Improved Balas and Mazzola Linearization for Quadratic 0 1 Programs with Application in a New CuttingPlane Algorithm

Improved Balas and Mazzola Linearization for Quadratic

0-1 Programs with Application in a New Cutting

Plane Algorithm

Wajeb Gharibi

Department of Computer Science, College of Computer Science & Information Systems, Jazan University, Jazan, KSA Email: [email protected]

Received December 9, 2011; revised December 26, 2011; accepted January 20, 2012

ABSTRACT

Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then considering the dual formulation. Additionally, a new cutting plane algorithm is proposed.

Keywords: Quadratic Program; Integer Program; Linearization; Cutting Plane Algorithm

1. Introduction

In this article, we consider the generalized quadratic 0-1 program given as follows

 

min

 

. . 0,1 ,

T T

n x Bx c x

s t x X



 



n n

= 0

b

2

=

b x b x

u

P (1.1)

where is an nonnegative matrix. With- out loss of generality we assume ii since

ii i ii i. Problem (P) is a generalization of uncon-

strained zero-one quadratic problems, zero-one quadratic knapsack problems, quadratic assignment problems and so on. It is a classical NP-hard problem [1].



= ij

B b 

Linearization strategies are to reformulate the zero-one quadratic programs as equivalent mixed-integer program- ming problems with additional binary variables and/or continuous variables and continuous constraints, see [2- 8]. Recently, Sherali and Smith [9] developed small linearizations for more generalized quadratic 0-1 pro- grams. Gueyea and Michelon [10] proposed a frame- work for unconstrained quadratic 0-1 programs. These linearizations are standard for employing exact algori- thms such as branch and bound. Balas and Mazzola pro- posed a small-size linearization [11] and then successfully applied it to devise exact or heuristic cutting plane al- gorithms.

In this article, we focus on new small-size tight lineari- zations. We first propose a primal version of Balas and Mazzola linearization (BML). By strengthening the line- arization and then considering the dual model, we obtain a new linearization which improves BML. As a direct application, a new cutting plane algorithm is proposed.

This article is organized as follows. In Section 2, we discuss Balas and Mazzola linearization (BML) [11] and the related primal linearization. In Section 3, we create a new approach to obtain a tighter linearization. It im- proves the primal linearization of BML in the sense that the linear programming relaxation often give tighter lower bound. In Section 4, we apply this dual lineariza- tion to devise cutting plane algorithm and compare the efficacy with that of BML. Concluding remarks are made in Section 5.

2. The Primal Model of Balas and Mazzola

Linearization

In this section, we show that Balas and Mazzola Linea- rization has a primal model.

Define a column vector with components

= max : , = 1, , ,

i ij j

j i

u b x x X i n



 



 





  (2.1)

where X is any suitable relaxation of X such that the problems (1) can be solved relatively easily.

We rewrite the objective function of (P) as

=1

.

n

i ij j i i

i j i

x b x c x



 



 

 

 

n

:= ,

i i ij j

j i

y x b x



Introducing continuous variables



(2.2)

we can obtain the following mixed 0-1 linear program





=1

1

min

. .

0, = 1, , , ,

n

i i i

i

i ij j i i

j i

i

y c x

, = 1, , ,

i

s t y b x u x

PL

y i n

x X





 

 







u i n

 

0

 bij 0

P PL

1

PL

=1

,

n n

i i i

i i

y c x

 _{ } 

 

 









(2.3)

where the two series inequality constraints follow from

the fact i i and , ,

respectively.



x 1







j ibjxjui



0 xi

Theorem 2.1. Problems ( ) and ( 1) are equivalent

in the sense that for each optimal solution to one prob- lem, there exists an optimal solution to the other problem having the same optimal objective value.

The proof is found in Appendix 1.

Remark 2.1. If we restrict (P) as the quadratic as- signment problem, the proposed linearization ( ) re- duces to Kaufman-Broeckx linearization [12,13].

Below we apply Benders’ decomposition approach to Problem (P), as in [14]. Firstly, (P) can be decomposed in the following way

( ) ₌₁

min min

x X y Y x

(2.4)

where

 

:=

, 0

n

i ij j i i i i

j i

Y x

y y b x u x u y





    





, = 1, 2,i n.

  

(2.5)

For fixed x, we dualize the first series constraints of the problem n ( ) ₌₁ using Lagrangian multi-

n y Y x



i yi

mi

pliers i (i= 1, 2,,n). We obtain the subproblem

= 1, 2, , .

j ui u xi i i

 

=1

max :

. . 0 1, ,

n ij

i j i

i

b x SP x

s t i j n







 

 

   

 



(2.6)

Note that the feasible solution region F of SP(x) does not depend on the chosen vector x. Let t be the incidence vectors of the extreme points of F (which is unit hypercube in n). Introducing

( ) ( )

,

t t t

i j bjii ui

2,, 2 := ,n T

( )

=1

,

n n

t t

i i

i i

( )

:=

j i

 





(2.7)

( ) ( )

=1

:= , = 1,

n

t t

i i i

u t



_

 (2.8)

we can see that Problem (4) is equivalent to

( )

1 ₌₁

max

min i i

x X t T

x   c x

   



  







(2.9)

by the fact that for any fixed x, the second-stage problem of (4) is a linear program-

one of the optimal solutions to the linear programming problem (6) is attained at an extreme point of F. Prob- lem (9) yields now the following mixed 0-1 linear pro- gram





=1

( ) ( ) 1

=1

min

: . . , 1 ,

.

n i i i

n

t t

i i

i

z c x

DL s t z x t T

x X

 



   







(2.10)

In some sense, linearization (DL1) can be regarded as

th

d ( quiva-

le

n see (D

1

DL ML

3. New Tight Primal and Dual

In se a new approach to establish e dual formulation of (PL1). Above we also obtained

the equivalence between (PL1) and (DL1): Theorem 2.2. Problems (PL1) an DL1) are e

nt in the sense that for each optimal solution to one problem, there exists an optimal solution to the other problem having the same optimal objective value.

Combining Theorem 2.1 with Theorem 2.2, we ca

1

L ) is equivalent to (P). In literature, linearization ( ) is known as Balas and Mazzola linearization (B ) [11].

Linearizations

this section, we propo new tight linearizations.

Define

= max : , = 0 , = 1, , ,

i ij j i

j i

v b x x X x i n



 



 





  (3.1)

= min : , = 1 , = 1, , .

i ij j i

j i

l b x x X x i n



 



 





  (3.2)

Let v and l be the vectors with components vi

an r

t x X ,1



n. For all i

= max , .

i ij j ij j i i i i i

j i j i

d li espectively, = 1,i ,n.

Lemma 3.1. [15] Le



0 = 1,,n,

( ) ₌₁

ny Y x



ni yi

ming whose dual formulation is just (6) and the fact that mi

x b x b x v x v l x

 

 

 

 

 





(3.3)

Therefore, the new linearization reads





=1

2

min

. . , = 1, , ,

, = 1, , . .

i i i

i

i ij j i i i

j i

i i i

y c x

n

s t y b x v x v i n

PL

y l x i n

x X





  

 









(3.4)

Under the linear transformations

i

x











=1

2

min

. .

0, = 1, , , .

n

i i i i

i

i ij j i i i

j i

i

t l c x



, = 1, , ,

i

s t t b x v l x

PL

t i n

x X

 _



  

 







v i n

 

(3.5)

As a corollary of Lemma 3.1, we have

2)) and (P)

ar o ptim

i



Theorem 3.1. Problems (PL2) (or (PL

e equivalent in the sense tha r each o al solution to one problem, there exists an optimal solution to the other problem having the same optimal objective value.

Continuously relaxing linearizations (PL1) and (PL2),

t f

.e., replacing X with X, we obtain linear program- ming lower bou s for (P enoted by v R



PL1



and



2



v RPL respectively. (PL2 ) is n than

following sense.

eorem 3.2. v R



PL1



 2

nd

weaker ( ) in t



.

ow any feasi solution of (P

a ), d

v R

ot



PL 1

PL

Th

he

Proof. It is sufficient to sh ble

2

L ) is also feasible in (PL1), which follows from the

fact that viui, xi[0,1] nd li0 since B=

 

bij

is nonneg

Next, we consid ative. □

er the odel of (PL2). As the

fo f (

the ).

dual m

rmulation of (PL2) is similar to that o PL1), we

immediately have dual model based on (DL2









=1

( )

=1

min

2 : . .

,

( )

, 1 ,

i i i

i

n _{t t}

i i

i

z l c x

DL s t z x t T

x X

 

 

   







(3.6)

where

n





( )

,

t t

i i i

b  v l (3.7)

( ) _{( )}

:=

t

i _{j ji}

j i

 





( ) _{( )}

=1

:= .

n

t _t

i i i

v





 (3.8)

Similarly to Theorem 2.2, we have

2

) and (DL2)

ar f utio

4. Cutting Plane Algorithms Based on Dual

W tting plane algorithm based on

obl

bset

Theorem 3.3. Problems (PL2)(or (PL

opt

e equivalent in the sense th or each imal sol n to one problem, there exists an optimal solution to the other problem having the same optimal objective value.

at

Linearizations

e first establish cu

(DL1). As in any decomposition approach the master

pr em (DL1) is not solved for all restrictions





( )

=1 , 1

n t

z



i i( )t xi  t T , but only for a su



t1 t r



of indices. We de lem by

soluti

note the restricted master

 

1

r

DL .

Getting a on

prob

 for the restricted master pro- blem, the subproblem P x

 

 is solved, which yields

( 1)

:= = 1, 2, , .

r

S

, ,

i x i ji )

(r 1) 

n

  _

(4.1



is an optimal solution of the subproblem

 

SP x

and the

because of the definition of the constants ui

constraints 0i 1. A new cut

( 1) ( 1)

=1

n

r r

i i i

z



  x   (4.2)

is added to the current

 

₁r

DL . Thus we get



1



1

r

DL . n valu

The objection functio e z of SP x





is an p- u per bound for (P), whereas the objective function value

z of the master problem

 

₁r

DL is a lower bound. If

= z, stop and return an op solution. s is the flow of cutting plane algorith

z timal

Thi m. Below we

sh

4.1. Assume that

ow the finite convergence. The proof is found in Ap- pendix 2.

Lemma x is an optimal solution of

 

1

r

L and (r 1)_:=

i i

D   x. For any s>r, x cannot be timal so

an op lution of

 

₁s

DL unle t i the optimal solution of (P).

From the abov

ss i s

e lemma, we have the convergence re- su

utting plane algorithm stops in

rithm based on (DL2 ) can be

si ) con

q





lt proved in Appendix 3.

Theorem 4.1. The above c

a finite number of steps and returns the optimal solution of (P).

Cutting plane algo

milarly devised. To compare with (DL1 veniently,

instead of (DL2), we use the following e uivalent model





=1

( ) ( )

2

=1

min

: . . , 1 ,

.

i i i

n _t

t

i i i

i

z c x

DL s t z l x t T

x X

 



     







(4.3)

Theorem 4.2. Assume l_i> 0 for all i= 1, 2,,n.

Th gram

1

r

DL

5. Conclusion

focus on the generalized quadratic 0-1

rm

an nuo

n

e restricted master pro ( 2

r

DL) gives a lower bound strictly better than that of ( ) until the cutting plane algorithm stops.

In this article, we

program, denoted by (P). We propose a linearization (PL1) for (P) and show that it can be regarded as a dual

fo ulation of Balas and Mazzola linearization (BML), denoted ( DL1 ). By applying a new approach, we

establish a tight linearization (PL2) of the same size. We

proved (PL2) is not weaker th (PL1) in the sense that

the conti us linear programming relaxation of (PL2)

gives tighter lower bound than that of (PL1). The dual

linearizations of (PL1) and (PL2), (DL1) and (DL2) are

in

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some weak assumptions.

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Appendix

1)Proof of Theorem 2.1

Let x be any feasible solution to (P). It is easy to verify that

 

x y, PL1

=

i i j iij j

y x



_b x PL



*_, *



is feasible in ( ) with the same objective value, where . As a conse-

quence, the optimal objective value of ( 1) gives a

lower bound for (P). It is sufficient to show that if x y is an optimal solution to (PL1),x





i ui





= 1

i

x

is optimal in (P) with the same objective value. We notice that

= max

i j i i

y



_b

=

i j i ij j

, 0

jxj u xi

_{ }

. If , we have

y



_b x

=

i i

y x



, otherwise, , . As a con-

clusion, which implies

= 0 y_i= 0



j i





i i j

  





i x



ij j

b x



=1 = =1

n n

i y i x





j ib xij . That is, x



is a feasible

solution to (P) whose objective value equals a lower bound. Therefore, x

 

is optimal in (P) and both the optimal objective values are equal. □

2) Proof of Lemma 4.1

Denote the optimal objective function value of any master problem

1

s

DL by zs, which is a lower bound

for (P). If x is also an optimal solution of

 

DL₁s for some s>r

( 1) ( 1)

,

r r

i x   _{ } 

 

, it follows that

=1

n

s i

i

z 



(2.1)

since DLs₁

( 1) ( 1)

=1

( 1) ( 1) ( 1)

=1 =1

=1 =1

=1

=

=

= .

n

r r

i i

i

n n

r r r

contains the constraint (2). The left-hand side of (4) is a lower bound for (P) while the right-hand side of (4) corresponds to a feasible objective function value of (P), which can be shown as follows:

j ji i i i i i

i j i i

n n

j ji i i i i i

i j i i

n

ji j i

i j i

x

b u x u

x b x u x x u

b x x

 

  

 

  









 

 

 

 

 

 

 

 

 

 

 



 



 



 





   

 

x Therefore (4) holds as equality and 

DL

must be the optimal solution of (P). □

3)Proof of Theorem 4.2

If the cutting plane algorithm based on ( 2) has not

stopped at step r, the optimal solution x

( )t

must be different from  for any , i.e., there exist index t such that

1 t r

i



( )



1 t > 0

t

x



 

2

r

DL

i_t i . Then the right-

hand side of ( ) satisfies













( ) ( )

=1

( ) ( ) ( )

=1 =1

( ) ( ) ( ) ( )

=1 =1 =1

( ) ( ) ( )

=1 =1

=

1 ,

> ,

n

t t

i i i

i

n n

t t t

j ji i i i i i i i

i j i i

n n n

t t t t

j ji i i i i i i i i

i j i i i

n n

t t t

j ji i i i i i

i j i i

l x

b v l l x v

b u x u l x

b u x u

 

  

   

  







 

 

   

 

 

 

 _  _   

 

 

 

 

 



 



 





 



1 t r

(3.1) for any  

2

r

DL

. Therefore the objective function value of ( ) is strictly larger than that of

 

1 . □

r

DL