Pivot and Gomory Cut. A MIP Feasibility Heuristic NSERC

Transcription

1 Pivot and Gomory Cut A MIP Feasibility Heuristic Shubhashis Ghosh Ryan Hayward shubhashis@randomknowledge.net hayward@cs.ualberta.ca NSERC CGGT 2007 Kyoto Jun page 1

2 problem given a MIP, find a feasible solution MIP mixed integer program min cx s.t. Ax b, u x l A,b,c,u,l integer some x j integer CGGT 2007 Kyoto Jun page 2

3 Gomory cut f k x k + k N Z :f k f j f j (1 f k ) x k + 1 f j k N Z :f k >f j k N :a jk 0 f j a jk 1 f j x k + k N :a jk >0 a jk x k f j wrt basic feasible x of simplex tableau x j = x j + k N a jk x k row corresponding to x j B,N N Z,N j f k f j basic,non-basic variables integer,continous non-basic variables index of infeasible basic variable a jk a jk x j x j CGGT 2007 Kyoto Jun page 3

4 Gomory s algorithm solve LP relaxation (simplex alg m) while not done add Gomory cut to formulation (continue with dual simplex pivots) CGGT 2007 Kyoto Jun page 4

5 Gomory s algorithm solve LP relaxation (simplex alg m) while not done add Gomory cut to formulation (continue with dual simplex pivots) performance? correct finite inefficient (cuts usually not deep) CGGT 2007 Kyoto Jun page 5

6 Gomory s alg m tuned for optimum guarantee reoptimize after each cut added dual simplex pivots cuts do not guide pivoting CGGT 2007 Kyoto Jun page 6

7 Gomory s alg m tuned for optimum guarantee reoptimize after each cut added dual simplex pivots cuts do not guide pivoting Pivot and Gomory Cut tuned for heuristic performance no reoptimization after adding cut primal simplex pivots cuts guide pivots/restarts CGGT 2007 Kyoto Jun page 7

8 Pivot and Gomory Cut: toy example min [1 1] x s.t x CGGT 2007 Kyoto Jun page 8

9 Pivot and Gomory Cut: toy example min [1 1] x s.t x integer: {x 1, x 2 } slack/continuous: {x 3, x 4, x 5 } CGGT 2007 Kyoto Jun page 9

10 Pivot and Gomory Cut: toy example min [1 1] x s.t x integer: {x 1, x 2 } slack/continuous: {x 3, x 4, x 5 } c 19/12 7/36 1/18 [ ] x 1 x 2 = 4/3 1/4 + 1/9 1/9 x3 1/12 1/6 x 4 x /3 1/3 CGGT 2007 Kyoto Jun page 10

11 Pivot and Gomory Cut: toy example min [1 1] x s.t x integer: {x 1, x 2 } slack/continuous: {x 3, x 4, x 5 } c 19/12 7/36 1/18 [ ] x 1 x 2 = 4/3 1/4 + 1/9 1/9 x3 1/12 1/6 x 4 x /3 1/3 x = e 1 = (4/3, 1/4)... infeasible CGGT 2007 Kyoto Jun page 11

12 Pivot and Gomory Cut: toy example min [1 1] x s.t x integer: {x 1, x 2 } slack/continuous: {x 3, x 4, x 5 } c 19/12 7/36 1/18 [ ] x 1 x 2 = 4/3 1/4 + 1/9 1/9 x3 1/12 1/6 x 4 x /3 1/3 x = e 1 = (4/3, 1/4)... infeasible x 1 cut: 1 9 x x CGGT 2007 Kyoto Jun page 12

13 Pivot and Gomory Cut: toy example min [1 1] x s.t x integer: {x 1, x 2 } slack/continuous: {x 3, x 4, x 5 } c 19/12 7/36 1/18 [ ] x 1 x 2 = 4/3 1/4 + 1/9 1/9 x3 1/12 1/6 x 4 x /3 1/3 x = e 1 = (4/3, 1/4)... infeasible x 1 cut: 1 9 x x fantastic pivot? x 1 x 4 CGGT 2007 Kyoto Jun page 13

14 after pivot: x = e 2 = (0, 9/4)... infeasible CGGT 2007 Kyoto Jun page 14

15 after pivot: x = e 2 = (0, 9/4)... infeasible crossed Gomory cut G 1, so slack variable x 6 c 9/4 1/2 1/4 x 2 9/4 3/2 1/4 [ ] x1 x 4 = x x x /6 CGGT 2007 Kyoto Jun page 15

16 after pivot: x = e 2 = (0, 9/4)... infeasible crossed Gomory cut G 1, so slack variable x 6 c 9/4 1/2 1/4 x 2 9/4 3/2 1/4 [ ] x1 x 4 = x x x /6 x 2 cut: 1 6 x x CGGT 2007 Kyoto Jun page 16

17 after pivot: x = e 2 = (0, 9/4)... infeasible crossed Gomory cut G 1, so slack variable x 6 c 9/4 1/2 1/4 x 2 9/4 3/2 1/4 [ ] x1 x 4 = x x x /6 x 2 cut: 1 6 x x fantastic pivot? no CGGT 2007 Kyoto Jun page 17

18 after pivot: x = e 2 = (0, 9/4)... infeasible crossed Gomory cut G 1, so slack variable x 6 c 9/4 1/2 1/4 x 2 9/4 3/2 1/4 [ ] x1 x 4 = x x x /6 x 2 cut: 1 6 x x fantastic pivot? no good pivot? x 5 x 3 CGGT 2007 Kyoto Jun page 18

19 after pivot: x = e 3 = (0, 9 2 )... infeasible CGGT 2007 Kyoto Jun page 19

20 after pivot: x = e 3 = (0, 9 2 )... infeasible crossed G 2, so slack variable x 7 c 9/2 1/4 1/4 x 2 9/2 3/4 1/4 x = + x x 6 5/2 1/2 1/6 x 7 1/2 5/12 1/12 [ x1 x 5 ] CGGT 2007 Kyoto Jun page 20

21 after pivot: x = e 3 = (0, 9 2 )... infeasible crossed G 2, so slack variable x 7 c 9/2 1/4 1/4 x 2 9/2 3/4 1/4 x = + x x 6 5/2 1/2 1/6 x 7 1/2 5/12 1/12 [ x1 x 5 ] x 2 cut: 1 4 x x CGGT 2007 Kyoto Jun page 21

22 after pivot: x = e 3 = (0, 9 2 )... infeasible crossed G 2, so slack variable x 7 c 9/2 1/4 1/4 x 2 9/2 3/4 1/4 x = + x x 6 5/2 1/2 1/6 x 7 1/2 5/12 1/12 [ x1 x 5 ] x 2 cut: 1 4 x x fantastic pivot? no CGGT 2007 Kyoto Jun page 22

23 after pivot: x = e 3 = (0, 9 2 )... infeasible crossed G 2, so slack variable x 7 c 9/2 1/4 1/4 x 2 9/2 3/4 1/4 x = + x x 6 5/2 1/2 1/6 x 7 1/2 5/12 1/12 [ x1 x 5 ] x 2 cut: 1 4 x x fantastic pivot? no good pivot? x 7 x 5 CGGT 2007 Kyoto Jun page 23

24 after pivot: x = e 3 = (0, 9 2 )... infeasible crossed G 2, so slack variable x 7 c 9/2 1/4 1/4 x 2 9/2 3/4 1/4 x = + x x 6 5/2 1/2 1/6 x 7 1/2 5/12 1/12 [ x1 x 5 ] x 2 cut: 1 4 x x fantastic pivot? no good pivot? x 7 x 5 after pivot: x = e 4 = (0, 3)... feasible CGGT 2007 Kyoto Jun page 24

25 x 2 3x 1-4x 2 <= 3 6x 1 +4x 2 >= 9 3x 1 +4x 2 <= 18 x 1 CGGT 2007 Kyoto Jun page 25

26 x 2 G1 x 1 CGGT 2007 Kyoto Jun page 26

27 x 2 G1 x 1 CGGT 2007 Kyoto Jun page 27

28 x 2 G2 x 1 CGGT 2007 Kyoto Jun page 28

29 x 2 G2 x 1 CGGT 2007 Kyoto Jun page 29

30 x 2 G3 x 1 CGGT 2007 Kyoto Jun page 30

31 x 2 x 1 CGGT 2007 Kyoto Jun page 31

32 Pivot and Gomory cut solve LP relaxation (simplex alg m) compute Gomory cut repeat if fantastic pivot then pivot if cut crossed, add to formulation and compute new cut else if good pivot then pivot if cut crossed, add to formulation and compute new cut else cross cut (via temporary objective function) add to formulation and compute new cut CGGT 2007 Kyoto Jun page 32

33 background 53 Dantzig simplex algorithm 58 Gomory Gomory cuts 73 Chvátal Chvátal-Gomory cuts 80 Balas Martin pivot and complement 88 Bixby Lowe Cplex 96 Balas Ceria Cornuéjols Natraj Gomory cuts revisited 05 Fischetti Glover Lodi feasibility pump 05 Ghosh-H pivot and Gomory cut CGGT 2007 Kyoto Jun page 33

34 comparing MIP feasibility heuristics algorithms: data set 1: data set 2: PC, PGC, Cplex, FP MIPLib et al. (77 instances) randomly generated (500 instances) CGGT 2007 Kyoto Jun page 34

35 feasibility-hard market share instances min m s i s.t. n a ij x j + s i = b i i=1 j=1 n a ij x j s i = b i i = (m 1 + 1),..., m j=1 x j {0, 1} [j = 1,..., n] s i 0 [i = 1,..., m] if a ij uniform integer [0,99] n b i = 0.5 a ij j=1 p = 0.5 m = n 1.5 m 1 = pm when 0 > p 0.5 m 1 = pm when 0.5 < p < 1 then hard with high probability feasible with high probability CGGT 2007 Kyoto Jun page 35

36 77 MIPLib et al. instances Cplex > PGC,FP >> PC Cplex LP solver >> PGC/FP LP solver (Gnu LPK) CGGT 2007 Kyoto Jun page 36

37 77 MIPLib et al. instances Cplex > PGC,FP >> PC Cplex LP solver >> PGC/FP LP solver (Gnu LPK) 500 randomly generated instances PGC >> FP >> Cplex CGGT 2007 Kyoto Jun page 37

38 140 Cplex-D Cplex-F FP PGC 120 Number of instances at which successful (50,100) (100,100) (150,100) (100,96) (100,95) (150,55) (200,85) (250,73) 20 0 (50,0) (100,0) (150,0) (200,10) (250,3) (200,0) (250,0) Number of variables CGGT 2007 Kyoto Jun page 38

39 ARIGATO! CGGT 2007 Kyoto Jun page 39