A Metaheuristic Optimization Algorithm for Binary Quadratic Problems

(1)

A Metaheuristic Optimization Algorithm for

Binary Quadratic Problems

Otto Nissfolk

CENTER OF EXCELLENCE IN

OPTIMIZATION AND SYSTEMS ENGINEERING

ÅBO AKADEMI UNIVERSITY

(2)

I

_{Problem Formulation}

.

The Quadratic Assignment Problem

.

QAP with rank-1 flow matrix

.

Convex QAP with rank-1 flow matrix

.

Metaheuristic Algorithm

I

_Testproblems

.

Taixxc

(3)

min

x∈X

n

¼

i=1

n

¼

j=1

n

¼

k=1

n

¼

l=1

f

ik

d

jl

x

ij

x

kl

+

n

¼

i=1

n

¼

j=1

c

ij

x

ij

X

= {x |

n

¼

j=1

x

ij

= 1

i ∈ N

n

¼

i=1

x

ij

= 1

j ∈ N

x

ij

∈ {0

, 1}

i

, j ∈ M }

(4)

Problem Formulation: QAP with rank-1 flow matrix

4 | 16

n

¼

i=1

n

¼

j=1

n

¼

k=1

n

¼

l=1

f

_ik

d

_jl

x

_ij

x

_kl

= trace(DXFX

T

)

= trace(DXqq

T

X

T

)

= trace(q

T

X

T

DXq)

= trace(Xq

T

DXq)

= trace(y

T

Dy) = y

T

Dy

(5)

Problem Formulation: QAP with rank-1 flow matrix

4 | 16

n

¼

i=1

n

¼

j=1

n

¼

k=1

n

¼

l=1

f

_ik

d

_jl

x

_ij

x

_kl

= trace(DXFX

T

)

F=qq

T

= trace(q

X

DXq)

= trace(Xq

T

DXq)

= trace(y

T

Dy) = y

T

Dy

(6)

Problem Formulation: QAP with rank-1 flow matrix

4 | 16

n

¼

i=1

n

¼

j=1

n

¼

k=1

n

¼

l=1

f

_ik

d

_jl

x

_ij

x

_kl

= trace(DXFX

T

)

F=qq

T

= trace(DXqq

T

X

T

)

= trace(q

T

X

T

DXq)

= trace(Xq

T

DXq)

(7)

n

¼

i=1

n

¼

j=1

n

¼

k=1

n

¼

l=1

f

_ik

d

_jl

x

_ij

x

_kl

= trace(DXFX

T

)

F=qq

T

= trace(DXqq

T

X

T

)

= trace(q

T

X

T

DXq)

= trace(Xq

T

DXq)

= trace(y

T

Dy) = y

T

Dy

(8)

min

x∈X

,y∈

n

y

T

_Dy

subject to

y

_i

=

n

¼

j=1

x

_ij

q

_j

∀ i

n

¼

i=1

y

i

=

n

¼

j=1

q

j

(9)

min

x∈X

, y,z∈

n

y

T

_{(D + Diag(u))y − u}

T

_z

subject to

y

_i

=

n

¼

j=1

x

_ij

q

_j

∀ i

z

i

=

n

¼

j=1

x

ij

q

j

2 ∀ i

n

¼

i=1

y

_i

=

n

¼

j=1

q

_j

(10)

min x

T

(D + diag(u))x − u

T

x

subject to

n

¼

i=1

x

i

= k

iteration constraint

x

iter

= 1

(11)

x

=







1

0

1

0

1

0

1

0 





(12)

x

=







1

0

1

0

1

0

1

0 





x

1 =







1

0

1

0

1

0

1

0 





(13)

x

=







1

0

1

0

1

0

1

0 





x

1 =







1

0

1

0

1

0

1

0 





x

2 =







1

0

1

0

1

0

1

0 





(14)

x

=







1

0

1

0

1

0

1

0 





x

1 =







1

0

1

0

1

0

1

0 





x

2 =







1

0

1

0

1

0

1

0 





x

3 =







1

0

1

0

1

0

1

0 





(15)

x

=







1

0

1

0

1

0

1

0 





x

1 =







1

0

1

0

1

0

1

0 





x

2 =







1

0

1

0

1

0

1

0 





x

3 =







1

0

1

0

1

0

1

0 





x

r1

=







1

0

1

0

1

0

1

0 





(16)

x

=







1

0

1

0

1

0

1

0 





x

1 =







1

0

1

0

1

0

1

0 





x

2 =







1

0

1

0

1

0

1

0 





x

3 =







1

0

1

0

1

0

1

0 





x

r1

=







1

0

1

0

1

0

1

0 





x

r

2 =







1

0

1

0

1

0

1

0 





(17)

T

rstu

=

max

v

,w∈{−1,0,1}

1 (r − t + nv)

2 _{+ (s − u + nw)}

2 f

ij

=

(

1 if i ≤ m and j ≤ m

0 otherwise

d

ij

= d

n(r−1)+s n(t−1)+u

= T

rstu

(18)

10

0

10

1

10

2

10

3

10

4

10

5

10

6

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Time in seconds

Objective

value

Objective function value vs. time

1 BK

(19)

10

0

10

1

10

2

10

3

10

4

10

5

10

6

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Time in seconds

Objective

value

Objective function value vs. time

1

5 BK

(20)

10

0

10

1

10

2

10

3

10

4

10

5

10

6

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Time in seconds

Objective

value

Objective function value vs. time

1

5

10 BK

(21)

10

0

10

1

10

2

10

3

10

4

10

5

10

6

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Time in seconds

Objective

value

Objective function value vs. time

1

5

10

12 BK

(22)

10

0

10

1

10

2

10

3

10

4

10

5

10

6

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Time in seconds

Objective

value

Objective function value vs. time

1

5

10

12

16 BK

(23)

10

0

10

1

10

2

10

3

10

4

10

5

10

6

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Time in seconds

Objective

value

Objective function value vs. time

1

5

10

12

16

18 BK

(24)

10

0

10

1

10

2

10

3

10

4

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

1 BK

(25)

10

0

10

1

10

2

10

3

10

4

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

1

5 BK

(26)

10

0

10

1

10

2

10

3

10

4

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

1

5

10 BK

(27)

10

0

10

1

10

2

10

3

10

4

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

1

5

10

12 BK

(28)

10

0

10

1

10

2

10

3

10

4

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

1

5

10

12

16 BK

(29)

10

0

10

1

10

2

10

3

10

4

4.47

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

1

5

10

12

16

18 BK

(30)

10

0

10

1

10

2

4.48

4.49

4.50 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

20 BK

(31)

10

0

10

1

10

2

4.48

4.49

4.50 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

20

22 BK

(32)

10

0

10

1

10

2

4.48

4.49

4.50 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

20

22

24 BK

(33)

10

0

10

1

10

2

4.48

4.49

4.50 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

20

22

24

26 BK

(34)

10

0

10

1

10

2

4.48

4.49

4.50 ·10

7 Number of iterations

Objective

value

Objective function value vs. iteration count

20

22

24

26

28 BK

(35)

10

5

10

6

4.48

4.49

4.50 ·10

7 Time

Objective

value

Objective function value vs. time

20 BK

(36)

10

5

10

6

4.48

4.49

4.50 ·10

7 Time

Objective

value

Objective function value vs. time

20

22 BK

(37)

10

5

10

6

4.48

4.49

4.50 ·10

7 Time

Objective

value

Objective function value vs. time

20

22

24 BK

(38)

10

5

10

6

4.48

4.49

4.50 ·10

7 Time

Objective

value

Objective function value vs. time

20

22

24

26 BK

(39)

10

5

10

6

4.48

4.49

4.50 ·10

7 Time

Objective

value

Objective function value vs. time

20

22

24

26

28 BK

(40)

1

10

100 1000

10000

4.48

4.49

4.50

4.51

4.52

4.53 ·10

7 Number of iterations

Objective

value

Solution spread

min

mean

max

(41)

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(42)

The end of the presentation

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