SDP based Branch & Bound for Max-Cut Franz Rendl, Giovanni Rinaldi, Angelika Wiegele Alpen-Adria-Universit¨ at Klagenfurt and IASI Rome
Overview The problem ▽ Angelika Wiegele, SDP based Branch & Bound – p.1/19
Overview The problem Some solution approaches ▽ Angelika Wiegele, SDP based Branch & Bound – p.1/19
Overview The problem Some solution approaches Bounding routine ▽ Angelika Wiegele, SDP based Branch & Bound – p.1/19
Overview The problem Some solution approaches Bounding routine Branching rules ▽ Angelika Wiegele, SDP based Branch & Bound – p.1/19
Overview The problem Some solution approaches Bounding routine Branching rules Numerical results on Max-Cut, unconstrained 0-1 programs, Equipartitioning Angelika Wiegele, SDP based Branch & Bound – p.1/19
the Max-Cut problem G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n ▽ Angelika Wiegele, SDP based Branch & Bound – p.2/19
the Max-Cut problem G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n −1 1 −1 1 −1 1 ▽ Angelika Wiegele, SDP based Branch & Bound – p.2/19
the Max-Cut problem G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n −1 −1 1 1 −1 1 ▽ Angelika Wiegele, SDP based Branch & Bound – p.2/19
the Max-Cut problem G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n max x T Lx z mc = x 2 s . t . 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) ▽ Angelika Wiegele, SDP based Branch & Bound – p.3/19
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
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
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
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) ▽ Angelika 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 ▽ Angelika 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 to the Branch & Bound tree ▽ Angelika 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 to the Branch & Bound tree ▽ Angelika 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, c ij , x ∈ {± 1 } n max x T Lx z mc = x 2 s . t . i = 1 , 1 ≤ i ≤ n ▽ Angelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: SDP relaxation G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n max x T Lx z mc = x 2 s . t . i = 1 , 1 ≤ i ≤ n X := xx T ▽ Angelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: SDP relaxation G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n max x T Lx z mc = x 2 s . t . i = 1 , 1 ≤ i ≤ n X := xx T ⇒ z mc = max � L, X � s . t . diag( X ) = e rank( X ) = 1 X � 0 , X ∈ S n ▽ Angelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: SDP relaxation G = ( V, E ) , V ( G ) = { 1 , . . . , n } , | E ( G ) | = m, c ij , x ∈ {± 1 } n max x T Lx z mc = x 2 s . t . i = 1 , 1 ≤ i ≤ n X := xx T ⇒ z mc − basic = max � L, X � s . t . diag( X ) = e ——————- rank( X ) = 1 X � 0 , X ∈ S n Angelika Wiegele, SDP based Branch & Bound – p.5/19
Bounding: triangle inequalities X ∈ MET ⇐ ⇒ ▽ Angelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: triangle inequalities X ∈ MET ⇐ ⇒ x ij + x ik + x jk ≥ − 1 , x ij − x ik − x jk ≥ − 1 , − x ij + x ik − x jk ≥ − 1 , − x ij − x ik + x jk ≥ − 1 , ∀ i < j < k. ▽ Angelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: triangle inequalities X ∈ MET ⇐ ⇒ x ij + x ik + x jk ≥ − 1 , x ij − x ik − x jk ≥ − 1 , − x ij + x ik − x jk ≥ − 1 , − x ij − x ik + x jk ≥ − 1 , ∀ i < j < k. z mc − met = max {� L, X � : X ∈ E , X ∈ MET } ▽ Angelika Wiegele, SDP based Branch & Bound – p.6/19
Bounding: triangle inequalities X ∈ MET ⇐ ⇒ x ij + x ik + x jk ≥ − 1 , x ij − x ik − x jk ≥ − 1 , − x ij + x ik − x jk ≥ − 1 , − x ij − x ik + x jk ≥ − 1 , ∀ i < j < k. z mc − 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 z mc − met = max {� L, X � : X ∈ E , A ( X ) ≤ b } ▽ Angelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: Lagrangian duality z mc − 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
Bounding: Lagrangian duality z mc − met = max {� L, X � : X ∈ E , A ( X ) ≤ b } Lagrangian L ( X ; γ ) := � L, X � + � γ, b − A ( X ) � and the dual functional X ∈E L ( X ; γ ) = b T γ + max X ∈E � L − A T ( γ ) , X � f ( γ ) := max ▽ Angelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: Lagrangian duality z mc − met = max {� L, X � : X ∈ E , A ( X ) ≤ b } Lagrangian L ( X ; γ ) := � L, X � + � γ, b − A ( X ) � and the dual functional X ∈E L ( X ; γ ) = b T γ + max X ∈E � L − A T ( γ ) , X � f ( γ ) := max z = min γ ≥ 0 f ( γ ) Angelika Wiegele, SDP based Branch & Bound – p.7/19
Bounding: dynamic bundle method f appr ( γ ) := max {L ( X ; γ ) : X ∈ conv( X 1 , . . . , X k ) } γ ≥ 0 f appr ( γ ) + 1 γ || 2 min 2 t || γ − ˆ ▽ Angelika Wiegele, SDP based Branch & Bound – p.8/19
Bounding: dynamic bundle method f appr ( γ ) := max {L ( X ; γ ) : X ∈ conv( X 1 , . . . , X k ) } γ ≥ 0 f appr ( γ ) + 1 γ || 2 min 2 t || γ − ˆ Computational effort: iteratively solve the minimization problem by solving a sequence of convex quadratic problems, giving a new trial point γ test evaluate f at γ test , i.e. solving an SDP of order n and n linear equalities Angelika Wiegele, SDP based Branch & Bound – p.8/19
Branching rules i and j such that they minimize � n k =1 (1 − | x ik | ) 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
Branching rules i and j such that they minimize � n k =1 (1 − | x ik | ) 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 | x ij | , i.e. we fix the most difficult decision ▽ Angelika Wiegele, SDP based Branch & Bound – p.9/19
Branching rules i and j such that they minimize � n k =1 (1 − | x ik | ) 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 | x ij | , 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
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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