The Strength of Multi-Row Relaxations
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1 The Strength of Multi-Row Relaxations Quentin Louveaux 1 Laurent Poirrier 1 Domenico Salvagnin 2 1 Université de Liège 2 Università degli studi di Padova August 2012
2 Motivations Cuts viewed as facets of relaxations of the problem In particular, multi-row relaxations Focus on exact separation Evaluate any relaxation
3 Plan A. Separation over arbitrary mixed-integer sets B. Application to two-row relaxations
4 A. Separation over arbitrary mixed-integer sets
5 Problem Given a mixed-integer set P R n, a point x R n, find (α, α 0 ) R n+1 such that α T x α 0 is a valid inequality for P that separates x, or show that x conv(p).
6 Problem Given a mixed-integer set P R n, a point x R n, find (α, α 0 ) R n+1 such that α T x α 0 is a valid inequality for P that separates x, or show that x conv(p).
7 General framework Solve the optimization problem min x T α s.t. x T α α 0 for all x P <norm.> Let (ᾱ, ᾱ 0 ) be the optimal solution. If x T ᾱ < ᾱ 0, then (ᾱ, ᾱ 0 ) separates x. If x T ᾱ ᾱ 0, then x conv(p).
8 General framework Solve the optimization problem min x T α s.t. x T α α 0 for all x P <norm.> Let (ᾱ, ᾱ 0 ) be the optimal solution. If x T ᾱ < ᾱ 0, then (ᾱ, ᾱ 0 ) separates x. If x T ᾱ ᾱ 0, then x conv(p).
9 Row generation 1. Consider the relaxation of the separation problem min x T α s.t. x T α α 0 for all x S P <norm.> (master) Let (ᾱ, ᾱ 0 ) be an optimal solution. 2. Now solve the MIP min s.t. ᾱ T x x P (slave) and let x p be a finite optimal solution. If ᾱ T x p ᾱ 0, then (ᾱ, ᾱ 0 ) is valid for P. If ᾱ T x p < ᾱ 0, then S := S {x p }.
10 Row generation 1. Consider the relaxation of the separation problem min x T α s.t. x T α α 0 for all x S P <norm.> (master) Let (ᾱ, ᾱ 0 ) be an optimal solution. 2. Now solve the MIP min s.t. ᾱ T x x P (slave) and let x p be a finite optimal solution. If ᾱ T x p ᾱ 0, then (ᾱ, ᾱ 0 ) is valid for P. If ᾱ T x p < ᾱ 0, then S := S {x p }.
11 Row generation 1. Consider the relaxation of the separation problem min x T α s.t. x T α α 0 for all x S P <norm.> (master) Let (ᾱ, ᾱ 0 ) be an optimal solution. 2. Now solve the MIP min s.t. ᾱ T x x P (slave) and let x p be a finite optimal solution. If ᾱ T x p ᾱ 0, then (ᾱ, ᾱ 0 ) is valid for P. If ᾱ T x p < ᾱ 0, then S := S {x p }.
12 Computational example Instance: bell3a Constraints: 123 Variables: 133 (71 integer: 32 general, 39 binaries) Models: 82 five-row models read from an optimal tableau Cuts: Gap closed: 37 (17 tight at the end) 59.02% (from 39.03% by GMIs) Time: s Iterations:
13 Two-phases: Phase one x between bounds {}}{ { x at bounds }} { x : x B } x N {{ } fix to bound α : α B α N. }{{}
14 Two-phases: Phase one x between bounds {}}{ { x at bounds }} { x : x B } x N {{ } fix to bound α : }{{}. } α B {{ } α N find
15 Two-phases: Phase two x between bounds {}}{ { x at bounds }} { x : x B } x N {{ } fix to bound α : }{{}. } α B {{ } } α N {{ } fixed lift
16 Two-phases summary The feasible region of phase-1 slave is P {x : x N = x N } phase-1 separates iff phase-2 separates whenever x conv(p), phase-2 is avoided Optimal objective function values are the same phase-2 master objective function is 0
17 Two-phases summary The feasible region of phase-1 slave is P {x : x N = x N } phase-1 separates iff phase-2 separates whenever x conv(p), phase-2 is avoided Optimal objective function values are the same phase-2 master objective function is 0
18 Two-phases summary The feasible region of phase-1 slave is P {x : x N = x N } phase-1 separates iff phase-2 separates whenever x conv(p), phase-2 is avoided Optimal objective function values are the same phase-2 master objective function is 0
19 Computational example (2-phases) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases Time: s s Iterations:
20 Computational example (2-phases) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases Time: s s Iterations:
21 Lifting binary variables x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x N α : } α B {{ } α Nbin α N fixed
22 Lifting binary variables x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x N α : } α B {{ } α Nbin }{{} α N fixed lift
23 Lifting binary variables x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x N α : } α B {{ } α Nbin }{{} α N fixed fixed
24 Computational example (lifting binaries) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting Time: s s s Iterations:
25 Computational example (lifting binaries) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting Time: s s s Iterations:
26 Sequential phase-2 ( phase-s ) x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x k x N }{{} (fixed to bnd) α : } α B {{ α Nbin } α k α N }{{} fixed zero
27 Sequential phase-2 ( phase-s ) x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x k x N }{{} (fixed to bnd) α : } α B {{ α Nbin } α k α N }{{} fixed zero
28 Sequential phase-2 ( phase-s ) x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x k x N }{{} (fixed to bnd) α : } α B {{ α Nbin } α k }{{} α N }{{} fixed lift zero
29 Sequential phase-2 ( phase-s ) x between bounds {}}{ { x at bounds }} { binary {}}{ x : x B x Nbin x k x N }{{} (fixed to bnd) α : } α B {{ α Nbin } α k }{{} α N }{{} fixed fixed zero
30 Computational example (phase S) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting phase S Time: s s s 5.84s Iterations:
31 Computational example (phase S) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting phase S Time: s s s 5.84s Iterations:
32 Solver tricks: callbacks Solving slave MIPs min ᾱ T x s.t. x P, Feasible solution ˆx with ᾱ T ˆx < ᾱ 0 ˆx can be added to S. Dual bound z reaches ᾱ 0, (ᾱ, ᾱ 0 ) is valid for P.
33 Solver tricks: callbacks Solving slave MIPs min ᾱ T x s.t. x P, Feasible solution ˆx with ᾱ T ˆx < ᾱ 0 ˆx can be added to S. Dual bound z reaches ᾱ 0, (ᾱ, ᾱ 0 ) is valid for P.
34 Solver tricks: callbacks Solving slave MIPs min ᾱ T x s.t. x P, Feasible solution ˆx with ᾱ T ˆx < ᾱ 0 ˆx can be added to S. Dual bound z reaches ᾱ 0, (ᾱ, ᾱ 0 ) is valid for P.
35 Computational example (solver tricks) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting phase S cb Time: s s s 5.84s 4.65s Iterations:
36 Computational example (solver tricks) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting phase S cb Time: s s s 5.84s 4.65s Iterations:
37 Computational example (summary) (bell3a, 82 five-row models, 37 cuts, 59.02%gc) original 2-phases lifting phase S cb Time: s s s 5.84s 4.65s Iterations:
38 B. Application to two-row relaxations
39 Objectives Mainly, evaluate and compare the intersection cut model a few strengthenings of it a full two-row model
40 Two-row relaxation: which models? We are still far from a closure What reasonable set of two-models can we select? All models read from a simplex tableau O(m 2 ) two-row models
41 Two-row relaxation: which models? We are still far from a closure What reasonable set of two-models can we select? All models read from a simplex tableau O(m 2 ) two-row models
42 Two-row relaxation: which models? We are still far from a closure What reasonable set of two-models can we select? All models read from a simplex tableau O(m 2 ) two-row models
43 all two-row models: separation loop Let x LP optimium Read the two-row models from optimal tableau. Read and add GMIs from that tableau. do { Let x new LP optimum. Separate x with the two-row models. } while (cuts were found).
44 all two-row models: separation loop Let x LP optimium Read the two-row models from optimal tableau. Read and add GMIs from that tableau. do { Let x new LP optimum. Separate x with the two-row models. } while (cuts were found).
45 all two-row models: separation loop Let x LP optimium Read the two-row models from optimal tableau. Read and add GMIs from that tableau. do { Let x new LP optimum. Separate x with the two-row models. } while (cuts were found).
46 all two-row models: results Computations on the 62 MIPLIB 3.0 (preprocessed) instances for which (a). the integrality gap is not zero, and (b). the optimal MIP solution is known.
47 all two-row models: results We have a result for 55/62 instances (4 numerical, 3 memory). cuts gc% GMI % All 2-row % For 13 instances, the separation is exact.
48 all two-row models: results We have a result for 55/62 instances (4 numerical, 3 memory). cuts gc% GMI % All 2-row % For 13 instances, the separation is exact.
49 all two-row models: results We have a result for 55/62 instances (4 numerical, 3 memory). cuts gc% GMI % All 2-row % For 13 instances, the separation is exact.
50 Heuristic selection of two-row models Issue: O(m 2 ) is already a large number of models Hypothesis: Not all models are necessary to achieve good separation Rationale: MIPLIB models are mostly sparse Multi-cuts from rows with no common support are linear combinations of the corresponding one-row cuts
51 Heuristic selection of two-row models Issue: O(m 2 ) is already a large number of models Hypothesis: Not all models are necessary to achieve good separation Rationale: MIPLIB models are mostly sparse Multi-cuts from rows with no common support are linear combinations of the corresponding one-row cuts
52 Heuristic selection of two-row models Issue: O(m 2 ) is already a large number of models Hypothesis: Not all models are necessary to achieve good separation Rationale: MIPLIB models are mostly sparse Multi-cuts from rows with no common support are linear combinations of the corresponding one-row cuts
53 Heuristic selection of two-row models: results With an arbitrary limit of m two-row models, we have a result for 58/62 instances (1 numerical, 3 memory). On the 55 common results, cuts gc% GMI % All 2-row % Heuristic % For 25 instances, the separation is exact.
54 Heuristic selection of two-row models: results With an arbitrary limit of m two-row models, we have a result for 58/62 instances (1 numerical, 3 memory). On the 55 common results, cuts gc% GMI % All 2-row % Heuristic % For 25 instances, the separation is exact.
55 Heuristic selection of two-row models: results With an arbitrary limit of m two-row models, we have a result for 58/62 instances (1 numerical, 3 memory). On the 55 common results, cuts gc% GMI % All 2-row % Heuristic % For 25 instances, the separation is exact.
56 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 2.4 x 2 x 3 x 5 +2 x 6 = 0.2 x 1, x 2, x 3, x 4, x 5, x 6 Z 0 x x x x x x 6 1
57 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 3 x 2 x 3 x 5 +2 x 6 = 1 x 1, x 2, x 3, x 4, x 5, x 6 Z 0 x x x x x x 6 1
58 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 2.4 x 2 x 3 x 5 +2 x 6 = 0.2 x 1,, x 3, x 4, x 5, x 6 Z 0 x x x x x x 6 1
59 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 2.4 x 2 x 3 x 5 +2 x 6 = 0.2 x 1, x 2, x 3, x 4, x 5, x 6 Z 0 x x x x x x 6 1
60 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 2.4 x 2 x 3 x 5 +2 x 6 = 0.2 x 1, x 2,,,, Z 0 x x x x x x 6 1
61 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 2.4 x 2 x 3 x 5 +2 x 6 = 0.2 x 1, x 2,,,, Z 0 x x x x 6 1
62 Two-row intersection cuts basic nonbasic {}}{{}}{ x 1 +2 x 3 x 4 + x 5 +3 x 6 = 2.4 x 2 x 3 x 5 +2 x 6 = 0.2 x 1, x 2,,,, Z 0 x 3 0 x 4 x 5 1 x 6 1
63 Two-row intersection cuts + strengthening P I S-free lifting P IU full 2-row basic nonbasic Z bnd. Z bnd. B B B : keep B: keep binding : drop
64 Two-row intersection cuts + strengthening P I S-free lifting P IU full 2-row basic nonbasic Z bnd. Z bnd. B B B : keep B: keep binding : drop
65 Two-row intersection cuts + strengthening P I S-free lifting P IU full 2-row basic nonbasic Z bnd. Z bnd. B B B : keep B: keep binding : drop
66 Two-row intersection cuts + strengthening P I S-free lifting P IU full 2-row basic nonbasic Z bnd. Z bnd. B B B : keep B: keep binding : drop
67 Two-row intersection cuts + strengthening P I S-free lifting P IU full 2-row basic nonbasic Z bnd. Z bnd. B B B : keep B: keep binding : drop
68 Two-row intersection cuts + strengthening P I S-free lifting P IU full 2-row basic nonbasic Z bnd. Z bnd. B B B : keep B: keep binding : drop
69 Two-row intersection cuts and strengthenings 51 common instances: cuts gc% exact GMI % all P I % 42 S-free % 29 lifting % 10 P IU % 25 full 2-row % 22
70 Two-row intersection cuts and strengthenings 51 common instances: cuts gc% exact GMI % all P I % 42 S-free % 29 lifting % 10 P IU % 25 full 2-row % 22
71 Two-row intersection cuts and strengthenings 51 common instances: cuts gc% exact GMI % all P I % 42 S-free % 29 lifting % 10 P IU % 25 full 2-row % 22
72 Two-row intersection cuts and strengthenings 51 common instances: cuts gc% exact GMI % all P I % 42 S-free % 29 lifting % 10 P IU % 25 full 2-row % 22
73 Two-row intersection cuts and strengthenings 15 common instances: cuts gc% exact GMI all P I all S-free all P IU all full 2-row all
74 Two-row intersection cuts and strengthenings 7 common instances: [bell5, blend2, egout, khb05250, misc03, misc07, set1ch] cuts gc% exact GMI all P I all S-free all lifting all P IU all full 2-row all
75 Bases We depend on the optimal basis Will the gap closed by two-row cuts survive more GMIs?
76 Bases We depend on the optimal basis Will the gap closed by two-row cuts survive more GMIs?
77 Bases We depend on the optimal basis Will the gap closed by two-row cuts survive more GMIs?
78 Relax and cut Convenient way to explore different (feasible) bases. Now trying to separate a point with a much stronger LP bound (obtained by adding GMIs).
79 Relax and cut Convenient way to explore different (feasible) bases. Now trying to separate a point with a much stronger LP bound (obtained by adding GMIs).
80 Relax and cut Convenient way to explore different (feasible) bases. Now trying to separate a point with a much stronger LP bound (obtained by adding GMIs).
81 Relax and cut: results 43 common instances: cuts gc% exact GMI all 2-row i.c full 2-row relax&cut GMI all relax&cut 2-row i.c relax&cut full 2-row
82 Future More rows More tricks More tests
83 Future More rows More tricks More tests
84 Future More rows More tricks More tests
85 Future More rows More tricks More tests
86
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