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 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
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}
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=
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
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)
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
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
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
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 =
Problem Formulation: Metaheuristic Algorithm 8 6 x =
Problem Formulation: Metaheuristic Algorithm 8 6 x = x =
Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 =
Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 = x 3 =
Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 = x 3 = x r =
Problem Formulation: Metaheuristic Algorithm 8 6 x = x = x 2 = x 3 = x r = x r2 =
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
Results 6 4.53 7 Objective function value vs. time 4.52 4.5 4.5 4.47 2 3 4 5 6 Time in seconds
Results 6 7 4.53 4.52 4.5 4.5 Objective function value vs. time 5 4.47 2 3 4 5 6 Time in seconds
Results 6 7 4.53 4.52 4.5 4.5 Objective function value vs. time 5 4.47 2 3 4 5 6 Time in seconds
Results 6 7 4.53 4.52 4.5 4.5 Objective function value vs. time 5 2 4.47 2 3 4 5 6 Time in seconds
Results 6 7 4.53 4.52 4.5 4.5 Objective function value vs. time 5 2 6 4.47 2 3 4 5 6 Time in seconds
Results 6 7 4.53 4.52 4.5 4.5 Objective function value vs. time 5 2 6 8 4.47 2 3 4 5 6 Time in seconds
Results 6 7 Objective function value vs. iteration count 4.53 4.52 4.5 4.5 4.47 2 3 4 Number of iterations
Results 6 7 Objective function value vs. iteration count 4.53 4.52 4.5 4.5 4.47 2 3 4 Number of iterations 5
Results 6 7 Objective function value vs. iteration count 4.53 4.52 4.5 4.5 4.47 2 3 4 Number of iterations 5
Results 6 7 Objective function value vs. iteration count 4.53 4.52 4.5 4.5 4.47 2 3 4 Number of iterations 5 2
Results 6 7 Objective function value vs. iteration count 4.53 4.52 4.5 4.5 4.47 2 3 4 Number of iterations 5 2 6
Results 6 7 Objective function value vs. iteration count 4.53 4.52 4.5 4.5 4.47 2 3 4 Number of iterations 5 2 6 8
Results 2 6 7 Objective function value vs. iteration count 4.5 2 2 Number of iterations
Results 2 6 7 Objective function value vs. iteration count 4.5 2 22 2 Number of iterations
Results 2 6 7 Objective function value vs. iteration count 4.5 2 22 24 2 Number of iterations
Results 2 6 7 Objective function value vs. iteration count 4.5 2 22 24 26 2 Number of iterations
Results 2 6 7 Objective function value vs. iteration count 4.5 2 22 24 26 28 2 Number of iterations
Results 3 6 7 4.5 Objective function value vs. time 2 5 6 Time
Results 3 6 7 4.5 Objective function value vs. time 2 22 5 6 Time
Results 3 6 7 4.5 Objective function value vs. time 2 22 24 5 6 Time
Results 3 6 7 4.5 Objective function value vs. time 2 22 24 26 5 6 Time
Results 3 6 7 4.5 Objective function value vs. time 2 22 24 26 28 5 6 Time
Results 4 6 7 4.53 4.52 4.5 4.5 Solution spread min mean max Number of iterations Figure : Spread of the solutions with m = 5
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 3. 998. 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. 53 76, 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 36 364. Elsevier, 22. É.D. Taillard. Comparison of iterative searches for the quadratic assignment problem. Location Science, 3(2):87 5, 995.
6 6 The end of the presentation Thank you for listening! Questions?