A fixed point iterative approach to integer programming and its distributed computation

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R E S E A R C H

Open Access

A ﬁxed point iterative approach to integer

programming and its distributed

computation

Chuangyin Dang

1*

_{and Yinyu Ye}

2

*_{Correspondence:}

[email protected] 1_{Department of Systems}

Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong, China

Full list of author information is available at the end of the article

Abstract

Integer programming is concerned with the determination of an integer or mixed-integer point in a polytope. It is an NP-hard problem and has many applications in economics and management. Although several popular methods have been developed for integer programming in the literature and extensively utilized in practices, it remains a challenging problem and appeals for more endeavors. By constructing an increasing mapping satisfying certain properties, we develop in this paper an alternative method for integer programming, which is called a ﬁxed point iterative method. Given a polytope, the method, within a ﬁnite number of iterations, either yields an integer or mixed-integer point in the polytope or proves no such point exists. As a very appealing feature, the method can easily be

implemented in a distributed way. Furthermore, the construction implies that determining the uniqueness of Tarski’s ﬁxed point is an NP-hard problem, and the method can be applied to compute all integer or mixed-integer points in a polytope and directly extended to convex nonlinear integer programming. Preliminary numerical results show that the method seems promising.

MSC: 90C10

Keywords: integer or mixed-integer point; polytope; integer programming; linear programming; self-dual embedding technique; increasing mapping; Tarski’s ﬁxed point theorem; ﬁxed point iterative method

1 Introduction

Integer programming is concerned with the determination of an integer or mixed-integer point in a polytope. As a powerful mechanism, integer programming has been extensively applied in economics [, ] and management []. Integer programming is an NP-complete problem []. To solve such a problem, several methods have been developed in the liter-ature. As an application of linear programming, the cutting plane method was pioneered in []. The method iteratively reﬁnes a feasible set or objective function by means of linear inequalities. The branch-and-bound method was formulated in []. The method gradu-ally improves upper and lower bounds of the objective function by solving linear programs and systematically enumerates candidate solutions in branches of a tree with the full set of candidate solutions at the root by checking against the upper and lower bounds. To test whether a given feasible integer point is optimal or not, the neighborhood method was

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proposed in [, ]. The method simply checks a minimal set of points in the neighbor-hood of a feasible point to determine whether one of them is in the polytope or yields a better objective function value. The basis-reduction method originates in [, ]. As that in a branch-and-bound method, the method searches for an integer point in a polytope along a set of vectors that forms a reduced basis. The simplicial method was developed in [, ]. The method starts from an arbitrary integer point in the space and follows a simplicial path that either leads to an integer point in a polytope or proves no such point exists when the polytope is in a speciﬁc form. Further developments of some of these methods and new methods can be found in the recent literature such as [, –], and the references therein. These methods play an extremely important role in the develop-ment of integer programming, however, it remains a challenging problem and appeals for more endeavors. Thus, developing alternative integer programming methods is always an active research area.

Integer programming can be cast as a fixed point problem of an increasing mapping. More precisely, letbe a binary relation on a nonempty setS. The pair (S,) is a par-tially ordered set ifis reflexive, transitive, and antisymmetric onS. A lattice is a partially ordered set (S,), in which any two elementsxandyhave a least upper bound (supre-mum),sup_S(x,y) =inf{z∈S|xzandyz}, and a greatest lower bound (infimum),

infS(x,y) =sup{z∈S|zxandzy}, in the set. A lattice (S,) is complete if every nonempty subset ofShas a supremum and an infimum inS. Letf be a mapping from Sinto itself.f is an increasing mapping iff(x)f(y) for anyxandyofSwithxy. When (S,) is a complete lattice andf is an increasing mapping, Tarski’s fixed point theorem [] asserts thatf has a fixed point inS. A significant feature of Tarski’s fixed point theorem is thatScan be a finite set and there is no restriction on its topological structures. This feature has a profound implication for integer programming as evidenced in this paper. The computational complexity of Tarski’s fixed point theorem on (S,≤) has been studied in [], and it is polynomial-time computable if the dimension is fixed. As an application of Tarski’s fixed point theorem to integer programming, an increasing-mapping approach was briefly described in []. However, the approach is very primitive and can only update one coordinate at each iteration.

LetN={, , . . . ,n}andN={, , , . . . ,n}. ForxandyofRn,x≤lyif eitherx=yor bothxi=yi,i= , , . . . ,k– , andxk<ykfor somek∈N, andx≤yifxi≤yifor alli∈N, where≤lis the lexicographic order onRnand≤is the componentwise order onRn. Let Sbe any given ﬁnite set ofRn_{. Then (}_S_,_≤

l) is a complete lattice. We now convert integer programming into the computation of fixed points of an increasing mapping from a finite lattice into itself, which leads to the fixed point iterative method proposed in this paper and is the driving force behind our research endeavors.

ConsiderP={x∈R_|_Ax_≤_b_}_with

A= ⎛ ⎜ ⎝

–    – –

⎞ ⎟ ⎠

andb= (–, , ). This polytope is illustrated in Figure .

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Figure 1 An illustration of the idea.

for any x∈D(P) withx∈/ P andx=xl _{and that}_h₍_x∗_{) =}_x∗ _{if and only if either}_x∗ _∈_P orx∗=xl_{. Such a mapping is illustrated in Figure . This simple example stimulates the} idea in this paper though the situation is far more complicated when the dimension is higher.

In this paper, with this constructing we develop a fixed point iterative method for in-teger programming. A self-dual technique is applied for a solution to a bounding linear program in the development. Given any polytope, within a finite number of iterations, the method either yields an integer or mixed-integer point in the polytope or proves no such point exists. Theoretically, one can make the method be a polynomial-time algo-rithm when the dimension is fixed. But a more appealing feature of the method is that it can easily be implemented in a distributed way. Furthermore, the construction implies that determining the uniqueness of Tarski’s fixed point is an NP-hard problem, and the method can be applied to compute all integer or mixed-integer points in a polytope and directly extended to convex nonlinear integer programming. Preliminary numerical re-sults show that the method is promising, and may offer a comparable solution to integer programming though a comprehensive comparison with the existing methods is beyond the scope of this paper.

The rest of the paper is organized as follows. A ﬁxed point iterative method is ﬁrst de-veloped for integer programming in Section . Then a distributed implementation of the method and the computation of all integer points in a polytope are discussed in Section . Preliminary numerical results are presented in Section .

2 A ﬁxed point iterative method

Let

P=x∈Rn|Ax+Gw≤bfor somew∈Rp ,

whereA∈Rm×nis anm×ninteger matrix withn≥,G∈Rm×panm×pmatrix, andb a vector ofRm_{. We assume throughout this paper that}_P_{is bounded and full dimensional.} For a real numberα, let αdenote the greatest integer less than or equal toα. Forx= (x,x, . . . ,xn)∈Rn_{, let} _x_{= (} _x

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Letxmax _{= (}_xmax

 ,xmax , . . . ,xmaxn ) withxmaxj =maxx∈Pxj, j= , , . . . ,n, andxmin= (xmin , xmin_ , . . . ,xmin

n )withxminj =minx∈Pxj,j= , , . . . ,n. Thenxmin≤x≤xmaxfor allx∈P. Let D(P) =x∈Zn|xl≤x≤xu ,

wherexu₌ _xmax_and_xl₌ _xmin_{. Thus,}_D₍_P_{) contains all integer points in}_P_{. We assume} without loss of generality thatxl_<_xmin _(let_xl

i=xmini –  ifxli=xmini for somei∈N) and that

xu_ –xl_≤xu_–xl_≤ · · · ≤xu_n–xl_n,

which can be obtained by interchanging the columns ofAif necessary. Forz∈Rn_and_k_∈_N_{, let}

P(z,k) ={x∈P|xi=zi, ≤i≤kandxi≤zi,k+ ≤i≤n}.

Given an integer point y∈D(P) withy >xl, we present in the following a ﬁxed point iterative method to determine whether there is an integer pointx∗∈Pwithx∗≤ly.

Initialization: Lety₌_y_,_k₌_n_{– }_{, and}_q_{= }_. Step: Ifyq_∈_P_or_yq₌_xl_,_Stop_{; else, go to}_Step_.

Step: Ifyq_i ≤xl_ifor somei∈NorP(yq_,_k_{) =}_∅_{, go to}_Step_{; else, go to}_Step_. Step: Solve the linear program

max

n

j=k+ xj_j

subject toxj∈Pyq,k, j=k+ ,k+ , . . . ,n,

to obtain the optimal value ofxj_j, denoted byxj_j(yq₎_,_j₌_k_{+ ,}_k_{+ , . . . ,}_n_{, and go to} Step.

Step: Ifyq_j >xj_j(yq₎_{for some}_j_≥_k_{+ }_{, let}_yq+_{= (}_yq+  ,y

q+  , . . . ,y

q+ n )with

yq+_i =

yq_i if≤i≤k, xi_i(yq) ifk+ ≤i≤n,

i= , , . . . ,n, andq=q+ , and go toStep; else, letk=k+ and go toStep.

Step: Ifk= , letyq+₌_xl_and_q₌_q_{+ }_{; else, let}_yq+_{= (}_yq+  ,y

q+  , . . . ,y

q+ n )with

yq+_i = ⎧ ⎪ ⎨ ⎪ ⎩

yq_i if≤i≤k– , yq_i –  ifi=k, xu_i ifk+ ≤i≤n,

i= , , . . . ,n,q=q+ , andk=k– . Go toStep.

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Figure 2 An illustration of the method.

Example  ConsiderP={x∈R_|_Ax_≤_b_}_with

A= ⎛ ⎜ ⎜ ⎜ ⎝

–    –  –  –

   ⎞ ⎟ ⎟ ⎟ ⎠

andb= (, , , , ). We havexu_{= (, , )}_and_xl_{= (–, –, –)}_. Lety=xu_,_y₌_y_{, and}_k_{=  –  = .}

Iteration: SinceP(y_{, ) =}_∅_{, we obtain from}_Step_

y=y_,y_– ,xu_= (, –, )

andk=k–  =  –  = .

Iteration: Solving

maxx_+x_

subject toxj∈Py, , j= , ,

we obtainx_(y) = –andx_(y) = –. Sincey_=  > – =x_(y), we obtain fromStep

y=y_,x_y,x_y= (, –, –),

which is an integer point inP.

An illustration ofy_,_y_{, and}_y_{can be found in Figure .}

Example  ConsiderP={x∈R|Ax≤b}with

A= ⎛ ⎜ ⎜ ⎜ ⎝

–   –  – – – –   

⎞ ⎟ ⎟ ⎟ ⎠

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Figure 3 An illustration of the method.

Lety=xu_,_y₌_y_{, and}_k₌_n_{–  = .}

Iteration: SinceP(y_{, ) =}_∅_{, we obtain from}_Step_

y=y_,y_– ,xu_= (, –, )

andk=k–  =  –  = .

Iteration: SinceP(y_{, ) =}_∅_{, we obtain from}_Step_

y=y_– ,xu_,xu_= (, , )

andk=k–  =  –  = .

Iteration: SinceP(y_{, ) =}_∅_{, we obtain from}_Step_

y=xl= (, –, –),

which shows that there is no integer point inP.

An illustration ofy,y,y, andycan be found in Figure .

Forq= , , . . . , letkqdenote the value ofkat which the method determinesyq. Clearly, xl_–_e_≤_yq_≤_xu_with_e_{= (, , . . . , )}_∈_Rn_.

Lemma  For q= , , . . . ,

yq+≤yq or yq+≤lyq

with yq₌_yq+_.

Proof This lemma is proved in two cases. Case:Suppose that P(yq_,_k

q) =∅. Then the method will performStep. Ifkq= , we obtain fromStep thatyq+₌_xl_{, and consequently,}_yq+_≤_yq_with_yq₌_yq+_{. Assume that} kq> . Then we obtain fromStep thatyq+= (yq+ ,y

q+  , . . . ,y

q+ n )with

yq+_i = ⎧ ⎪ ⎨ ⎪ ⎩

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i= , , . . . ,n. Thus,yq+_kq <yq_kq. Therefore,yq+_≤

lyqwithyq=yq+.

Case:Suppose that P(yq_,_k_q)₌_∅_{. Then the method will repeatedly perform}_Steps__and  right before it goes toStep. Sinceyq_∈_/_P_{, as}_k_{increases one by one, it will reach a value} kq+ such thatyqj >x

j

j(yq) for somej≥kq++ . When this occurs, we obtain fromStep thatyq+_{= (}_yq+

 ,y q+  , . . . ,y

q+ n )with

yq+_i =

yq_i if ≤i≤kq+, xi

i(yq) ifkq++ ≤i≤n,

i= , , . . . ,n. Moreover, one can see fromStep that

xj_jyq≤yq_j, j=kq++ ,kq++ , . . . ,n.

Therefore,yq+≤yqwithyq=yq+. The proof is completed.

Theorem  Given an integer point y∈D(P)with y>xl,the method,within a ﬁnite num-ber of iterations,either yields an integer point x∗∈P with x∗≤ly or proves no such point exists.

Proof Letybe any given integer point inD(P) withy>xl. Suppose thaty∈/Pand there is some integer pointz_∈_P_with_z_≤

ly. We assume without loss of generality thatzis the largest integer point ofPsatisfying thatz_≤

ly. Applying mathematical induction, we show in the following thatz≤lyq,q= , , . . . .

. Consider the case ofq= . From the method, we know thatk=n– .

(a)Suppose that P(y_,_k_{) =}_∅_{. Then the method will perform}_Step_{, and we obtain from} Step thaty_{= (}_y

,y, . . . ,yn)with

y_i= ⎧ ⎪ ⎨ ⎪ ⎩

y_i if ≤i≤n– , y

i –  ifi=n– , xu

i ifi=n,

i= , , . . . ,n, andk=n– . Ifyi >zi for somei≤n– , thenz≤lywithzj<yjfor some j≤k. Assume thatyi =zi for alli≤n– . Then, fromP(y,k) =∅andz≤ly, we derive thatz_n– <y_n–since otherwiseP(y_,_k_)₌_∅_{. Therefore,}_z_≤_y_.

(b)Suppose that P(y_,_k_)₌_∅_{. Then the method will perform}_Steps__and_{. Since}_y_∈_/_P_, y

n>xnn(y). Thus, we obtain fromStepy= (y,y, . . . ,yn)with

y_i=

y_i if ≤i≤n– , xn

n(y) ifi=n,

i= , , . . . ,n, andk=n– . Ifyi >zi for somei≤n– , thenz≤lywithzj <yj for some j≤k. Assume thatyi =zi for alli≤n– . Then we derive fromz≤lythatz∈P(y,k). Thus,z

n≤ xnn(y). Therefore,z≤y.

.Induction hypothesis: For any given ≤h≤q, we assume thatyh_∈_/_P_{and that}_z_≤_yh orz≤lyhwithzj<yhj for somej≤kh.

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Case:Suppose that P(yq_,_k_{q) =}_∅_{. Then the method will perform}_Step_{. Assume that} kq= . From the method, we know thatStep must be performed at least once beforeyq is generated. Letyq_{be the point obtained by the method in the last performance of}_Step_

beforeyq_{is generated. Thus,}_k

q=kq= ,kq–= , andP(yq–,kq–) =∅. FromStep, we

know thatyq_{= (}_yq

 ,y q

 , . . . ,y q

n)with

yq

i =

yq–

i –  ifi= , xu_i if ≤i≤n,

i= , , . . . ,n. This together withP(yq–_,_k

q–) =∅and the induction hypothesis forh= q–  leads to thatz≤yq. Ifq=q, thenz≤yq=yq. Suppose thatq>q. Thenq=q+  and the method will performStepsand to generateyqright afteryq_{is generated. Since} kq=kq, the method will perform onceStepsand to generatey

q_{. With}_k

q= , we obtain fromStep thatyq= (yq,y

q , . . . ,y

q n)with

yq_i =xi_iyq_, _i_{= , , . . . ,}_n_.

Thus, it follows fromz∈P(yq_,_k

q) thatz

_≤_yq_{. Therefore,}_z_∈_P₍_yq_,_k_{q). It contradicts} withP(yq,kq) =∅. So, we must havekq> .

FromStep, we obtainyq+= (yq+ ,y q+  , . . . ,y

q+ n )with

yq+_i = ⎧ ⎪ ⎨ ⎪ ⎩

yq_i if ≤i≤kq– , yq_i –  ifi=kq, xu_i ifkq+ ≤i≤n,

i= , , . . . ,n, andkq+=kq– . Ifzi <y q

i for somei≤kq– , thenz≤lyq+withzj <y q+ j for somej≤kq+. Suppose thatyqi =zi,i= , , . . . ,kq– . Hence,zkq <y

q

kq from the induction hypothesis since otherwiseP(yq_,_k

q)=∅. Therefore,z≤yq+. Case :Suppose that P(yq_,_k

q)=∅. Then the method will repeatedly performSteps and right before it goes toStep. Sinceyq_∈_/_P_{, as}_k_{increases one by one, it will reach} a valuekq+≥kq such thatyqj >x

j

j(yq) for somej≥kq++ . When this occurs, we obtain fromStep thatyq+_{= (}_yq+

 ,y q+  , . . . ,y

q+ n )with

yq+_i =

yq_i if ≤i≤kq+, xi

i(yq) ifkq++ ≤i≤n,

i= , , . . . ,n. Ifz i <y

q

i for somei≤kq, thenz≤lyq+with zj <y q+

j for somej≤kq+. Suppose thatyq_i =z

i for alli≤kq. Thus,z≤yqfrom the induction hypothesis forh=q.

• Considerkq+=kq. Sincez∈P(yq,kq),zi≤ xii(yq)for allkq++ ≤i≤n. Therefore, z_≤_yq+_.

• Considerkq+>kq. Ifzi <y q

i for somekq<i≤kq+, then it follows fromStepsand

thatz_≤

lyq+withzj <y q+

j for somej≤kq+. Suppose thatyqi =zifor all kq<i≤kq+. Sincez≤yq,z∈P(yq,kq+)and consequently,zi ≤ xii(yq)for all kq++ ≤i≤n. Therefore,z≤yq+.

The above results together with mathematical induction show that

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From Lemma , we know thatyq+_≤_yq_or_yq+_≤

lyq withyq+=yq. Therefore, within a ﬁnite number of iterations, the method meetsz_{since there are only a ﬁnite number of} integer points in the set

z∈Zn|z≤lz≤lyandxl–e≤z≤xu .

This completes the proof.

As a corollary of Theorem , we come to the following conclusion.

Corollary  Starting from y₌_xu_,_{the method}_,_{within a ﬁnite number of iterations}_,_either yields an integer point in P or proves no such point exists.

3 Distributed computation and computing all integer points in a polytope

For any given positive integer ν, letxi, i= , , . . . ,ν, be a sequence of diﬀerent integer points inD(P) withxl_≤_x_≤

lx≤l· · · ≤lxν=xu. Then the method can easily be imple-mented in a distributed way by starting fromxi_,_i_{= , , . . . ,}_ν_{, simultaneously.}

The method can also be applied to compute all integer points inP, which is as follows.

Step: Use the method starting fromxu_{to compute an integer point in}_P_{. If no integer}

point has been found,Stop. Otherwise, lets_{be the solution found by the method}

andg= , and go toStep.

Step: Lety= (y,y, . . . ,yn)with

y_i =

sg_i ifi<n, sg_i–  ifi=n,

i= , , . . . ,n, and go toStep.

Step: Ify_∈_P_{, let}_sg+₌_y_and_g₌_g_{+ }_{, and go to}_Step_{. Otherwise, go to}_Step_. Step: Use the method starting fromy to compute an integer point inP. If no integer

point has been found,Stop. Otherwise, letsg+_{be the solution found by the method}

andg=g+ , and go toStep.

4 Numerical results

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Table 1 The market split problem

Prob. p q The method The best CPLEX strategy NumLPs Feasibility NumLPs Feasibility

1 5 40 9,640 Feasible 87,136 Feasible

6 5 40 54,525 Infeasible 677,463 Infeasible

7 5 40 105,670 Infeasible 1,644,945 Infeasible

22 5 40 16,170 Feasible 1,136,200 Feasible

a full-dimensional polytope given byP={x∈Rn|Ax≤b}, we apply the basis-reduction algorithm of [] in the same way as that in [] and in the appendix withN= , and N= ,.

Example (The market split problem) The market split problem given in [] is to deter-mine whether the system,Cx=d, has a - integer solution, whereC= (cij) is ap×q(e.g., q= (p– )) nonnegative integer matrix andd= (d,d, . . . ,dp)is an integer vector given bydi=

n

j=cij/,i= , , . . . ,p. In our numerical experiments,cij∈[, ],i= , , . . . ,p, j= , , . . . ,q, are generated randomly.

For the problem withp=  andq= , we have solved  instances using the method and the best CPLEX strategy. Numerical results for  instances of the problem are given in Table .

To demonstrate the capability of distributed computation of the method, we have im-plemented the method in a distributed way to solve the market split problem withp=  andq= . We divide the problem space into  parts and run  subproblems simultane-ously on the workstation. In the presentation of numerical results, MAX NumLPs stands for either the largest number of linear programs consumed by the method for any of the  subproblems when an instance is infeasible or the smallest number of linear programs consumed by the method for the subproblem in which a feasible solution is found. Nu-merical results for ﬁve instances of the problem are given in Table .

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Table 2 The market split problem

Prob. p q NumLPs Feasibility

1 6 50 1.25E+07 Feasible 2 6 50 4.13E+07 Infeasible 3 6 50 3.46E+07 Feasible 4 6 50 4.17E+07 Feasible 5 6 50 3.73E+07 Feasible

Table 3 The 0-1 knapsack feasibility problem

Prob. n The method The best CPLEX strategy NumLPs Feasibility NumLPs Feasibility

1 1,000 1,002 Feasible 3,023 Feasible

5 1,000 1,118 Feasible 883 Feasible

7 1,000 1,000 Infeasible 1,280 Infeasible

17 1,000 999 Infeasible 1,315 Infeasible

This paper has no intention to make a comprehensive comparison of the proposed method with the existing methods. Nevertheless, one can see from these preliminary nu-merical results that the numbers of linear programs solved by the method for most in-stances of two speciﬁc problems are less than those of the best CPLEX strategy: branch and cut. An eﬃcient implementation of the method requires a considerable amount of additional research, which is beyond the scope of this paper and will be carried out in another research project.

Appendix: Basis reduction and preconditioning

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bj,j= , , . . . ,l, the orthogonal vectorsb∗_j,j= , , . . . ,l, by the following procedure:

b∗_=b,

b∗_j =bj– j–

k=

μjkb∗k, ≤j≤l,

μjk= b_jb∗_k

b∗_k b∗_k, ≤k<j≤l.

For y∈Rn_{, let} _y _{denote the Euclidean norm of} _y_{. The following deﬁnition comes} from [].

Deﬁnition  A basisb,b, . . . ,blis called reduced if the following two conditions are sat-isﬁed:

Condition: |μjk| ≤_ for≤k<j≤l, and

Condition: b∗_j +μj,j–b∗j–≥b∗j–for <j≤l.

Given this deﬁnition, Lováz’s basis reduction algorithm in [] can be stated as follows:

Step  (Size reduction): If, for any pair ofjandkwith≤k<j≤l, Condition  is violated, then replacebjbybj–μjkbk, whereμjk=μjk–.

Step  (Interchange): If Condition  is violated for somejwith <j≤l, then interchange

bj–andbj.

Step  (Repeat): Repeat the above two steps till there is no violation of either of Conditions  and .

LetH={y∈Rm

+ |Cy=d}, whereC= (cij) is ann×minteger matrix anddis an inte-ger point ofRn_{. Without loss of generality we assume that}_gcd₍_c_i,_c_{i, . . . ,}_c_{im) =  for all} ≤i≤m. This assumption can be met by directly dividing the GCD (greatest common divisor) to each row ofC, where the GCD can be found by an extended GCD algorithm. To convert this problem into an equivalent problem of determining whether there is an integer point in a polytope given byP={x∈Rn_|_Ax_≤_b_}_{, we use the same procedure as} in []. Let

B= ⎛ ⎜ ⎝

Im 

 N NC –Nd

⎞ ⎟ ⎠,

whereIm is anm×midentity matrix and N andN are two suﬃciently large positive integers (e.g.,N= , andN= ,). Applying Lováz’s basis reduction algorithm to B, we obtain

ˆ B=

⎛ ⎜ ⎝

A b G

 N    NIn

⎞ ⎟ ⎠,

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Given any polytopeP={x∈Rn_|_Ax_≤_b_}_{, one can precondition}_A_{by applying Lováz’s} basis reduction algorithm. If there are continuous variables, one can apply Gram-Schmidt orthogonalization to the corresponding matrix.

Given positive integerspi,i= , , . . . ,m, the extended GCD with the basis reduction [] can be employed to ﬁnd their common greatest divisor, which is as follows.

Initialization: Letm= ,n= ,U= (uij)m×m=Im,di= ,i= , , . . . ,m+ ,T= (tij)m×m= m, andk= .

Step: Leti=k– and performReduce(k,i). Let

r=n

dk–dk++tk,k–

–mdk.

Ifpk–= orpk–= ,pk= , andr< , then performSwap(k)and letk=k– if k> . Otherwise, fori=k– ,k– , . . . , , performReduce(k,i). Letk=k+ and go toStep.

Step: Ifk>m,Stop. Otherwise, go toStep.

Finalization: Ifpm< , letpm= –pmandujm= –ujm,j= , , . . . ,m. LetGCD =pmand, for j= , , . . . ,m, let

h=uj,

uj=ujm,

ujm=h.

Reduce(k,i)

Ifpi= , let

r=

pk/pi–  

.

Otherwise, let

r=

tki/di+–_ if |tki|>di+,  otherwise.

Ifr= , let

pk=pk–rpi,

ujk=ujk–ruji, j= , , . . . ,m,

tki=tki–rdi+,

tkj=tkj–rtij, j= , , . . . ,i– .

Swap(k)

h=pk,

pk=pk–,

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Forj= , , . . . ,m,

h=ujk,

ujk=uj,k–,

uj,k–=h.

Forj= , , . . . ,k– ,

h=tkj,

tkj=tk–,j,

tk–,j=h.

Forj=k+ ,k+ , . . . ,m,

h=tj,k–tk,k–+tjkdk–,

h=tj,k–dk+–tjktk,k–,

tj,k–=h/dk,

tjk=h/dk,

dk=

dk–dk++tk,k–

/dk.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong}

Kong, China.2_{Department of Management Science and Engineering, Stanford University, Stanford, CA 94305-4026, USA.}

Acknowledgements

The authors are very grateful to the editor and reviewers for their valuable comments and suggestions and would like to thank Dr. Zhengtian Wu for numerical implementation of the method. This work was partially supported by GRF: CityU 101113 of Hong Kong SAR Government.

Received: 24 June 2015 Accepted: 24 September 2015

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