A multicriterion linear combinatorial problem with a parametric principle of optimality is considered. This principle is defined by a partitioning of partial criteria onto Pareto preference relation groups 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 semi-continuity of the multiple-valued mapping that defines the choice function. A formula of quasistability radius is derived for the case of the metric l∞. Some known results are stated as corollaries. 



Mathematics Subject Classification 2000: 90C05, 90C10, 90C29, 90C31

Sergey E. Bukhtoyarov

Vladimir A. Emelichev

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Computer Science Journal of Moldova, vol.13, no.1(37), 2005On quasistability radius of a vector trajectorialproblem with a principle of optimalitygeneralizing Pareto and lexicographic principles∗Sergey E. Bukhtoyarov, Vladimir A. EmelichevAbstractA multicriterion linear combinatorial problem with a para-metric principle of optimality is considered. This principle is de-fined by a partitioning of partial criteria onto Pareto preferencerelation groups within each group and the lexicographic prefer-ence relation between them. Quasistability of the problem isinvestigated. This type of stability is a discrete analog of Haus-dorff lower semi-continuity of the multiple-valued mapping thatdefines the choice function. A formula of quasistability radius isderived for the case of the metric l∞. Some known results arestated as corollaries.Mathematics Subject Classification 2000: 90C05, 90C10,90C29, 90C31.Key words and phrases: vector trajectorial problem,Pareto set, set of lexicographically optimal trajectories, quasista-bility, quasistability radius.1 IntroductionTraditionally stability of an optimization problem is understood as con-tinuous dependence of solutions on parameters of the problem. Themost general approaches to stability analysis of optimization problemsare based on properties of multiple-valued mappings that define opti-mality principles [1–4].c©2005 by S.E. Bukhtoyarov, V.A. Emelichev∗This work was supported by Fundamental Research Foundation of the Republicof Belarus ”Mathematical structures 29” (Grant 913/28)47S.E. Bukhtoyarov, V.A. EmelichevMathematical analysis does not present methods sufficient to inves-tigate stability of a discrete optimization problem. It is greatly due tocomplexity of discrete models, which can behave unpredictably undersmall variations of initial data [4, 5]. At the same time, if terminol-ogy of general topology is not used, then the formulation of a stabilityproblem can be significantly simplified in the case of a space of ac-nodes. There are different types of stability of discrete optimizationproblems (e. g. [4–9]). Stability of a discrete problem in the broadsense means that there exists a neighborhood of the initial data in thespace of problem parameters such that any problem with parametersfrom this neighborhood possesses some invariance with respect to theinitial problem. In particular, upper (lower) semicontinuity of an op-timal mapping is equivalent to nonappearance of new (preserving ofinitial) optimal solutions under ”small” perturbations of the mappingparameters. So concepts of stability [4–8] and quasistability [6–8, 10,11] of discrete optimization problems arise.In this article we consider an n-criterion trajectorial linear prob-lem with partitioning of criteria into groups according to given Paretopreference relation within each group and the lexicographic preferencerelation between them. Two special cases of such partitioning corre-spond to Pareto and lexicographic optimality principles. A formula forquasistability radius of this problem is derived for the case of indepen-dent perturbations of initial data in the metric l∞. Some known resultsare stated as corollaries.Note that similar formulas were derived earlier in [12–16] for sta-bility and quasistability radii of vector trajectorial and game-theoreticproblems with other parametric principles of optimality (”from Con-dorset to Pareto”, ”from Pareto to Slater”, ”from Pareto to Nash”).2 Basic definitions and notationsLet a vector criterionf(t, A) = (f1(t, A1), f2(t, A2), . . . , fn(t, An)) → mint∈T48On quasistability radius of a vector trajectorial problem with . . .with partial criterionfi(t, Ai) =∑j∈N(t)aij , i ∈ Nn = {1, 2, ..., n}, n ≥ 1,be defined on a system of subsets (trajectories) T ⊆ 2E , |T | ≥ 2,E = {e1, e2, . . . , em}, m ≥ 2. Here N(t) = {j ∈ Nm : ej ∈ t}, Ai is thei-th row of a matrix A = [aij ] ∈ Rn×m. Put fi(∅, Ai) = 0.Let s ∈ Nn, I = {I1, I2, . . . , Is} be a partitioning of the set Nn intos nonintersecting nonempty sets, i. e.Nn =⋃r∈NsIr,where Ir 6= ∅, r ∈ Ns; p 6= q ⇒ Ip ∩ Iq = ∅. To any such partitioningwe put in correspondence the binary relation ΩnI of strict preference inthe space Rn between different vectors y = (y1, y2, . . . , yn) and y′ =(y′1, y′2, . . . , y′n) as followsy ΩnI y′ ⇔ yIk Â y′Ik ,where k = min{i ∈ Ns : yIi 6= y′Ii}; yIk and y′Ik are the projectionsof the vectors y and y′ correspondingly onto the coordinate axes ofthe space Rn with numbers from the group Ik; Â is a relation, whichinduces Pareto optimality principle in the space R|Ik| :yIk Â y′Ik ⇔ yIk 6= y′Ik & yIk ≥ y′Ik .The introduced binary relation ΩnI determines ordering of theformed groups of criteria such that any previous group is significantlymore important that any consequent group. This relation generatesthe set of I-optimal trajectoriesTn(A, I) = {t ∈ T : ∀t′ ∈ T (f(t, A) ΩnI f(t′, A))},where ΩnI is the negation of the relation ΩnI .It is evident that Tn(A, IP ), IP = {Nn} (s = 1), is Pareto set, i. e.the set of efficient trajectoriesPn(A) = {t ∈ T : ∀t′ ∈ T (f(t, A) Â f(t′, A))},49S.E. Bukhtoyarov, V.A. Emelichevand Tn(A, IL), IL = {{1}, {2}, . . . , {n}} (s = n), is the set of lexico-graphically optimal trajectoriesLn(A) = {t ∈ T : ∀t′ ∈ T (f(t, A) ` f(t′, A))},where ` is the lexicographic order in the space Rn. This order is definedas followsy ` y′ ⇔ yk > y′k,k = min{i ∈ Nn : yi 6= y′i}.So under the parametrization of optimality principle we understandassigning the characteristic of binary relation that in special cases in-duces well-known Pareto and lexicographic optimality principles.It is easy to show that the binary relation ΩnI is antireflexive, asym-metric, transitive, and hence it is acyclic. And since the set T is finite,the set Tn(A, I) is non-empty for any matrix A and any partitioningI of the set Nn.Hereinafter by Zn(A, I) we denote the problem of finding the setTn(A, I).Clearly, T 1(A, {1}) is the set of optimal trajectories of the scalarlinear trajectorial problem Z1(A, {1}), where A ∈ Rm. Many extremecombinatoric problems on graphs, boolean programming and schedul-ing problems and others are reduced to Z1(A, {1}) [7, 9, 10, 17]).The following properties follow directly from the above definitions.Property 1. Tn(A, I) ⊆ P1(A) ⊆ T, whereP1(A) = {t ∈ T : ∀t′ ∈ T (fI1(t, A) Â fI1(t′, A))}.Property 2. If fI1(t, A) Â fI1(t′, A), then f(t, A) ΩnI f(t′, A).Property 3. If f(t, A) ΩnI f(t′, A), then fI1(t, Ai) ≥ fI1(t′, Ai).Property 4. A trajectory t 6∈ Tn(A, I) if and only if there existsa trajectory t′ such that f(t, A) ΩnI f(t′, A).Property 5. A trajectory t ∈ Tn(A, I) if and only if for any tra-jectory t′ the relation f(t, A) ΩnI f(t′, A) holds.50On quasistability radius of a vector trajectorial problem with . . .DenoteS1(A) = {t ∈ P1(A) : ∀t′ ∈ T \ {t}(fI1(t, A) 6= fI1(t′, A))}.Property 6. S1(A) ⊆ Tn(A, I).Proof. Assume the converse, i. e. t ∈ S1(A) and t 6∈ Tn(A, I).Then according to property 4 there exists a trajectory t′ 6= t such thatf(t, A) ΩnI f(t′, A).Hence due to property 3 we havefI1(t, A) ≥ fI1(t′, A).Taking into account the inclusion t ∈ P1(A) we obtainfI1(t, A) = fI1(t′, A),i. e. t 6∈ S1(A), which contradicts the assumption.Property 7. ∀ t ∈ S1(A) ∀ t′ ∈ T \ {t} ∃ i ∈ I1(fi(t′, Ai) >fi(t, Ai)).For any number ε > 0, define the set of perturbation matrixesB(ε) = {B ∈ Rn×m : ||B|| < ε},where ||B|| = max{|bij | : (i, j) ∈ Nn ×Nm}, B = [bij ].As in [8, 10, 14, 17], under the quasistability radius of the problemZn(A, I) we understand the numberρn(A, I) ={supKn(A, I) if Kn(A, I) 6= ∅,0 if Kn(A, I) = ∅,whereKn(A, I) = {ε > 0 : ∀ B ∈ B(ε) (Tn(A, I) ⊆ Tn(A + B, I))}.51S.E. Bukhtoyarov, V.A. Emelichev3 LemmasFor any trajectories t and t′ we define the numbers∆(t, t′) = |(t ∪ t′) \ (t ∩ t′)|,dn(t, t′, A) = maxi∈I1fi(t′, Ai)− fi(t, Ai)∆(t, t′).Lemma 1. If dn(t, t′, A) ≥ ϕ > 0, then the following relation holdsfor any perturbation matrix B ∈ B(ϕ) :f(t, A + B) ΩnI f(t′, A + B).Proof. Directly from the definition of the number dn(t, t′, A) wehave∃k ∈ I1(fk(t′, Ak)− fk(t, Ak) ≥ ϕ∆(t, t′)). (1)Further suppose that the assertion of the lemma is false, i. e. thereexists matrix B∗ = [b∗ij ] ∈ B(ϕ) such that f(t, A+B∗) ΩnI f(t′, A+B∗).Then by virtue of property 3 and linearity of the functions fi(t, A), i ∈Nn, we derive0 ≥ fi(t′, Ai + B∗i )− fi(t, Ai + B∗i ) == fi(t′, Ai)− fi(t, Ai) + fi(t′, B∗i )− fi(t, B∗i ) ≥≥ fi(t′, Ai)− fi(t, Ai)− ||Bi||∆(t, t′) >> fi(t′, Ai)− fi(t, Ai)− ϕ∆(t, t′), i ∈ I1,i. e.∀i ∈ I1 (fi(t′, Ai)− fi(t, Ai) < ϕ∆(t, t′)),which contradicts (1).Lemma 2. Let t ∈ Tn(A, I), t′ ∈ T \ {t}. For any number α >dn(t, t′, A) there exists a matrix B∗ ∈ Rn×m with norm ||B∗|| = α suchthatf(t, A + B∗) ΩnI f(t′, A + B∗). (2)52On quasistability radius of a vector trajectorial problem with . . .Proof. We construct the perturbation matrix B∗ = [b∗ij ] ∈ Rn×mby the formulab∗ij =−α if i ∈ I1, ej ∈ t′ \ t,α if i ∈ I1, ej ∈ t \ t′,0 otherwise.Then ||B∗|| = α andfi(t′, B∗i )− fi(t, B∗i ) = −α∆(t, t′), i ∈ I1.From here we get1∆(t, t′)(fi(t′, Ai + B∗i )− fi(t, Ai + B∗i )) =fi(t′, Ai)− fi(t, Ai)∆(t, t′)− α ≤≤ dn(t, t′, A)− α < 0, i ∈ I1,i. e. fI1(t, A + B∗) Â fI1(t′, A + B∗). This implies (2) by virtue ofproperty 2.4 TheoremTheorem. For any partitioning I of the set Nn, n ≥ 1, into sgroups, s ∈ Nn, the quasistability radius ρn(A, I) of a problem Zn(A, I)is expressed by the formulaρn(A, I) = mint∈T n(A,I)mint′∈T\{t}dn(t, t′, A). (3)Proof. Denote the right hand side of (3) by ϕ for short. Beforeproving the theorem we note that since the sets Tn(A, I) and T \ {t}are non-empty, the number ϕ is correctly defined and nonnegative.First we prove the inequalityρn(A, I) ≥ ϕ. (4)Without loss of generality assume that ϕ > 0 (otherwise inequality (4)is obvious). From the definition of the number ϕ, it follows that forany trajectories t ∈ Tn(A, I) and t′ 6= t the inequalitiesdn(t, t′, A) ≥ ϕ > 053S.E. Bukhtoyarov, V.A. Emelichevhold. Applying lemma 1 we get∀ B ∈ B(ϕ) ∀ t ∈ Tn(A, I) ∀ t′ ∈ T (f(t, A + B) ΩnI f(t′, A + B)).Therefore t ∈ Tn(A + B, I) by virtue of property 5. Thus we conclude∀ B ∈ B(ϕ) (Tn(A, I) ⊆ Tn(A + B, I)).This formula proves (4).It remains to show thatρn(A, I) ≤ ϕ. (5)Let ε > α > ϕ and trajectories t ∈ Tn(A, I), t′ 6= t be such thatdn(t, t′, A) = ϕ. Then according to lemma 2 there exists a matrix B∗with norm ||B∗|| = α such that (2) holds, i. e. t 6∈ Tn(A + B∗, I).Hence we have∀ε > ϕ ∃B∗ ∈ B(ε) (Tn(A, I) 6⊆ Tn(A + B∗, I)) .This proves inequality (5). Summarizing (4) and (5) we obtain (3).5 CorollariesCorollary 1 [10]. The quasistability radius of the problemZn(A, IP ), n ≥ 1, of finding Pareto set Pn(A) is expressed by theformulaρn(A, IP ) = mint∈P n(A)mint′∈T\{t}maxi∈Nnfi(t′, Ai)− fi(t, Ai)∆(t, t′).Corollary 2 [18]. The quasistability radius of the problemZn(A, IL), n ≥ 1, of finding the set of lexicographically optimal trajec-tories Ln(A) is expressd by the formulaρn(A, IL) = mint∈Ln(A)mint′∈T\{t}f1(t′, A1)− f1(t, A1)∆(t, t′).54On quasistability radius of a vector trajectorial problem with . . .A problem Zn(A, I) is called quasistable if ρn(A, I) > 0. Thusquasistability of a problem Zn(A, I) is the property of preserving opti-mality by all I-efficient trajectories under small variation of matrix A.In other words, quasistability is a discrete analog of Hausdorff lowersemi-continuity of the multiple-valued mapping that assigns the set ofI-efficient trajectories to each set of the problem parameters.Corollary 3. For any partitioning I of the set Nn, n ≥ 1, into sgroups, s ∈ Nn, the following statements are equivalent for a problemZn(A, I), n ≥ 1:(i) the problem Zn(A, I) is quasistable,(ii) ∀ t ∈ Tn(A, I) ∀ t′ ∈ T \ {t} ∃ i ∈ I1(fi(t′, Ai) > fi(t, Ai)),(iii) Tn(A, I) = S1(A).Proof. Equivalence of statements (i) and (ii) follows directly fromthe theorem.The implication (ii) ⇒ (iii) is proved by contradiction. Supposethat (ii) holds but (iii) does not.From properties 1 and 6 we getS1(A) ⊆ Tn(A, I) ⊆ P1(A).Then (since Tn(A, I) 6= S1(A) is assumed) there exists a trajectoryt ∈ Tn(A, I) ⊆ P1(A) such that t 6∈ S1(A). It follows that there existstrajectory t′ ∈ P1(A) such thatt′ 6= t, fI1(t, A) = fI1(t′, A).This contradicts statement (ii).The implication (iii) ⇒ (i) is obvious by virtue of property 7.From corollary 3, we easily get the following known result (e. g.see [10]).Corollary 4. The problem Zn(A, IP ), n ≥ 1, of finding Pareto setPn(A) is quasistable if and only if Pn(A) = Sn(A).55S.E. Bukhtoyarov, V.A. EmelichevHere Sn(A) is Smale set [19], i. e. the set of strictly efficienttrajectories:Sn(A) = {t ∈ Pn(A) : ∀t′ ∈ T \ {t} (f(t, A) 6= f(t′, A))}.Corollary 3 also impliesCorollary 5 [18]. The problem Zn(A, IL), n ≥ 1, of finding theset Ln(A) of lexicographically optimal trajectories is quasistable if andonly if|Ln(A)| =∣∣∣∣Arg mint∈T f1(t, A1)∣∣∣∣ = 1.References[1] Belousov E. G., Andronov V. G. Solvability and stability of poly-nomial programming problems. Moscow: Izd. MGU, 1993, 172 p.(in Russian)[2] Dubov Yu. A., Travkin S. I., Yakimets V. N. Multicriterion mod-els of forming and choosing variants of systems. Moscow: Nauka,1986, 296 p. (in Russian)[3] Molodtsov D. A. Stability of optimality principles. Moscow:Nauka, 1987, 272 p. (in Russian)[4] Sergienko I. V., Kozerackaja L. N., Lebedeva T. T. Investigation ofstability and parametric analisys of discrete optimization problems.Kiev: Naukova Dumka, 1995, 169 p. (in Russian)[5] Kozerackaja L. N., Lebedeva T. T., Sergienko I. V. Investigation ofstability of discrete optimization problems. Kibernetika i SistemniyAnaliz, 1993, no. 3, pp. 78 –93. (in Russian)[6] Lebedeva T. T., Sergienko T. I. 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On one type ofstability of a vector combinatorial problem with Σ-MINMAX andΣ-MINMIN partial criteria. Izv. Vuzov, Matem., 2004, no. 12, pp.17 –27. (in Russian)[12] Emelichev V. A., Bukhtoyarov S. E. Stability of generally efficientsituation in finite cooperative games with parametric optimalityprinciple (”from Pareto to Nash”). Computer Science Journal ofMoldova, 2003, v. 11, no. 3, pp. 316-323.[13] Bukhtoyarov S. E., Emelichev V. A. Parametrization of principleof optimality (”from Pareto to Slater”) and stability of multicrite-rion trajectory problems. Diskretniy Analiz i Issled. Operaciy, ser.2, 2003, v. 10, no. 2, pp. 3–18. (in Russian)[14] Bukhtoyarov S. E., Emelichev V. A. On quasistability of vectortrajectorial problem with the parametric principle of optimality.Izv. Vuzov, Matem., 2004, no. 1, pp. 25–30 (in Russian).[15] Bukhtoyarov S. E., Emelichev V. A., Stepanishina Yu. V. Stabilityof discrete vector problems with the parametric principle of opti-mality. Cybernetics and Systems Analysis, 2003, v. 39, no. 4, pp.604-614.57S.E. Bukhtoyarov, V.A. Emelichev[16] Emelichev V. A., Stepanishina Yu. V. Multicriteria combinatoriallinear problems : parametrization of the optimality principle andthe stability of the effective solutions. Discrete Math. Appl., 2001,v. 11, no. 5, pp. 435-444.[17] Emelichev V. A., Kuz’min K. G., Leonovich A. M. Stability inthe combinatorial vector optimization problems. Automation andRemote Control, 2004, v. 65, no. 2, pp. 227–240.[18] Emelichev V. A., Berdysheva R. A. On stability and quasistabilityof trajectorial problem of sequential optimization. Dokl. Nazhion.Akad. Nauk Belarusi, 1999, v. 43, no. 3, pp. 41–44. (in Russian)[19] Smale S. Global analysis and economics, V. Pareto theory withconstraints. J. Math. Econom., 1974, v. 1, no. 3, pp. 213–221.S.E. Bukhtoyarov, V.A. Emelichev, Received April 7, 2005Belarusian State Universityave. Fr. Skoriny, 4,Minsk, 220050, Belarus.E–mail : emelichev@bsu.by58

On quasistability radius of a vector trajectorial problem with a principle of optimality generalizing Pareto and lexicographic principles

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On quasistability radius of a vector trajectorial problem with a principle of optimality generalizing Pareto and lexicographic principles

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