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 MarieChristine 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 koopmansbeckmann 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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