C g Rmxn:

Em(C) = Sm(C)\(Pm(C)\Lm(C)) = Lm(C) U (Sm(C)\Pm(C)),

Em(C) H Pm(C) = Lm(C) = 0, Lm(C) Ç Pm(C) Ç Sm(C),

Lm(C) Ç Em(C) Ç Sm(C),

where Pm(C) denotes the Pareto set [31], Sm(C) denotes the Slater set [32], and Lm(C) denotes the lexicographic set (see e.g. [33, 34]).

Below we define all the three sets in a traditional way:

Pm(C) = {x G X : $ x0 G X (Cx > Cx0 & Cx = Cx0)},

Sm(C) = {x G X : $ x0 G X Vk G Nm (Ck(x) > Ck(x°))},

Lm(C) = U L(C, s), L(C, s) = {x G X : Vx' G X (Cx <s Cx'^. «enm

Here nm is the set of all m! permutations of numbers 1,2, ...,m; s = (s1, s2,..., sm) G nm; and the binary relation of lexicographic order between

«TaepunecKuu eecmuuK urnfiopMamuKU u MameMamuKU», № 2 (39)' 2018

a - ' 3eN-

two vectors y — (yl,y2,..., ym) E Rm and y' — (y' ,y'2,..., y'm) E Rm is defined as follows y <s y' ^ (y — y') V (3k E Nm Vi E Nk-i (ySk < y'sfc & ysi — y'si)),

where N0 — 0. Obviously all the sets, Pm(C), Sm(C), Lm(C) and Em(C), are non-empty for any matrix C E Rmxn due to the finite number of alternatives in X.

We will perturb the elements of matrix C E Rmxn by adding elements of the perturbing matrix C E Rmxn. Thus the perturbed problem Zm(C + C') of finding extremum solutions has the following form

(C + C')x ^ min .

xex

The set of extremum solutions of the perturbed problem is denoted by Em(C + C'). In the solution space Rn, we define an arbitrary Holder's norm lp, p E [1, oo], i. e. the norm of vector a — (al,a2,..., an)T E Rn is defined by the number

(E ai^l/p if i < p< o,

_ View /

'P — 1

max{|aj| : j E Nn} if p — o.

In the criterion space Rm, we define another Holder's norm lq, q E [1, o], The norm of matrix C E Rmxn is defined by the number

IIC llpq — IKHClHp, ||C2|P,..., \\Cm\\p)\\q,

It is well-known that lp norm, defined in Rn, induces conjugated lp* norm in (Rn)*. For p and p*, the following relations hold

1 + — — 1, 1 < p < o. p p*

In addition, if p — 1 then p* — o. Obviously, if p* — 1 then p — o. Also notice that p and p* belong to the same range [1, o]. We also set p — 0 if p — o.

It is easy to see that for any vector a — (al,a2, ...,an) E Rn with |aj| — a, j E Nn it holds

\a\p — np a (1)

for any p E [1, o]. For any two vectors a and b of the same dimension, the following Holder's inequalities are well-known

|aTb| < \\a|

pN^Np*

Using the well-known condition (see [35]) that transforms (2) into equality, the validity of the following statements becomes transparent

Vb e Rn Va> 0 За e Rn (|aTb| = a\\b\\p. & \\a\\p = a). (3)

Given e > 0, let

Qpq(e) = jC e Rmxn : \\C'\\pq < e]

be the set of perturbing matrices C with rows C'k e Rn, k e Nm, and \\C/\pq is the norm of C/ = [cj] e Rmxn. Denote

SPq = {e > 0: 3x e Em(C) VC/ e (e) (x e Em(C + C/))}. Following [4, 16], the number

(sup Epq if Epq = 0,

0 if Spq = 0

is called the T2-stability radius (strong quasistability radius in terminology [10, 24]) of problem Zm(C), m e N, with Holder's norms lp and lq in the spaces Rn and Rm respectively. Thus, the Testability radius of problem Zm(C) defines the extreme level of independent perturbations of the elements of matrix C in the metric space Rmxn such that there exists at least one of the extremum solutions of Zm(C) preserving a property of being an extremum solution for each perturbed problem Zm(C + C/).

The same concept of the Testability radius of Zm(C) can also be introduced using the definition of the stability kernel, known earlier in [4]. Indeed, it is easy to see that

pm(p, q) = sup{e > 0 : Krm(C, e) = 0},

where

Krm(C,e) = {x e Em(C) : VC/ e ^(e) (x e Em(C + CO)}. Here Krm(C,e) is a e-stability kernel of Zm(C), and

Krm(C) = {x e Em(C) : 3e > 0 VC/ e ^(e) (x e Em(C + C/))}

is a stability kernel of Zm(C). Thus, the problem Zm(C) is T2-stable (pm(p, q) > 0) if and only if the stability kernel is nonempty.

Bounds on stability radius

Given the multicriteria ILP problem Zm(C), m e N, for any p e [1, ro] we set

. ( ) • [Ci(x - x/)]+

0m(p) = max max mm —---,

x'eEm(c) ieNm x€X\(x'} \\x — x/\p*

lp* •

where [a] + — max{0,a} is a nonnegative projection of a E R. Obviously, 0m(p) > 0.

Theorem 1. Given p,q E [1, o] and m E N, for the Testability radius pm(p,q) of multicriteria ILP problem Zm(C), the following lower and upper bounds are valid

0m(p) < pm (p,q) < mq 0 (p).

Proof. First, we prove that pm(p, q) > 0 :— 0m(p). If 0 — 0, then it is evident. Let 0 > 0. Then according to the definition of 0, there exist a solution x0 E Em(C) and an index k E Nm such that for any solution x E X\{x0} the following inequality holds

[Ck(x - x0)] + > 0\\x — x0L*

Since 0 > 0, we have Ck(x — x0) > 0 for x — x0. Assuming C E Qpq(0), taking into account

HCk\p < \\C'\\pq <0

and Holder's inequalities (2), we deduce

(Ck + C'k)(x — x0) — [Ck(x — x0)]+ + C'k(x — x0) >

(0 —\\C'k\p)\x — x0\\p* > 0

for any x — x0 i. e. x E Em(C + C') for C E Qpq(0), and hence pm(p, q) > 0.

Further, we prove that pm(p,q) < mq0. According to the definition of number 0, for any solution x E Em(C) and any index i E Nm, there exists a solution x(i) E X\{x} such that

[Ci(x(i) — x)]+ < 0\x(i) — x\\p*. (4)

Setting a with a condition

£

— > a > 0, (5)

m q

according to formula (3) for any index i E Nm there exists C0 E Rn such that

C0(x(i) — x) — — a\\x(i) — x\\p*, (6)

HCX — a.

Therefore, due to (1), the norm of matrix C0 containing rows C0, i E Nm, is calculated as

WC0Wpq — m 1 a,

i. e. C E Qpq(e). Using sequentially (6), (4) and (5) we get for any index i E Nm we deduce (Ci + C°)(x(i) — x) — Ci(x(i) — x) + C0(x(i) — x) < [Ci(x(i) — x)]+ — a\\x(i) — x\\p* <

(0 — a) \ x(i) — x\ p* < 0.

Thus, x G Em(C + C0) for x G Em(C). Hence, the following formula is valid Ve > m10 Vx G Em(C) 3C0 G (e) (x G Em(C + C0)),

i. e.

pm(p,q) < m 10. □

The following corollary from Theorem 1 illustrates the attainability of the lower bound for the T2-stability radius.

Corollary 1. If q = to, then for any p G [1, to) and m G N for the Testability radius pm(p, to) of ILP problem Zm(C) the following formula holds

/ \ / / \ • [Ci(x — x0]+ pm(p, to) = 0m(p) = max max mm —---.

x'eem(c) iewm xex\{x'} ||x - x'||p*

The next Theorem illustrates the attainability of the upper bound for the Testability radius specified in Theorem 1.

Theorem 2. Given p = to, q G [1, to] and m G N, there exists a class of multicriteria ILP problem problems such that for any Zm(C) belonging to that class the T2-stability radius of Zm(C) can be expressed by the following formula

Pm(p,q) = m1 0to(to) . (7)

Proof. According to Theorem 1, in order to prove equation (7), it suffices to specify a class of problems Zm(C) with pTO(<ro,q) > mq 0to(to). Let X = {x^x2, ...,xn} C En = {0,1}, where n = m + 1, and every solution xj, j G Nn, be a unit vector, i.e. a column of identity matrix of size n x n. Let matrix C = [cj] G Rmxn with rows Cj G Rn, i G be constructed as follows

/0 M ... M -2a \

M 0 ... M -2a

C —

M M ... 0 -2a where M >> a>0, and M is a number large enough. Then we have

Cx1 = (0,M,...,M, M)T G Rm, Cx2 = (M, 0,..., M, M)T G Rm,

Cxn-1 = (M, M,..., M, 0)T e Rm, Cxn = (—2a, —2a,..., —2a, —2a)T e Rm,

Thus, xn E Em(C), xj E Em(C), j E Nm. Moreover, the following equality is evident

. . . Ci(x x ) ©m(oo) = max mm -= a.

ieNm jeNm 2

Let C' = [cj] E Q^q(m 1 a) be an arbitrary perturbing matrix with rows C[,C'2,...,C'm i.e. C E Rmxn, ||C'||œq < m 1 a. Proving by contradiction, it is easy to show that there exists an index k E Nm with ||Ck||œ < a. Therefore, lcCkj| < a for any j E Nn. So, we deduce

(Ck + C'k)(xk - xn) = 2a + ckk - ckn > 2a - ^| - |cknl > 0, and hence for any index i E Nm\{k} we obtain

(Ci + Ci)(xk - xn) = Ct(xk - xn) + C[{xk - xn) = M + 2a + 4 - c'm > 0.

As a result we conclude that xn E Em(C + C') for any perturbing matrix C E (m 1 a) the following inequality holds

pm(œ,q) > mq$m(<x),

and hence, taking into account Theorem 1, we get that equality (7) is true, i.e. Theorem 2 is now proven. □

The problem Zm(C) is called T2-stable if pm(p, q) > 0. We introduce a set of strict extremum solutions of Zm(C):

SEm(C) = {x E X : 3k E Nm V x' E X\{x} {Ck(x) < Ck(x'))}.

From Theorem 1 we get the following result.

Corollary 2. Given the ILP problem Zm(C), the following statements are equivalent

• The problem Zm(C) is T2-stable;

• Krm(C ) = SEm (C) = 0;

• &m(p) > 0.

Due to equivalence of any two norms in a finite dimensional linear space (see e.g. [36, 37]), the result of Corollary 2 is true for any norms specified in the parameter space Rmxn of the problem Zm(C).

At the end to compare the result of theorem 1, we present here a formula to calculate the Testability radius of multicriteria ILP problem Zm(C) consisting in finding the set of Pareto optimal solutions Pm(C):

P ( ) . \\[C (x — x' )]+\\q pm(p,q) — max m.n —n-n-,

x'ePm(o) xex\{x'} \x — x'Np*

where [a]+ = (a+, a+,..., am)T is а nonnegative projection of vector a = (ab a2,..., am)T e Rm, i.e. a+ = max{0,aj}, i e Nm. This formula is clearly follows from the results of [25].

Conclusion

Since the initial data of real problems are usually given with a certain degree of uncertainty (inaccuracy), there is a need to study the stability of the optimal solutions to perturbations of the problem parameters. It is worth mentioning that such investigations are done not only in optimization theory but also in other areas of operations research and applied mathematics, in particular scheduling theory (see e.g. [38, 39]).

The quantitative characteristic of Testability of the multicriteria integer linear programming problem, consisting in the search for all extremum solutions, is investigated. These studies were carried out on the assumption that different Holder's norms are given in the space of solutions and criteria. The following results are obtained: 1) the lower and upper bounds of the Testability radius are found; 2) the attainability of these estimates is indicated; 3) the stability criteria of the problem are specified.

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