[PPT] - SDP based Branch & Bound for Max-Cut Franz Rendl, Giovanni PowerPoint Presentation

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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

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Overview

The problem

▽Angelika Wiegele, SDP based Branch & Bound – p.1/19

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Overview

The problem Some solution approaches

▽Angelika Wiegele, SDP based Branch & Bound – p.1/19

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Overview

The problem Some solution approaches Bounding routine

▽Angelika Wiegele, SDP based Branch & Bound – p.1/19

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Overview

The problem Some solution approaches Bounding routine Branching rules

▽Angelika Wiegele, SDP based Branch & Bound – p.1/19

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Overview

The problem Some solution approaches Bounding routine Branching rules Numerical results on Max-Cut, unconstrained 0-1 programs, Equipartitioning

Angelika Wiegele, SDP based Branch & Bound – p.1/19

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the Max-Cut problem

G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij, x ∈ {±1}n

▽Angelika Wiegele, SDP based Branch & Bound – p.2/19

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the Max-Cut problem

G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij, x ∈ {±1}n

−1 −1 1 1 1 −1

▽Angelika Wiegele, SDP based Branch & Bound – p.2/19

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the Max-Cut problem

G = (V, E), V (G) = {1, . . . , n}, |E(G)| = m, cij, x ∈ {±1}n

−1 −1 1 1 1 −1

▽Angelika Wiegele, SDP based Branch & Bound – p.2/19

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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

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Solution approaches

LP based methods (Barahona, Jünger, Reinelt, 89)

▽Angelika Wiegele, SDP based Branch & Bound – p.3/19

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Solution approaches

LP based methods (Barahona, Jünger, Reinelt, 89) 2nd order cone programming (Muramatsu, Suzuki, 03)

▽Angelika Wiegele, SDP based Branch & Bound – p.3/19

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Solution approaches

LP based methods (Barahona, Jünger, Reinelt, 89) 2nd order cone programming (Muramatsu, Suzuki, 03) Branch & Bound with preprocessing (Pardalos, Rodgers, 90)

▽Angelika Wiegele, SDP based Branch & Bound – p.3/19

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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)

▽Angelika Wiegele, SDP based Branch & Bound – p.3/19

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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

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Branch & Bound

At each node of the Branch & Bound tree: compute a feasible solution (lower bound)

▽Angelika Wiegele, SDP based Branch & Bound – p.4/19

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Branch & Bound

At each node of the Branch & Bound tree: compute a feasible solution (lower bound) compute an upper bound

▽Angelika Wiegele, SDP based Branch & Bound – p.4/19

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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 to the Branch & Bound tree

▽Angelika Wiegele, SDP based Branch & Bound – p.4/19

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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

to the Branch & Bound tree

▽Angelika Wiegele, SDP based Branch & Bound – p.4/19

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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

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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

▽Angelika Wiegele, SDP based Branch & Bound – p.5/19

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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

▽Angelika Wiegele, SDP based Branch & Bound – p.5/19

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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 = maxL, X s.t. diag(X) = e rank(X) = 1 X 0, X ∈ Sn

▽Angelika Wiegele, SDP based Branch & Bound – p.5/19

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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 = maxL, X s.t. diag(X) = e rank(X) = 1

——————-

X 0, X ∈ Sn

Angelika Wiegele, SDP based Branch & Bound – p.5/19

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Bounding: triangle inequalities

X ∈ MET ⇐ ⇒

▽Angelika Wiegele, SDP based Branch & Bound – p.6/19

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Bounding: triangle inequalities

X ∈ MET ⇐ ⇒ xij + xik + xjk ≥ −1, xij − xik − xjk ≥ −1, −xij + xik − xjk ≥ −1, −xij − xik + xjk ≥ −1, ∀i < j < k.

▽Angelika Wiegele, SDP based Branch & Bound – p.6/19

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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}

▽Angelika Wiegele, SDP based Branch & Bound – p.6/19

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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

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Bounding: Lagrangian duality

zmc−met = max{L, X : X ∈ E, A(X) ≤ b}

▽Angelika Wiegele, SDP based Branch & Bound – p.7/19

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Bounding: Lagrangian duality

zmc−met = max{L, X : X ∈ E, A(X) ≤ b}

Lagrangian

L(X; γ) := L, X + γ, b − A(X)

▽Angelika Wiegele, SDP based Branch & Bound – p.7/19

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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) := max

X∈E L(X; γ) = bT γ + max X∈E L − AT (γ), X

▽Angelika Wiegele, SDP based Branch & Bound – p.7/19

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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) := max

X∈E L(X; γ) = bT γ + max X∈E L − AT (γ), X

z = min

γ≥0 f(γ)

Angelika Wiegele, SDP based Branch & Bound – p.7/19

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Bounding: dynamic bundle method

fappr(γ) := max{L(X; γ) : X ∈ conv(X1, . . . , Xk)} min

γ≥0 fappr(γ) + 1

2t||γ − ˆ γ||2

▽Angelika Wiegele, SDP based Branch & Bound – p.8/19

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Bounding: dynamic bundle method

fappr(γ) := max{L(X; γ) : X ∈ conv(X1, . . . , Xk)} min

γ≥0 fappr(γ) + 1

2t||γ − ˆ γ||2

Computational effort: iteratively solve the minimization problem by solving a sequence of convex quadratic problems, giving a new trial point γtest evaluate f at γtest, i.e. solving an SDP of order n and n linear equalities

Angelika Wiegele, SDP based Branch & Bound – p.8/19

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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 are closest to a {±1} vector.

▽Angelika Wiegele, SDP based Branch & Bound – p.9/19

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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 are closest to a {±1} vector. edge ij which minimizes |xij|, i.e. we fix the most difficult decision

▽Angelika Wiegele, SDP based Branch & Bound – p.9/19

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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 are closest to a {±1} vector. edge ij which minimizes |xij|, i.e. we fix the most difficult decision “strong branching.” forecast on some potential branching edges and choose the most promising.

Angelika Wiegele, SDP based Branch & Bound – p.9/19

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Numerical results

until the early nineties...

n\d

10 20 30 40 50 60 70 80 90 100 20

30
40
50
60
70
80
90
100
110

120

Angelika Wiegele, SDP based Branch & Bound – p.10/19

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