Some Properties of Multiple Parameters Linear Programming

Volume 2010, Article ID 204263,13pages doi:10.1155/2010/204263

Research Article

Some Properties of Multiple Parameters

Linear Programming

Maoqin Li,

_{Shanlin Li,}

_{and Hong Yan}

1_{Department of Mathematics, Taizhou University, Taizhou, Zhejiang 317000, China} 2_{Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University,}

Kowloon, Hong Kong

Correspondence should be addressed to Hong Yan,[email protected]

Received 20 August 2009; Revised 9 March 2010; Accepted 12 May 2010

Academic Editor: Kok Lay Teo

Copyrightq2010 Maoqin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a linear programming problem in which the right-hand side vector depends on multiple parameters. We study the characters of the optimal value function and the critical regions based on the concept of the optimal partition. We show that the domain of the optimal value functionfcan be decomposed into finitely many subsets with disjoint relative interiors, which is different from the result based on the concept of the optimal basis. And any directional derivative offat any point can be computed by solving a linear programming problem when only an optimal solution is available at the point.

1. Introduction

Parametric and sensitivity analyses are classic subject in linear programming problems. They are of great importance in the analysis of practical linear models. Almost any textbook includes a section about them and many commercial optimization package offer an option to perform postoptimal analysis. Over the years we have learned to use an optimal basic solution to perform parametric and sensitivity analyses. However, this approach has led to the existing misuse of parametric optimization in commercial packages1. This misuse

is of course a shortcoming of the packages and by no means a shortcoming in the model existing theoretical literature. In 2–4, an alternative optimal partition approach to

behavior of the optimal solutions and the optimal objective function value of a parametric linear programming, and avoids any misunderstanding. Goldfarb and Scheinberg5extend the optimal partition approach to one-parameter semidefinite programming and Yildirim 6to one-parameter conic optimization. They investigate mainly the range of perturbations for which the optimal partition remains constant. In this paper, we extend this approach to multiple parameters linear programming. Our special attention is paid to investigate some properties of the whole range of perturbations for which the given problem has a finite optimal solution and the optimal value function on it.

The paper is organized as follows. In the next section we introduce the related concepts. In Section 3 the property of optimal value function is discussed. In Section 4

the character of the critical region is described. In the last section our conclusions are summarized.

2. Preliminaries

In this paper we deal with a problemPin standard format:

mincT_{x:Axb, x≥0, P}

and the dual problemDis written as

maxbTy:ATysc, s≥0, D

where matrixA∈Rm×n_{with rankm, vectorx,c,s∈R}n_andy,b∈Rm_{.x≥0 means that each} coordinate ofxis greater than or equal to zero. We assume thatPandDare both feasible hereafter. The feasible regions ofPandDare denoted, respectively, by

P:{x:Axb, x≥0},

D:y, s:ATysc, s≥0.

2.1

The optimal solutions set ofPandDare denoted byP∗andD∗,respectively. We define the index setsBandNby

B:{i:xi>0 for somex∈P∗},

N:i:si>0 for some

y, s∈D∗. 2.2

Then from4, we haveB∩N ∅andB∪N{1,2, . . . , n}. ThusBandNform a partition of the full index set. This partition, denoted byπ B, N,is called the optimal partition of Pand\orD.

Given the optimal partitionπ B, N ofPand\orD, the optimal solutions x

complementary optimal solutionsxandy, sgenerated from the interior point method are called the central solutions ofPandD, respectively.

It is well known that the optimal partition is uniquely determined by the central solution and the converse is true. We have a one-to-one correspondence between the optimal partition and the central solution. The following lemmas come from 4 and are stated without proof.

Lemma 2.1. Letx∗∈P∗andy∗, s∗∈D∗. Then

P∗x:x∈P, xTs∗0,

D∗y, s:y, s∈D, sTx∗0.

2.3

Lemma 2.2. Letπ B, Nbe the optimal partition of PandD. Then

P∗{x:x∈P, xN0},

D∗y, s:y, s∈D, sB0

, 2.4

wherexNandsBrefer to the restriction of vectorsxandsto the coordinate setsNandB,respectively.

3. The Optimal Value Function

In this section we consider multiple parameters perturbation ofband investigate the effect of

change inbon the optimal value function.

Suppose thatbt bHt,where matrix H ∈ Rm×s _{with ranks, t ∈ R}s_{. Parametric} linear programming problems are defined as follows:

mincTx:Axbt, x≥0, Pt

maxbtTy:AT_{ysc, s≥0. D} t

The feasible regions ofPtandDtare denoted byPtandDt, and the optimal solutions set byP_t∗ andD_t∗, respectively. The optimal value ofPtandDtis denoted byftwhich is

a function of the parametert, withft −∞ifPtis unbounded andDtinfeasible; and ft ∞ifDtis unbounded andPtinfeasible. IfPtandDtare both infeasible then ftis undefined. The region, in whichftis finite, is called the domain off, denoted byK. By the Linear Programming theory, we have thatftis finite if and only ifPtandDtare

both feasible. Thus

K{t∈Rs:Pt/∅, Dt/∅},

ft mincTxt:t∈K, xt∈Pt

maxbtTyt:t∈K,

yt, st

∈Dt

cTx∗_t btTy∗_t,

3.1

To characterizeK, we use the following sets, The polyhedral convex set inRn_{is defined} as the intersection of finitely many closed half-spaces ofRn_{, that is, as the set of the form}

{x∈Rn:Bx≤b}, 3.2

whereB ∈Rp×n_{, b ∈R}p_{. The polyhedral convex cone inR}n_{is defined as the set which is a} polyhedral convex set and a cone. It is clear that a set is a polyhedral convex cone if and only if it can be expressed as the set of the form

{x∈Rn:Bx≤0}, 3.3

whereB∈Rp×n_.

Here is the assumption that P0 and D0 are both feasible. It follows that Dt is

feasible for all values oft. Therefore, the domain K off consists of allt for which Ptis

feasible.

Theorem 3.1. The domainKoffis a polyhedral convex set inRs_.

Proof. Note that the domainKoffconsists of alltfor whichPtis feasible.Ptis feasible if

and only ifbHt∈ {Ax:x≥ 0}.Since the set{Ax:x≥0}is a polyhedral convex cone, it may be represented in the form of{x∈Rm_{:Bx≤0},whereB∈R}p×m_{. Thust∈Kif and only} ifBbHt Bb BHt≤0. This means that

K{t:BHt≤ −Bb}, 3.4

whereBH ∈Rp×s_{and -Bb∈R}p_{.The result now follows by the definition of the polyhedral} convex set.

It is convenient to introduce another notation. Let K be a convex subset ofRs_{. We} define a function f to be piecewise linear onK if there exist finitely many convex subsets

Ri, i1,2, . . . , pofKsuch thatKp_i1Riandfis an affine function on everyRi.

Theorem 3.2. The optimal value functionfis continuous, convex, and piecewise linear onK.

Proof. By definition,

ft maxbtTyt:yt, st∈Dt

. 3.5

For eacht ∈K, due to the feasibilities ofPtandDt, we havePt∗/∅andD∗t/∅. Further, there is a unique optimal partition of Ptand\or Dt, and then a unique central solution

y∗t, s∗tofDt. We may assume now that the maximum value is attained at the central

solution and write

Noting that the number of partition of the full index{1,2, . . . , n}is finite and that there is a unique optimal partition ofDtfor anyt∈K, we may define the index setΓby

Γ:i:πi Bi, Niis an optimal partition for someDtwith t∈K

. 3.7

It is obvious thatΓis a finite set. Due to the definition ofDt, the feasible region ofDtis

constant whentvaries. We have thatDtD0andy∗t, s∗t∈D0for allt. ByLemma 2.2,

we have that if central solutions y∗t1, s∗t1and y∗t2, s∗t2 associate with sameπi, then these two central solutions are central solutions ofDt1andDt2each other. Thus, we may take a representative, sayyi, si, among all the central solutions associated with same

πi. Further, the optimal solution ofDtmust be attained at someyi, sifor anyt ∈K. The set{yi, si:i∈Γ}is a finite subset ofD0clearly. We may write

ft maxbtTyi:i∈Γ

. 3.8

For eachi∈Γ, we have

btTyibTyi HtTyi, 3.9

which is an affine function oft. Thusftis the maximum of a finite set of affine functions.

Lett1_{, t}2 _{∈ K andt λt}1 _1−λt2_{, where λ ∈ 0,1.Recalling that K is convex,}

we havet ∈K. By3.8, there existi1, i2, j ∈ Γsuch thatft1 bt1Tyi1,ft2 bt2Tyi2,

ft btTyj,and

f t1_{≥b t}1T_y

i, ∀i∈Γ,

f t2≥b t2Tyi, ∀i∈Γ.

3.10

SincebtTyiis an affine function oftfor eachi∈Γ, we have

f tb tTyj

b λt1 1−λt2Tyj

λb t1Tyj 1−λb t2

T

yj

≤λb t1Tyi1 1−λb t2

T

yi2

λf t1 1−λf t2.

3.11

For eachi∈Γ, let

Ri

t:ft btTyi, t∈K

. 3.12

Since, for anyt ∈K, there exists ani∈Γsuch thatft btTyi, we haveK

i∈ΓRi.For anyt∈Ri, by3.8, we obtain

btTyi−btTyj≥0, j∈Γ, j /i. 3.13

This means thatt∈Ei :{t:btTyi−yj≥0,for allj∈Γ, j /i}.In turn, ift∈Kandt∈Ei, thenftis finite andft btTyifollows from the definition offt. Thus, we conclude thatRi Ei∩K.Note thatbtTyi−yjis an affine function oftfor everyj ∈Γ j /i. SoEi is a polyhedral convex set. ByTheorem 3.1,Riis a polyhedral convex set, of course a convex set. Thus,fis an affine function onRi. It follows thatfis piecewise linear onK.

Letε >0.For eacht0 _{∈K, sincebt}T_y

iis continuous at the pointt0for everyi∈Γand Γis a finite set, there exists a positive numberδsuch that

b t0Tyi−ε < btTyi < b t0

T

yiε 3.14

for alli∈Γand all pointstinKwitht−t0_{< δ. So from the inequalities above, we have}

f t0_{−ε < ft< f t}0_{ε 3.15}

for all pointstinKwitht−t0_{< δ. Thus,fis continuous att}0_{. It follows thatfis continuous}

onK.

Summarizing the above results, the proof of the theorem is completed.

From the arguments above, we have known that the domainK off is a polyhedral convex set and the union of finite polyhedral convex sets. To explore further the construction properties of the domainK, we do the following. It is possible of course that the dimension ofRiis less than the dimension ofK. For this case, we have the result below.

Lemma 3.3. IfdimRi<dimK, thenK_{j∈Γ,j /i}Rj.

Proof. For anyt0_∈R

iand natural numbern, since dim{t0 1/nB∩K}dimK,whereB is the Euclidean unit ball inRs_{thus there existst}

n∈Kwithtn/∈Risuch thattn∈t0 1/nB. This means thatt0_{is the limit point of the sequence{t}

n}in_{j∈Γ,j /i}Rj. As the set_{j∈Γ,j /i}Rjis closed, we havet0_∈

j∈Γ,j /iRj,as required. We now define the new index set

Γ i:i∈Γ,dimRi dimK

and callRi {t : ft btTyi, t ∈ K}with dimRi dimK as a critical region off. By

Lemma 3.3, we haveK_j∈ΓRj, andΓis a finite set clearly. The following result describes a construction property ofK.

Lemma 3.4. IfRiandRjare two different critical regions off, then riRi∩riRj∅.

Proof. To see this, we argue by contradiction. Suppose that there existst0 _{∈Ksuch thatt}0 _∈

riRi∩riRj.By affKaffRiaffRj, we may choose a positive numberεsuch thatt0εB∩ affK ⊂Riandt0εB∩affK⊂Rj.For anyt∈K,as1−λt0λt∈K ⊂affKfor any number 0 ≤ λ ≤ 1,we may choose a number 0≤ λ0 ≤ 1 such that1−λ0t0λ0t ∈Riand 1−λ0t0λ0t∈Rj.Due to definitions ofRiandRj, we have

f 1−λ0t0λ0t

bH 1−λ0t0λ0t

T

yi,

f 1−λ0t0λ0t

bH 1−λ0t0λ0t

T

yj,

3.17

that is,

bH 1−λ0t0λ0t

T

yi bH 1−λ0t0λ0t

T

yj. 3.18

UsingbHt0T_y

i bHt0Tyj,we havebHtTyi bHtTyj. ThusRi Rjfollows from definitions of them. This contradicts the supposition ofRiandRjbeing different.

Summarizing the above results, we have the following consequence.

Theorem 3.5. Every critical region offis a polyhedral convex set inRs_{and the number of critical} regions is finite. The domainK off can be expressed as the union of all critical regions. Different critical regions have disjoint relative interiors.

4. The Optimal Solution Sets on Critical Regions

We established in the previous section that optimal value functionftis continuous, convex, and piecewise linear and that the domainKand every critical regionRiare polyhedral convex sets. In this section we will see some characters of optimal solution set at the points in some critical regionRi. Before proceeding, we introduce several notations. LetK be a nonempty convex subset ofRs_andd∈Rs_{withd /0. We calldas an admissible direction ofKat pointt} inK, ifK∩ {tλd:λ >0}/∅. Letfbe a convex function fromRs_{to−∞,∞, and lettbe a} point wherefis finite. The directional derivative offattwith respect to a directiondd /0

is defined to be the limit

ft;d lim

λ↓0

ftλd−ft

λ . 4.1

Theorem 4.1. LetRibe a critical region off. Then the dual optimal setD∗_tis constant (i.e., invariant) over relative interior region riRiofRi.

Proof. Lett1, t2 ∈riRiandt1/t2.SinceRiis convex, there are two pointst1andt2ofRisuch thatt1,t2 are relative interiors of the line segmentt1, t2included inRi.The fact that f is linear onRiimplies thatfis linear ont1, t2. Supposing thatgλ fλt2 1−λt1, gλis

a linear function on0,1.Letλ ∈0,1be arbitrary and lety, s∈D∗

t be arbitrary as well, wheretλt2 1−λt1.Sincey, sis optimal forDt,we have

g λf t bHtTybTy λHt2 1−λ

Ht1

T

y, 4.2

and, sincey, sis dual feasible for allt,

bHt1

T

y≤f t1

g0,

bHt2

T

y≤f t2

g1.

4.3

Hence we find that

g1−g λ≥ 1−λ H t2−t1

T

y,

g λ−g0≤λ H t2−t1

T

y.

4.4

The linearity ofgon0,1implies that

g1−g λ

1−λ

g λ−g0

λ .

4.5

Hence, the last two inequalities are equalities. This means that the derivative ofgwith respect toλon the interval0,1satisfies

gλ H t2−t1

T

y, ∀λ∈0,1, 4.6

or equivalently

gλ g0 λgλ bTy Ht1

T

yλ H t2−t1

T

y

bT_{y H λt}

2 1−λt1

T

ybtTy, ∀λ∈0,1,

4.7

wheretλt2 1−λt1. We conclude thatyis optimal for anyDtwithtbeing an interior

of the line segmentt1, t2.Sinceλandyare arbitrary, it follows thatD∗_t1 D_t2∗.The theorem

Corollary 4.2. One has

ft bTy HtTy, ft;d HTyTd, ∀t∈Ri, 4.8

wheredis an admissible direction ofRiatt,y, sis inD∗_t, andtis in riRi.

Proof. The above theorem reveals thatD_t1∗ D∗_t2for allt1, t2∈riRi.This implies that

ft btTybTy HtTy, ∀t∈riRi, ∀

y, s∈D∗_t. 4.9

By continuity off, we conclude that

ft btTybTy HtTy, ∀t∈Ri, ∀

y, s∈D∗_t for somet∈riRi. 4.10

Moreover, ifdis an admissible direction ofRiatt, we have

ft;d HTyTd, ∀y, s∈D∗_t for somet∈riRi, 4.11

as required.

Corollary 4.3. Lettbe an arbitrary relative interior ofRi, and lettbe an arbitrary boundary point of

Ri. ThenD∗_t ⊆Dt∗.

Proof. Lety, s ∈ D∗

t. Sincey, sis dual feasible for allt,y, s ∈ Dt. Using t ∈ riRi and 4.10, we haveft btTy. That is,y, s∈D∗_t. The proof is completed.

From the argument of the theorem above, we have the following consequences.

Corollary 4.4. Ifftis linear on the line segmentt1, t2, wheret1/t2, then the dual optimal setD∗t is constant fort∈t1, t2and the slope offtont1, t2is equal toHt2−t1Ty∗for anyt∈t1, t2

and anyy∗, s∗∈D∗_t.

Theorem 4.5. Lett1andt2be any two different points of the domainKoffsuch thatD∗t1∩D∗t2/∅. ThenD∗_t is constant for allt∈t1, t2andftis linear on the line segmentt1, t2.

Proof. Lety, s∈D∗_t1∩D_t2∗. Then

ft1 bt1Ty, ft2 bt2Ty. 4.12

Consider the following linear functionh:

hcoincides withfatt1andt2. Sinceftis convex, this implies that

ft≤ht, ∀t∈t1, t2. 4.14

Nowy, sis dual feasible for allt∈t1, t2. Sinceftis the optimal value ofDt, it follows

that

ft≥btTy bHtTyht. 4.15

Therefore, f coincides with h on t1, t2. As a consequence, f is linear on t1, t2. By

Corollary 4.4, we have thatD∗_tis constant ont1, t2, and we complete the proof.

Theorem 4.6. IfRiandRjare any two different critical regions, then

D∗_ti∩D∗_tj ∅, ∀ti∈riRi, ∀tj∈riRj. 4.16

Proof. Letπi Bi, Niandπj Bj, Njbe the optimal partitions ofDtiandDtj,yi, si andyj, sjbe the central solutions, respectively. ByLemma 2.2, we have

D_ti∗ y, s:y, s∈Dti, sBi 0

,

D∗_tj y, s:y, s∈Dtj, sBj0

. 4.17

Bi/Bj andti/tj follow fromRi andRj being different. Further, eithersiBj/0 orsjBi/0 holds, wheresiBj andsjBi are the restrictions ofsiandsjto the coordinate setsBjandBi, respectively. Otherwise, by the definition of the central solution,siBi 0,siNi >0, sjBj 0, sjBj > 0, siBj 0, and sjBi 0 hold simultaneously. This implies thatBj ⊆ Bi and

Bi⊆Bj, which contradictsBi/Bj.

Since dimRi dimRj dimK, we have affRi affRj affK.The inclusive relation

{t : t λti 1−λtj, λ ∈ R} ⊂ affRi affRj follows fromti, tj ∈ K, ti/tj. Due to tiand

tjbeing the relative interiors ofRiandRj separately, there exists a numberλ0 > 1 such that ti λ0ti 1−λ0tjandtj λ0tj 1−λ0tiare relative interiors ofRiandRjseparately. By

Theorem 4.1, it holds thatD_ti∗ D∗

ti and D

∗

tj D_tj∗. In order to prove the theorem, we now

argue by contradiction. IfD_ti∗∩D_tj∗/∅, thenD∗

ti∩D

∗

tj/∅. UsingTheorem 4.5andti, tj∈ti, tj, we conclude thatD_ti∗ D∗_tj. Hence we haveyi, si∈ D∗tj andyj, sj∈ D∗ti. This contradicts the definition above ofD∗_ti ifsjBi/0 or the definition above ofDtj∗ ifsiBj/0. The theorem is proved.

Theorem 4.7. Lettbe an arbitrary point ofKandx∗an arbitrary optimal solution ofP_t. Then for

any directiondd /0,

f t;dmax

y,s

HdTy:y, s∈D∗

t

max y,s

HdTy:ATysc, s≥0, sTx∗0.

Proof. The second equality obviously holds owing to

D∗_t y, s:ATysc, s≥0, sTx∗0. 4.19

Below we proceed by considering two cases separately.

Case 1. One hasK∩ {tλd:λ >0}/∅.

SinceKis the union of finitely many critical regions and each of them is polyhedral, there is certainly a critical regionRisuch thatt, tλd ⊆ Ri,whereλ > 0.Lety, s ∈ D∗_t, wheret∈riRi.FromCorollary 4.2, we have

ft bHtTy, ∀t∈Ri. 4.20

By the definition offt;d, we easily obtain that

f t;d HdTy. 4.21

Sincey, sis optimal forD_tλdand anyy, s ∈ D∗

t is feasible forDtλdwith respect to

λ∈0, λ,so we have

bH tλdTy≥ bH tλdTy, ∀y, s∈D∗_t. 4.22

We also havey, s∈D∗

t.Therefore

bHtTy bHtTy, ∀y, s∈D∗_t. 4.23

Subtracting both sides of this equality from the corresponding sides in the last inequality, we get

λHdTy≥λHdTy, ∀y, s∈D_t∗. 4.24

Dividing both sides by the positive numberλ,we obtain

HdTy≥HdTy, ∀y, s∈D∗_t, 4.25

thus proving that

f t;dmax

y,s

HdTy:y, s∈D∗_t HdTy. _4.26

Case 2. One hasK∩ {tλd:λ >0}∅.

In this case, we point out first thatP_tλdis infeasible for any positive numberλand

ft;d ∞.SinceP_thas an optimal solutionx∗,D_thas an optimal solution as well. This implies that the problem

max y,s

HdTy:ATysc, s≥0, sTx∗0 _4.27

is feasible. Hence, if the problem is not unbounded, the problem and its dual have optimal solutions. The dual problem is given by

min ξ,λ

cTξ:AξHd, ξλx∗≥0. _4.28

We conclude that there are a vectorξ∈Rn_{and a real numberλsuch thatAξHd, ξλx}∗_≥0.

This implies that we cannot haveξi<0 andx∗_i 0 for 1≤i≤n. In other words,

x∗_i 0⇒ξi ≥0, ∀1≤i≤n. 4.29

Therefore, there is a positive numberεsuch thatx:x∗εξ≥0.Now we have

AxAx∗εξ Ax∗εAξbHtεHdbH tεd. 4.30

Thus we find thatP_tεdadmitsxas a feasible point. This contradicts the fact thatP_tλdis infeasible for any positive numberλ. We conclude that the problem is unbounded, proving the theorem.

5. Conclusions

Using the properties of the optimal partition, we give some description to a multiple parameters linear programming problem. The results in Section 3 show the geometric structures of the optimal value function and its domain. In Section 4, we point out that the character of the domain Kof f is completely decided by the structure of dual optimal solutions and the directional derivative offat any point can be obtained by solving a linear programming problem. Similarly, we may study multiple parameters perturbation of the cost coefficient vector problem or other parameter values problem. Our results maybe become as a theoretical foundation of summarizing critical regions.

Acknowledgments

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