# A Metaheuristic Optimization Algorithm for Binary Quadratic Problems

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1 OSE SEMINAR 22 A Metaheuristic Optimization Algorithm for Binary Quadratic Problems Otto Nissfolk CENTER OF EXCELLENCE IN OPTIMIZATION AND SYSTEMS ENGINEERING ÅBO AKADEMI UNIVERSITY ÅBO, NOVEMBER 29 th 22

2 2 6 Table of contents Problem Formulation The Quadratic Assignment Problem QAP with rank- flow matrix Convex QAP with rank- flow matrix Metaheuristic Algorithm Testproblems Taixxc Results

3 Problem Formulation: The Quadratic Assignment Problem 3 6 min x X n n n n n n f ik d jl x ij x kl + c ij x ij i= j= k= l= i= j= n X = {x x ij = i N j= n x ij = i= j N x ij {,} i,j M}

4 Problem Formulation: QAP with rank- flow matrix 4 6 n n n n f ik d jl x ij x kl = trace(dxfx T ) i= j= k= l=

5 Problem Formulation: QAP with rank- flow matrix 4 6 n n n n f ik d jl x ij x kl = trace(dxfx T ) i= j= k= l= F=qq T

6 Problem Formulation: QAP with rank- flow matrix 4 6 n n n n f ik d jl x ij x kl = trace(dxfx T ) i= j= k= l= F=qq T = trace(dxqq T X T ) = trace(q T X T DXq) = trace(xq T DXq)

7 Problem Formulation: QAP with rank- flow matrix 4 6 n n n n f ik d jl x ij x kl = trace(dxfx T ) i= j= k= l= 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 Problem Formulation: QAP with rank- flow matrix 5 6 min x X,y n y T Dy subject to n y i = x ij q j i j= n y i = n i= j= q j

9 Problem Formulation: Convex QAP with rank- flow matrix 6 6 min x X, y,z n y T (D + Diag(u))y u T z subject to n y i = x ij q j i j= n z i = x ij qj 2 i j= n y i = n i= j= q j

10 Problem Formulation: Metaheuristic Algorithm 7 6 minx T (D + diag(u))x u T x subject to n x i = k i= iteration constraint x iter =

11 Problem Formulation: Metaheuristic Algorithm 8 6 x =

12 Problem Formulation: Metaheuristic Algorithm 8 6 x = x =

13 Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 =

14 Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 = x 3 =

15 Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 = x 3 = x r =

16 Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 = x 3 = x r = x r2 =

17 Testproblems: Taixxc 9 6 T rstu = max v,w {,,} (r t + nv) 2 + (s u + nw) 2 { if i m and j m f ij = otherwise d ij = d n(r )+s n(t )+u = T rstu where (r,s) are the coordinates for i and (t,u) are the coordinates for j

18 Results Objective function value vs. time Time in seconds

19 Results Objective function value vs. time Time in seconds

20 Results Objective function value vs. time Time in seconds

21 Results Objective function value vs. time Time in seconds

22 Results Objective function value vs. time Time in seconds

23 Results Objective function value vs. time Time in seconds

24 Results 6 7 Objective function value vs. iteration count Number of iterations

25 Results 6 7 Objective function value vs. iteration count Number of iterations 5

26 Results 6 7 Objective function value vs. iteration count Number of iterations 5

27 Results 6 7 Objective function value vs. iteration count Number of iterations 5 2

28 Results 6 7 Objective function value vs. iteration count Number of iterations 5 2 6

29 Results 6 7 Objective function value vs. iteration count Number of iterations

30 Results Objective function value vs. iteration count Number of iterations

31 Results Objective function value vs. iteration count Number of iterations

32 Results Objective function value vs. iteration count Number of iterations

33 Results Objective function value vs. iteration count Number of iterations

34 Results Objective function value vs. iteration count Number of iterations

35 Results Objective function value vs. time Time

36 Results Objective function value vs. time Time

37 Results Objective function value vs. time Time

38 Results Objective function value vs. time Time

39 Results Objective function value vs. time Time

40 Results Solution spread min mean max Number of iterations Figure : Spread of the solutions with m = 5

41 Some references 5 6 Alain Billionnet, Sourour Elloumi, and Marie-Christine Plateau. Improving the performance of standard solvers for quadratic - programs by a tight convex reformulation: The qcr method. Discrete Appl. Math., 57:85 97, March 29. R.E. Burkard, E. Cela, P.M. Pardalos, and L.S. Pitsoulis. Handbook of Combinatorial Optimization, volume C. S. Edwards. A branch and bound algorithm for the koopmans-beckmann quadratic assignment problem. Combinatorial Optimization II, 3:35 52, 98. Tjalling C. Koopmans and Martin Beckmann. Assignment problems and the location of economic activities. Econometrica, 25():pp , 957. Otto Nissfolk, Ray Pörn, Tapio Westerlund, and Fredrik Jansson. A mixed integer quadratic reformulation of the quadratic assignment problem with rank- matrix. In Iftekhar A. Karimi and Rajagopalan Srinivasan, editors, th International Symposium on Process Systems Engineering, volume 3 of Computer Aided Chemical Engineering, pages Elsevier, 22. É.D. Taillard. Comparison of iterative searches for the quadratic assignment problem. Location Science, 3(2):87 5, 995.

42 6 6 The end of the presentation Thank you for listening! Questions?

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