SDP based Branch & Bound for Max-Cut - CNR · SDP based Branch & Bound for Max-Cut Franz Rendl,...
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SDP based Branch & Bound for Max-Cut Franz Rendl, Giovanni Rinaldi, Angelika Wiegele Alpen-Adria-Universit ¨ at Klagenfurt and IASI Rome
Transcript of SDP based Branch & Bound for Max-Cut - CNR · SDP based Branch & Bound for Max-Cut Franz Rendl,...
SDP based Branch & Bound forMax-Cut
Franz Rendl, Giovanni Rinaldi, Angelika Wiegele
Alpen-Adria-Universitat Klagenfurt and IASI Rome
Overview
The problem
Some solution approaches
Bounding routine
Branching rules
Numerical results on Max-Cut, unconstrained 0-1programs, Equipartitioning
OAngelika Wiegele, SDP based Branch & Bound – p.1/19
Overview
The problem
Some solution approaches
Bounding routine
Branching rules
Numerical results on Max-Cut, unconstrained 0-1programs, Equipartitioning
OAngelika Wiegele, SDP based Branch & Bound – p.1/19
Overview
The problem
Some solution approaches
Bounding routine
Branching rules
Numerical results on Max-Cut, unconstrained 0-1programs, Equipartitioning
OAngelika Wiegele, SDP based Branch & Bound – p.1/19
Overview
The problem
Some solution approaches
Bounding routine
Branching rules
Numerical results on Max-Cut, unconstrained 0-1programs, Equipartitioning
OAngelika Wiegele, SDP based Branch & Bound – p.1/19
Overview
The problem
Some solution approaches
Bounding routine
Branching rules
Numerical results on Max-Cut, unconstrained 0-1programs, Equipartitioning
Angelika Wiegele, SDP based Branch & Bound – p.1/19
the Max-Cut problem
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
OAngelika Wiegele, SDP based Branch & Bound – p.2/19
the Max-Cut problem
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
−1
−1
1
1
1
−1
OAngelika Wiegele, SDP based Branch & Bound – p.2/19
the Max-Cut problem
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
−1
−1
1
1
1
−1
OAngelika Wiegele, SDP based Branch & Bound – p.2/19
the Max-Cut problem
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
−1
−1
1
1
1
−1
Angelika Wiegele, SDP based Branch & Bound – p.2/19
Solution approaches
LP based methods (Barahona, Jünger, Reinelt, 89)
2nd order cone programming (Muramatsu, Suzuki, 03)
Branch & Bound with preprocessing (Pardalos, Rodgers,90)
SDP based methods (Helmberg, Rendl, 98)
Using a MIQP solver (Billionnet, Elloumi, 06)
OAngelika Wiegele, SDP based Branch & Bound – p.3/19
Solution approaches
LP based methods (Barahona, Jünger, Reinelt, 89)
2nd order cone programming (Muramatsu, Suzuki, 03)
Branch & Bound with preprocessing (Pardalos, Rodgers,90)
SDP based methods (Helmberg, Rendl, 98)
Using a MIQP solver (Billionnet, Elloumi, 06)
OAngelika Wiegele, SDP based Branch & Bound – p.3/19
Solution approaches
LP based methods (Barahona, Jünger, Reinelt, 89)
2nd order cone programming (Muramatsu, Suzuki, 03)
Branch & Bound with preprocessing (Pardalos, Rodgers,90)
SDP based methods (Helmberg, Rendl, 98)
Using a MIQP solver (Billionnet, Elloumi, 06)
OAngelika Wiegele, SDP based Branch & Bound – p.3/19
Solution approaches
LP based methods (Barahona, Jünger, Reinelt, 89)
2nd order cone programming (Muramatsu, Suzuki, 03)
Branch & Bound with preprocessing (Pardalos, Rodgers,90)
SDP based methods (Helmberg, Rendl, 98)
Using a MIQP solver (Billionnet, Elloumi, 06)
OAngelika Wiegele, SDP based Branch & Bound – p.3/19
Solution approaches
LP based methods (Barahona, Jünger, Reinelt, 89)
2nd order cone programming (Muramatsu, Suzuki, 03)
Branch & Bound with preprocessing (Pardalos, Rodgers,90)
SDP based methods (Helmberg, Rendl, 98)
Using a MIQP solver (Billionnet, Elloumi, 06)
Angelika Wiegele, SDP based Branch & Bound – p.3/19
Branch & Bound
At each node of the Branch & Bound tree:
compute a feasible solution (lower bound)
compute an upper bound
choose an edge (ij) and add the two nodes(ij) ∈ cut and(ij) /∈ cut
to the Branch & Bound tree
OAngelika Wiegele, SDP based Branch & Bound – p.4/19
Branch & Bound
At each node of the Branch & Bound tree:
compute a feasible solution (lower bound)
compute an upper bound
choose an edge (ij) and add the two nodes(ij) ∈ cut and(ij) /∈ cut
to the Branch & Bound tree
OAngelika Wiegele, SDP based Branch & Bound – p.4/19
Branch & Bound
At each node of the Branch & Bound tree:
compute a feasible solution (lower bound)
compute an upper bound
choose an edge (ij) and add the two nodes
(ij) ∈ cut and(ij) /∈ cut
to the Branch & Bound tree
OAngelika Wiegele, SDP based Branch & Bound – p.4/19
Branch & Bound
At each node of the Branch & Bound tree:
compute a feasible solution (lower bound)
compute an upper bound
choose an edge (ij) and add the two nodes(ij) ∈ cut and
(ij) /∈ cut
to the Branch & Bound tree
OAngelika Wiegele, SDP based Branch & Bound – p.4/19
Branch & Bound
At each node of the Branch & Bound tree:
compute a feasible solution (lower bound)
compute an upper bound
choose an edge (ij) and add the two nodes(ij) ∈ cut and(ij) /∈ cut
to the Branch & Bound tree
Angelika Wiegele, SDP based Branch & Bound – p.4/19
Bounding: SDP relaxation
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
X := xxT ⇒
zmc−basic = max〈L, X〉
s.t. diag(X) = e
rank(X) = 1——————-X � 0, X ∈ Sn
OAngelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: SDP relaxation
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
X := xxT
⇒
zmc−basic = max〈L, X〉
s.t. diag(X) = e
rank(X) = 1——————-X � 0, X ∈ Sn
OAngelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: SDP relaxation
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
X := xxT ⇒
zmc = max〈L, X〉
s.t. diag(X) = e
rank(X) = 1
X � 0, X ∈ Sn
⇒
zmc−basic = max〈L, X〉
s.t. diag(X) = e
rank(X) = 1——————-X � 0, X ∈ Sn
OAngelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: SDP relaxation
G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij , x ∈ {±1}n
zmc = max xT Lx
s.t. x2
i = 1, 1 ≤ i ≤ n
X := xxT ⇒
zmc−basic = max〈L, X〉
s.t. diag(X) = e
rank(X) = 1——————-X � 0, X ∈ Sn
Angelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: triangle inequalities
X ∈ MET ⇐⇒
xij + xik + xjk ≥ −1,
xij − xik − xjk ≥ −1,
−xij + xik − xjk ≥ −1,
−xij − xik + xjk ≥ −1, ∀i < j < k.
zmc−met = max{〈L, X〉 : X ∈ E , X ∈ MET}
E = {X : diag(X) = e, X � 0}
OAngelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: triangle inequalities
X ∈ MET ⇐⇒
xij + xik + xjk ≥ −1,
xij − xik − xjk ≥ −1,
−xij + xik − xjk ≥ −1,
−xij − xik + xjk ≥ −1, ∀i < j < k.
zmc−met = max{〈L, X〉 : X ∈ E , X ∈ MET}
E = {X : diag(X) = e, X � 0}
OAngelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: triangle inequalities
X ∈ MET ⇐⇒
xij + xik + xjk ≥ −1,
xij − xik − xjk ≥ −1,
−xij + xik − xjk ≥ −1,
−xij − xik + xjk ≥ −1, ∀i < j < k.
zmc−met = max{〈L, X〉 : X ∈ E , X ∈ MET}
E = {X : diag(X) = e, X � 0}
OAngelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: triangle inequalities
X ∈ MET ⇐⇒
xij + xik + xjk ≥ −1,
xij − xik − xjk ≥ −1,
−xij + xik − xjk ≥ −1,
−xij − xik + xjk ≥ −1, ∀i < j < k.
zmc−met = max{〈L, X〉 : X ∈ E , X ∈ MET}
E = {X : diag(X) = e, X � 0}
Angelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: Lagrangian duality
zmc−met = max{〈L, X〉 : X ∈ E , A(X) ≤ b}
Lagrangian
L(X; γ) := 〈L, X〉 + 〈γ, b −A(X)〉
and the dual functional
f(γ) := maxX∈E
L(X; γ) = bT γ + maxX∈E
〈L −AT (γ), X〉
z = minγ≥0
f(γ)
OAngelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: Lagrangian duality
zmc−met = max{〈L, X〉 : X ∈ E , A(X) ≤ b}
Lagrangian
L(X; γ) := 〈L, X〉 + 〈γ, b −A(X)〉
and the dual functional
f(γ) := maxX∈E
L(X; γ) = bT γ + maxX∈E
〈L −AT (γ), X〉
z = minγ≥0
f(γ)
OAngelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: Lagrangian duality
zmc−met = max{〈L, X〉 : X ∈ E , A(X) ≤ b}
Lagrangian
L(X; γ) := 〈L, X〉 + 〈γ, b −A(X)〉
and the dual functional
f(γ) := maxX∈E
L(X; γ) = bT γ + maxX∈E
〈L −AT (γ), X〉
z = minγ≥0
f(γ)
OAngelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: Lagrangian duality
zmc−met = max{〈L, X〉 : X ∈ E , A(X) ≤ b}
Lagrangian
L(X; γ) := 〈L, X〉 + 〈γ, b −A(X)〉
and the dual functional
f(γ) := maxX∈E
L(X; γ) = bT γ + maxX∈E
〈L −AT (γ), X〉
z = minγ≥0
f(γ)
Angelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: dynamic bundle method
fappr(γ) := max{L(X; γ) : X ∈ conv(X1, . . . , Xk)}
minγ≥0
fappr(γ) +1
2t||γ − γ||2
Computational effort: iteratively
solve the minimization problem by solving a sequence ofconvex quadratic problems, giving a new trial point γtest
evaluate f at γtest, i.e. solving an SDP of order n and nlinear equalities
OAngelika Wiegele, SDP based Branch & Bound – p.8/19
Bounding: dynamic bundle method
fappr(γ) := max{L(X; γ) : X ∈ conv(X1, . . . , Xk)}
minγ≥0
fappr(γ) +1
2t||γ − γ||2
Computational effort: iteratively
solve the minimization problem by solving a sequence ofconvex quadratic problems, giving a new trial point γtest
evaluate f at γtest, i.e. solving an SDP of order n and nlinear equalities
Angelika Wiegele, SDP based Branch & Bound – p.8/19
Branching rules
i and j such that they minimize∑n
k=1(1 − |xik|)
2,i.e. find two rows i and j in the primal matrix X, that areclosest to a {±1} vector.
edge ij which minimizes |xij |,i.e. we fix the most difficult decision
“strong branching.” forecast on some potential branchingedges and choose the most promising.
OAngelika Wiegele, SDP based Branch & Bound – p.9/19
Branching rules
i and j such that they minimize∑n
k=1(1 − |xik|)
2,i.e. find two rows i and j in the primal matrix X, that areclosest to a {±1} vector.
edge ij which minimizes |xij |,i.e. we fix the most difficult decision
“strong branching.” forecast on some potential branchingedges and choose the most promising.
OAngelika Wiegele, SDP based Branch & Bound – p.9/19
Branching rules
i and j such that they minimize∑n
k=1(1 − |xik|)
2,i.e. find two rows i and j in the primal matrix X, that areclosest to a {±1} vector.
edge ij which minimizes |xij |,i.e. we fix the most difficult decision
“strong branching.” forecast on some potential branchingedges and choose the most promising.
Angelika Wiegele, SDP based Branch & Bound – p.9/19
Numerical results
until the early nineties...
n\d 10 20 30 40 50 60 70 80 90 10020 � � � � � � � � � �
30 � � � � � � � � � �
40 � � � � � � � �
50 � � � � � �
60 � � � �
70 � � �
80 � �
90 �
100 �
110120
Angelika Wiegele, SDP based Branch & Bound – p.10/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671
10 122 331 853
120 0.3 10 168 1263 4667
10 48 339 1016
120 0.8 6 322 3909 9898
10 386 2650 7577
150 0.3 1 6789
6 342 1492 3305
150 0.8 0 –
6 4451 13214 17081
200 0.3 0 –
1 2345
OAngelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671 10 122 331 853120 0.3 10 168 1263 4667
10 48 339 1016
120 0.8 6 322 3909 9898
10 386 2650 7577
150 0.3 1 6789
6 342 1492 3305
150 0.8 0 –
6 4451 13214 17081
200 0.3 0 –
1 2345
OAngelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671 10 122 331 853120 0.3 10 168 1263 4667 10 48 339 1016120 0.8 6 322 3909 9898
10 386 2650 7577
150 0.3 1 6789
6 342 1492 3305
150 0.8 0 –
6 4451 13214 17081
200 0.3 0 –
1 2345
OAngelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671 10 122 331 853120 0.3 10 168 1263 4667 10 48 339 1016120 0.8 6 322 3909 9898 10 386 2650 7577150 0.3 1 6789
6 342 1492 3305
150 0.8 0 –
6 4451 13214 17081
200 0.3 0 –
1 2345
OAngelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671 10 122 331 853120 0.3 10 168 1263 4667 10 48 339 1016120 0.8 6 322 3909 9898 10 386 2650 7577150 0.3 1 6789 6 342 1492 3305150 0.8 0 –
6 4451 13214 17081
200 0.3 0 –
1 2345
OAngelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671 10 122 331 853120 0.3 10 168 1263 4667 10 48 339 1016120 0.8 6 322 3909 9898 10 386 2650 7577150 0.3 1 6789 6 342 1492 3305150 0.8 0 – 6 4451 13214 17081200 0.3 0 –
1 2345
OAngelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: UQBP
Comparison with Billionnet, Elloumi 06
CPU time (sec) CPU time (sec)n d solved min avg. max solved min avg. max
100 1.0 10 27 372 1671 10 122 331 853120 0.3 10 168 1263 4667 10 48 339 1016120 0.8 6 322 3909 9898 10 386 2650 7577150 0.3 1 6789 6 342 1492 3305150 0.8 0 – 6 4451 13214 17081200 0.3 0 – 1 2345
Angelika Wiegele, SDP based Branch & Bound – p.11/19
Numerical results: Max-Cut
min-time avg-time max-time min avg maxn d solved h:min h:min h:min nodes
G0.5
100 0.5 10 9 1:07 5:29 52 617 3036G
−1/0/1
100 0.99 10 10 1:52 3:47 66 640 1810G[−10,10]
100 0.5 10 11 44 1:35 58 406 866100 0.9 10 3 1:25 5:20 14 686 2568G[1,10]
100 0.5 10 16 1:02 2:36 104 599 1546100 0.9 10 22 1:02 2:08 168 465 998
Angelika Wiegele, SDP based Branch & Bound – p.12/19
Numerical results: Physics
torus graphs with gaussian distribution
Problem Branch & Cut B & Bnumber times (sec) times (sec)
2 dimensional10 × 10 0.15 0.14 0.18 9 43 1615 × 15 0.44 0.78 0.67 443 480 48520 × 20 1.70 3.50 2.61 – – –
3 dimensional5 × 5 × 5 2.68 3.29 3.07 16 60 596 × 6 × 6 20.56 37.74 27.30 200 1531 1797 × 7 × 7 95.25 131.34 460.01 828 539 –
Angelika Wiegele, SDP based Branch & Bound – p.13/19
Numerical results: Physics
Ising instances (dense graphs)
Problem Branch & Cut & Price B & Bnumber times (h:min:sec) times (h:min:sec)3.0-100 4:52 0:24 7:31 3:58 1:31 1:233.0-150 2:36:46 4:49:05 3:48:41 11:59 13:32 9:183.0-200 9:22:03 32:48:03 8:53:26 29:17 1:14:55 1:01:283.0-250 21:17:07 7:42:25 17:30:13 5:11:26 2:48:24 4:18:453.0-300 17:20:54 10:21:40 18:33:49 6:32:19 3:47:56 17:49:49
Problem Branch & Cut & Price B & Bnumber times (h:min:sec) times (h:min:sec)2.5-100 18:22 6:27 10:08 3:21 1:30 592.5-150 21:28:39 23:35:11 31:40:07 14:17 14:57 28:192.5-200 24:34 57:30 1:45:45
OAngelika Wiegele, SDP based Branch & Bound – p.14/19
Numerical results: Physics
Ising instances (dense graphs)
Problem Branch & Cut & Price B & Bnumber times (h:min:sec) times (h:min:sec)3.0-100 4:52 0:24 7:31 3:58 1:31 1:233.0-150 2:36:46 4:49:05 3:48:41 11:59 13:32 9:183.0-200 9:22:03 32:48:03 8:53:26 29:17 1:14:55 1:01:283.0-250 21:17:07 7:42:25 17:30:13 5:11:26 2:48:24 4:18:453.0-300 17:20:54 10:21:40 18:33:49 6:32:19 3:47:56 17:49:49
Problem Branch & Cut & Price B & Bnumber times (h:min:sec) times (h:min:sec)2.5-100 18:22 6:27 10:08 3:21 1:30 592.5-150 21:28:39 23:35:11 31:40:07 14:17 14:57 28:192.5-200 24:34 57:30 1:45:45
Angelika Wiegele, SDP based Branch & Bound – p.14/19
Numerical results: OR library
n 100 200d 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50BHT05 � � � �
BiEl06 � � � � � � � � � �
B & B
� � � � � � � � � � � � � �
n 250d 10 10 10 10 10 10 10 10 10 10BHT05 � �
BiEl06B & B
� � � � � � � � �
BHT05. . . Boros, Hammer, Tavares 05 BiEl06. . . Billionnet, Elloumi 06
OAngelika Wiegele, SDP based Branch & Bound – p.15/19
Numerical results: OR library
n 100 200d 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50BHT05 � � � �
BiEl06 � � � � � � � � � �
B & B � � � � � � � � � � � � � �
n 250d 10 10 10 10 10 10 10 10 10 10BHT05 � �
BiEl06B & B
� � � � � � � � �
BHT05. . . Boros, Hammer, Tavares 05 BiEl06. . . Billionnet, Elloumi 06
OAngelika Wiegele, SDP based Branch & Bound – p.15/19
Numerical results: OR library
n 100 200d 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50BHT05 � � � �
BiEl06 � � � � � � � � � �
B & B � � � � � � � � � � � � � �
n 250d 10 10 10 10 10 10 10 10 10 10BHT05 � �
BiEl06B & B � � � � � � � � �
BHT05. . . Boros, Hammer, Tavares 05 BiEl06. . . Billionnet, Elloumi 06
Angelika Wiegele, SDP based Branch & Bound – p.15/19
Numerical results: Equipartitioning
best knownn d bound |Ecut|
124 0.02 12.01 13 �
124 0.04 61.22 63
�
124 0.08 170.93 178
�
124 0.16 440.08 449
�
250 0.01 26.06 29
�
OAngelika Wiegele, SDP based Branch & Bound – p.16/19
Numerical results: Equipartitioning
best knownn d bound |Ecut|
124 0.02 12.01 13 �
124 0.04 61.22 63 �
124 0.08 170.93 178 �
124 0.16 440.08 449 �
250 0.01 26.06 29 �
Angelika Wiegele, SDP based Branch & Bound – p.16/19
Numerical results
n\d 10 20 30 40 50 60 70 80 90 10020 � � � � � � � � � �
30 � � � � � � � � � �
40 � � � � � � � �
� �
50 � � � � � �
� � � �
60 � � � �
� � � � � �
70 � � �
� � � � � � �
80 � �
� � � � � � � �
90 �
� � � � � � � � �
100 �
� � � � � � � � �
110
� � � � � � � �
120
� � � � � � � �
OAngelika Wiegele, SDP based Branch & Bound – p.17/19
Numerical results
n\d 10 20 30 40 50 60 70 80 90 10020 � � � � � � � � � �
30 � � � � � � � � � �
40 � � � � � � � � � �
50 � � � � � � � � � �
60 � � � � � � � � � �
70 � � � � � � � � � �
80 � � � � � � � � � �
90 � � � � � � � � � �
100 � � � � � � � � � �
110 � � � � � � � �
120 � � � � � � � �
Angelika Wiegele, SDP based Branch & Bound – p.17/19
Summary
strong branching too expensive
SDP based Branch & Bound code, that
. . . solves any instance of size up to n = 100
. . . solves sparse instances and instances of specialstructure up to n = 300
OAngelika Wiegele, SDP based Branch & Bound – p.18/19
Summary
strong branching too expensive
SDP based Branch & Bound code, that
. . . solves any instance of size up to n = 100
. . . solves sparse instances and instances of specialstructure up to n = 300
OAngelika Wiegele, SDP based Branch & Bound – p.18/19
Summary
strong branching too expensive
SDP based Branch & Bound code, that
. . . solves any instance of size up to n = 100
. . . solves sparse instances and instances of specialstructure up to n = 300
Angelika Wiegele, SDP based Branch & Bound – p.18/19
the end
Thank you for your attention!
Angelika Wiegele, SDP based Branch & Bound – p.19/19
