R. Sadykov and F. Vanderbeck - CNR · R. Sadykov and F. Vanderbeck Column Generation for Extended...
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Column Generation for Extended Formulations
R. Sadykov and F. Vanderbeck
INRIA team ReAlOpt & University of Bordeaux
(inputs from L.A. Wolsey)
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulations: why & how
Reformulation involving extra variables
⇓
tighter relations between variables
Variable Splitting (binary or unary expansion)Network Flow (Multi-Commodity)Dynamic Programming Solver [Martin et al]Union of Polyhedra [Balas]Polyhedral Branching Systems [Kaibel, Loos]. . .
often rely on problem decomposition
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulations: why & how
Reformulation involving extra variables
⇓
tighter relations between variables
Variable Splitting (binary or unary expansion)Network Flow (Multi-Commodity)Dynamic Programming Solver [Martin et al]Union of Polyhedra [Balas]Polyhedral Branching Systems [Kaibel, Loos]. . .
often rely on problem decomposition
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulations: why & how
Reformulation involving extra variables
⇓
tighter relations between variables
Variable Splitting (binary or unary expansion)Network Flow (Multi-Commodity)Dynamic Programming Solver [Martin et al]Union of Polyhedra [Balas]Polyhedral Branching Systems [Kaibel, Loos]. . .
often rely on problem decomposition
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulation in practice
1 Use a direct MIP-solver approach: size is an issue.
2 Use projection tools: Benders’ cuts.→ dynamic outer approximation of the intended form.
3 Use of an approximation [Van Vyve & Wolsey MP06]Drop some of the constraintsAggregate commoditiesPartial reformulation
→ static outer approximation of the intended form.
4 Use column generation (and row management)→ dynamic inner approximation of the intended form.
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulation in practice
1 Use a direct MIP-solver approach: size is an issue.
2 Use projection tools: Benders’ cuts.→ dynamic outer approximation of the intended form.
3 Use of an approximation [Van Vyve & Wolsey MP06]Drop some of the constraintsAggregate commoditiesPartial reformulation
→ static outer approximation of the intended form.
4 Use column generation (and row management)→ dynamic inner approximation of the intended form.
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulation in practice
1 Use a direct MIP-solver approach: size is an issue.
2 Use projection tools: Benders’ cuts.→ dynamic outer approximation of the intended form.
3 Use of an approximation [Van Vyve & Wolsey MP06]Drop some of the constraintsAggregate commoditiesPartial reformulation
→ static outer approximation of the intended form.
4 Use column generation (and row management)→ dynamic inner approximation of the intended form.
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Extended formulation in practice
1 Use a direct MIP-solver approach: size is an issue.
2 Use projection tools: Benders’ cuts.→ dynamic outer approximation of the intended form.
3 Use of an approximation [Van Vyve & Wolsey MP06]Drop some of the constraintsAggregate commoditiesPartial reformulation
→ static outer approximation of the intended form.
4 Use column generation (and row management)→ dynamic inner approximation of the intended form.
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Outline
1 Assumption 1: ∃ Extended Formulation for a SPMachine SchedulingCapacitated Network Design
2 Assumption 2: ∃ Tight Reformulation for a SPBin Packing
3 Column-and-Row Generation
4 Recombination PropertyNetwork Flow based reformulationsDyn. Prog. based reformulationsUnion of Polyhedra based reformulations
5 Numerical experiments
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPOutline
1 Assumption 1: ∃ Extended Formulation for a SPMachine SchedulingCapacitated Network Design
2 Assumption 2: ∃ Tight Reformulation for a SPBin Packing
3 Column-and-Row Generation
4 Recombination PropertyNetwork Flow based reformulationsDyn. Prog. based reformulationsUnion of Polyhedra based reformulations
5 Numerical experiments
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPAssumption 1: ∃ Extended Formulation for a SP
[F] ≡ min{c xA x ≥ aB x ≥ b
x ∈ INn}
Subproblem : P = {x ∈ Rn+ : Bx ≥ b} and X = P ∩ Zn
Extended Formulation for a Subproblem
∃ a polyhedron Q = {z ∈ Re+ : Hz ≥ h, z ∈ Re
+} and transformation T s.t.:Q defines an extended formulation for conv(X):
conv(X) = projx Q = {x = Tz : Hz ≥ h, z ∈ Re+}
Special Case: Dantzig-Wolfe reformulation, let X = {xg}g∈G
conv(X) = {x =X
g
xg λg :X
g
λg = 1, λg ≥ 0}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPAssumption 1: ∃ Extended Formulation for a SP
[F] ≡ min{c xA x ≥ aB x ≥ b
x ∈ INn}
Subproblem : P = {x ∈ Rn+ : Bx ≥ b} and X = P ∩ Zn
Extended Formulation for a Subproblem
∃ a polyhedron Q = {z ∈ Re+ : Hz ≥ h, z ∈ Re
+} and transformation T s.t.:Q defines an extended formulation for conv(X):
conv(X) = projx Q = {x = Tz : Hz ≥ h, z ∈ Re+}
Special Case: Dantzig-Wolfe reformulation, let X = {xg}g∈G
conv(X) = {x =X
g
xg λg :X
g
λg = 1, λg ≥ 0}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPAssumption 1: ∃ Extended Formulation for a SP
[F] ≡ min{c xA x ≥ aB x ≥ b
x ∈ INn}
Subproblem : P = {x ∈ Rn+ : Bx ≥ b} and X = P ∩ Zn
Extended Formulation for a Subproblem
∃ a polyhedron Q = {z ∈ Re+ : Hz ≥ h, z ∈ Re
+} and transformation T s.t.:Q defines an extended formulation for conv(X):
conv(X) = projx Q = {x = Tz : Hz ≥ h, z ∈ Re+}
Special Case: Dantzig-Wolfe reformulation, let X = {xg}g∈G
conv(X) = {x =X
g
xg λg :X
g
λg = 1, λg ≥ 0}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPExtended or Dantzig-Wolfe reformulation
[F] ≡ min{c xA x ≥ aB x ≥ b
x ∈ INn}
[R] ≡ min{c T zA T z ≥ a
H z ≥ hz ∈ Ze
+}
[M] ≡ min{P
g∈Gc xg λgPg∈GA xg λg ≥ aP
g∈Gλg = 1
λ ∈ {0, 1}|G|}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPRestricted reformulations
[F] ≡ min{c xA x ≥ aB x ≥ b
x ∈ INn}
[R] ≡ min{c T zA T z ≥ a
H z ≥ hz ∈ Ze
+}
[M] ≡ min{P
g∈Gc xg λgPg∈GA xg λg ≥ aP
g∈Gλg = 1
λ ∈ {0, 1}|G|}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPHybrid Approach: two points of view
1 An alternative to a direct extended formulation approach
Dynamic generation of the variables of [R]:generated in bunch by optimizing over the SP
Adding rows that become active.
2 An alternative to standard column generation
Perform Column Generation for [M]
“Project” the Master Program in [R]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: reformulations
[F] ≡ min{X
j
c(Sj ) : Sj + pj ≤ Si or Si + pi ≤ Sj ∀(i, j) ∈ J × J}
timeB4 C3 D5C3A8
Sa Sb Sc Sd
[R] ≡ min{X
jt
cjt zjt
T−pj +1Xt=1
zjt = 1 ∀j
Xj
tXτ=t−pj +1
zjτ ≤ 1 ∀t
zjt ∈ {0, 1} ∀j, t}
[M] ≡ min{Xg∈G
cgλg :Xg∈G
T−pj +1Xt=1
zgjtλg = 1 ∀j,
Xg∈G
λg = 1, λg ∈ {0, 1} ∀g ∈ G}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: reformulations
[F] ≡ min{X
j
c(Sj ) : Sj + pj ≤ Si or Si + pi ≤ Sj ∀(i, j) ∈ J × J}
timeB4 C3 D5C3A8
Sa Sb Sc Sd
[R] ≡ min{X
jt
cjt zjt
T−pj +1Xt=1
zjt = 1 ∀j
Xj
tXτ=t−pj +1
zjτ ≤ 1 ∀t
zjt ∈ {0, 1} ∀j, t}
[M] ≡ min{Xg∈G
cgλg :Xg∈G
T−pj +1Xt=1
zgjtλg = 1 ∀j,
Xg∈G
λg = 1, λg ∈ {0, 1} ∀g ∈ G}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: reformulations
[F] ≡ min{X
j
c(Sj ) : Sj + pj ≤ Si or Si + pi ≤ Sj ∀(i, j) ∈ J × J}
timeB4 C3 D5C3A8
Sa Sb Sc Sd
[R] ≡ min{X
jt
cjt zjt
T−pj +1Xt=1
zjt = 1 ∀j
Xj
tXτ=t−pj +1
zjτ ≤ 1 ∀t
zjt ∈ {0, 1} ∀j, t}
[M] ≡ min{Xg∈G
cgλg :Xg∈G
T−pj +1Xt=1
zgjtλg = 1 ∀j,
Xg∈G
λg = 1, λg ∈ {0, 1} ∀g ∈ G}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: Hybrid Approach
1 Generate a column for [M]:
[SP] ≡ min{X
jt
(cjt−πj )zjt :X
j
tXτ=t−pj +1
zjτ ≤ 1 ∀t, zjt ∈ {0, 1} ∀j, t},
1 2 3 4 5 6 7 8
2 Disaggregate the SP solution in arc variables z for [R]
3 Add the associated flow conservation constraints to [R]
4 Solve the restricted [R] to obtain dual prices
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: Hybrid Approach
1 Generate a column for [M]:
[SP] ≡ min{X
jt
(cjt−πj )zjt :X
j
tXτ=t−pj +1
zjτ ≤ 1 ∀t, zjt ∈ {0, 1} ∀j, t},
1 2 3 4 5 6 7 8
2 Disaggregate the SP solution in arc variables z for [R]
3 Add the associated flow conservation constraints to [R]
4 Solve the restricted [R] to obtain dual prices
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: Hybrid Approach
1 Generate a column for [M]:
[SP] ≡ min{X
jt
(cjt−πj )zjt :X
j
tXτ=t−pj +1
zjτ ≤ 1 ∀t, zjt ∈ {0, 1} ∀j, t},
1 2 3 4 5 6 7 8
2 Disaggregate the SP solution in arc variables z for [R]
3 Add the associated flow conservation constraints to [R]
4 Solve the restricted [R] to obtain dual prices
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPSingle Machine Scheduling: Hybrid Approach
1 Generate a column for [M]:
[SP] ≡ min{X
jt
(cjt−πj )zjt :X
j
tXτ=t−pj +1
zjτ ≤ 1 ∀t, zjt ∈ {0, 1} ∀j, t},
1 2 3 4 5 6 7 8
2 Disaggregate the SP solution in arc variables z for [R]
3 Add the associated flow conservation constraints to [R]
4 Solve the restricted [R] to obtain dual prices
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPStandard Column Gen. versus Hybrid Approach
Iteration Subproblem solution
Initial solution
· · · · · ·
Final solution
Column generation for [M]
1
2
3
10
11
Column-and-rowgeneration for [R]
Subproblem solution
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPInterest of the Hybrid Approach: Flow recombination
T = 7 pj = j
for g = 1, xgjt = 1 for (j, t) ∈ {(3, 1), (2, 4), (2, 6)};
for g = 2, xgjt = 1 for (j, t) ∈ {(5, 1), (1, 6), (1, 7)}.
In the associated [R] formulation, these solutions can be recombined:
1 2 3 4 5 6 7 8
But such recombinations are not feasible in the restricted master [M].
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPInterest of the Hybrid Approach: Numerical Tests
pj ∈ U[1, 10]
n Iter[R] Time[R] Iter[M] Time[M]12 29 0.4 92 1.625 50 3.7 351 36.250 87 33.3 1336 762.6
Iteration decrease: factor 15; time decrease: factor 22.
pj ∈ U[1, 50]
n Iter[R] Time[R] Iter[M] Time[M]12 116 16.6 121 46.025 174 150.6 399 876.4
Iteration decrease: factor 2.3; time decrease: factor 6.
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPMulti-Commodity Capacitated Network Design
[F] ≡ min{Xijk
ckij xk
ij +X
ij
fij yijXj
xkji −
Xj
xkij = dk
i ∀i, k
Xk
xkij ≤ uij yij ∀i, j
xkij ≥ 0 ∀i, j, k
yij ∈ IN ∀i, j}
[SP ij ] ≡ min{X
k
ck xk + f y :Xk
xk ≤ u y
xk ≤ min{dk , u y}∀k}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: Extended form. for the SPs
Let ysij = 1 and xks
ij = xkij if yij = s.
[SP ij ] ≡ min{Xks
ckij xks
ij +X
s
fij s ysij :X
s
ysij ≤ 1
(s − 1) uij ysij ≤
Xk
xksij ≤ s uij ys
ij ∀s
xksij ≤ min{dk , s uij} ys
ij ∀k , s}
Extended f. for SP [Croxton, Gendron and Magnanti OR07]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: extended formulation
[R] ≡ min{Xijks
ckij xks
ij +Xijs
fij s ysijX
js
xksji −
Xjs
xksij = dk
i ∀i, k
(s − 1) uij ysij ≤
Xk
xksij ≤ s uij ys
ij ∀i, j, s
0 ≤ xksij ≤ dk ys
ij ∀i, j, k , sXs
ysij = 1 ∀i, j
ysij ∈ {0, 1} ∀i, j, s}
solved by col-and-row generation [Frangioni & Gendron DAM09]better performance than by adding Benders’ cut to [F]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: standard col. gen. formulation
[M] ≡ min{X
i,j,s,g∈Gij
(ckij xg
ks + fij s ygs ) λij
g
Xjs
Xg∈Gij
xgks λ
ijg −
Xjs
Xg∈Gij
xgks λ
ijg = dk
i ∀i, k
Xg∈Gij
λijg ≤ 1 ∀i, j
λijg ∈ {0, 1} ∀i, j, g ∈ Gij}
col-and-row generation for [R] outperforms standard col gen for [M][Frangioni & Gendron WP10]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: Union of Polyhedra with [R]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: Union of Polyhedra with [R]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: Union of Polyhedra with [R]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 1: ∃ Extended Formulation for a SPNetwork Design: Union of Polyhedra with [M]
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPOutline
1 Assumption 1: ∃ Extended Formulation for a SPMachine SchedulingCapacitated Network Design
2 Assumption 2: ∃ Tight Reformulation for a SPBin Packing
3 Column-and-Row Generation
4 Recombination PropertyNetwork Flow based reformulationsDyn. Prog. based reformulationsUnion of Polyhedra based reformulations
5 Numerical experiments
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPAssumption 2: ∃ Tight Reformulation for a SP
[F] ≡ min{c xA x ≥ aB x ≥ b
x ∈ INn}
Subproblem : P = {x ∈ Rn+ : Bx ≥ b} and X = P ∩ Zn
Reformulation for a Subproblem
∃ a polyhedron Q = {z ∈ Re+ : Hz ≥ h, z ∈ Re
+} and transformation T s.t.:Q defines an tighter formulation for X :
conv(X)⊂projx Q = {x = Tz : Hz ≥ h, z ∈ Re+}⊂P
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPBin Packing
[F] ≡ min{X
k
δk :Xk
xik = 1 ∀iXi
si xik ≤ C δk ∀k
xik ∈ {0, 1} ∀i, kδk ∈ {0, 1} ∀k}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPBin Packing: standard column generation
[F] ≡ min{X
k
δk :Xk
xik = 1 ∀iXi
si xik ≤ C δk ∀k
xik ∈ {0, 1} ∀i, kδk ∈ {0, 1} ∀k}
[SP] ≡ min{δ −X
i
πixi :X
i
sixi ≤ C δ, (x, δ) ∈ {0, 1}n+1}
[M] ≡ min{X
g
λg :
Xg
xgi λg = 1 ∀i ∈ I
λg ∈ {0, 1} ∀g ∈ G.}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPBin Packing: Network Flow Reformulation
[SP] ≡ min{δ −Xiuv
πi f iuv : (f , δ) ∈ {0, 1}n∗m+1
Xi,v
f i0v + f0C = δ
Xi,u
f iuv =
Xi,u
f ivu + fv,C v = 1, . . . ,C − 1
Xi,u
f iuC +
Xv
fvC = δ
0 ≤ f iuv ≤ 1 ∀i, u, v = u + si }
60 1 2 3 4 5
A relaxation to an unbounded knapsack Problem
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPBin Packing: Network Flow ReformulationLet F i
uv =P
k f ikuv , FvC =
Pk f k
vC , and ∆ =P
k δk .
60 1 2 3 4 5
[R] ≡ min{∆ :X(u,v)
F iuv = 1 ∀i
Xi,v
F i0v + F0C = ∆
Xi,u
F iuv =
Xi,u
F ivu + FvC v = 1, . . . ,C − 1
Xi,u
F iuC +
Xv
FvC = ∆
F iuv ∈ {0, 1} ∀i, (u, v) : v = u + si }.
[Valerio de Carvalho AOR99]R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Assumption 2: ∃ Tight Reformulation for a SPPros and Cons of the Hybrid Approach
“+”Recombinations of SP solutionsExtra variables for branching [Valerio de Carvalho AOR99]Extra variables for defining cuts [Uchoa et al]
“-”need to handle dynamic row generationrequires a specific pricing oracle in the extended spacesymmetries
60 1 2 3 4 5
60 1 2 3 4 5
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Column-and-Row GenerationOutline
1 Assumption 1: ∃ Extended Formulation for a SPMachine SchedulingCapacitated Network Design
2 Assumption 2: ∃ Tight Reformulation for a SPBin Packing
3 Column-and-Row Generation
4 Recombination PropertyNetwork Flow based reformulationsDyn. Prog. based reformulationsUnion of Polyhedra based reformulations
5 Numerical experiments
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Column-and-Row GenerationRestricted reformulations
X = {xg}g∈G = {T zg}g∈G
[R] ≡ min{c T zA T z ≥ a π
H z ≥ hz ∈ Ze
+}
[M] ≡ min{P
g∈Gc xg λgPg∈GA xg λg ≥ aP
g∈Gλg = 1
λ ∈ {0, 1}|G|}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Column-and-Row GenerationHybrid column generation: convergence
vRLP
iteration
restricted master Lp values
Master LP value
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Column-and-Row GenerationHybrid column generation: convergence
L(π) = π a + minx∈X{(c − πA) x}
iteration
restricted master Lp values
intermediate Lagrangian bounds
Master LP value
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Column-and-Row GenerationHybrid column generation procedure
Step 0: Initialize the dual bound, β := −∞, and the subproblemsolution set G so that the linear relaxation of [R] is feasible.
Step 1: Solve [RLP ] and collect the dual solution π.
Step 2: Solve the pricing problem: z∗ ←min{(c− πA) T z : z ∈ Z} = min{(c− πA) x : x ∈ X}.
Step 3: Compute the Lagrangian dual bound:L(π) = π a + (c − πA) T z∗, and update the dual boundβ := max{β, L(π)} (lagrangian dual value estimator).If vR
LP ≤ β, STOP.
Step 4: Update the current bundle G by adding solution zs := z∗ andupdate the resulting restricted reformulation [R]. Goto Step 1.
Proposition
Either vRLP ≤ β (stopping condition), or [(c − πA) T − σ H]z∗ < 0 and
some of the components of z∗ have negative reduced cost in [RLP ].
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Column-and-Row GenerationHybrid column generation procedure
Step 0: Initialize the dual bound, β := −∞, and the subproblemsolution set G so that the linear relaxation of [R] is feasible.
Step 1: Solve [RLP ] and collect the dual solution π.
Step 2: Solve the pricing problem: z∗ ←min{(c− πA) T z : z ∈ Z} = min{(c− πA) x : x ∈ X}.
Step 3: Compute the Lagrangian dual bound:L(π) = π a + (c − πA) T z∗, and update the dual boundβ := max{β, L(π)} (lagrangian dual value estimator).If vR
LP ≤ β, STOP.
Step 4: Update the current bundle G by adding solution zs := z∗ andupdate the resulting restricted reformulation [R]. Goto Step 1.
Proposition
Either vRLP ≤ β (stopping condition), or [(c − πA) T − σ H]z∗ < 0 and
some of the components of z∗ have negative reduced cost in [RLP ].
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Recombination PropertyOutline
1 Assumption 1: ∃ Extended Formulation for a SPMachine SchedulingCapacitated Network Design
2 Assumption 2: ∃ Tight Reformulation for a SPBin Packing
3 Column-and-Row Generation
4 Recombination PropertyNetwork Flow based reformulationsDyn. Prog. based reformulationsUnion of Polyhedra based reformulations
5 Numerical experiments
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Recombination PropertyWhy consider Col Gen for [R] instead of [M]
RLP ⊃ MLP
Property (recombination)
Given G ⊂ G, ∃z ∈ RLP(G), such that z 6∈ conv(Z (G)).
1 2 3 4 5 6 7 8
True for Network Flow based reformulations: w = z1 − z2
is a cycle flow; w decomposes into elementary cycle flow wA,wB, . . .; z = z1 + αwA ∈ Q; but, z 6∈ conv(z1, z2)
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Recombination PropertyWhy consider Col Gen for [R] instead of [M]
RLP ⊃ MLP
Property (recombination)
Given G ⊂ G, ∃z ∈ RLP(G), such that z 6∈ conv(Z (G)).
1 2 3 4 5 6 7 8
True for Network Flow based reformulations: w = z1 − z2
is a cycle flow; w decomposes into elementary cycle flow wA,wB, . . .; z = z1 + αwA ∈ Q; but, z 6∈ conv(z1, z2)
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Recombination PropertyTrue for DP based reformulations[Martin et al OR90]Consider a solution to dynamic programming recursion
γ(l) = min(J,l)∈A
{Xj∈J
γ(j) + c(J, l)}
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R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Recombination PropertyTrue for DP based reformulations
The recombination of DP sol 1 and DP sol 2 into DP sol 3
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R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Recombination PropertyFor Union of Polyhedra based reformulations
Qk = {zk ∈ Re+ : Hkzk ≥ hk ; zk ≤ uk}
Q = {z =∑
k
zk ;∑
k
δk = 1; zk ≤ uk δk∀k ,Hkzk ≥ hk δk∀k}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Numerical experimentsOutline
1 Assumption 1: ∃ Extended Formulation for a SPMachine SchedulingCapacitated Network Design
2 Assumption 2: ∃ Tight Reformulation for a SPBin Packing
3 Column-and-Row Generation
4 Recombination PropertyNetwork Flow based reformulationsDyn. Prog. based reformulationsUnion of Polyhedra based reformulations
5 Numerical experiments
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Numerical experimentsMulti Item Lot Sizing – small bucket
Periods 1 to 5
Machine
Item 2
Item 1
[F] ≡ min{X
kt
(ckt xk
t + f kt yk
t ) :Xk
ykt ≤ 1 ∀t
tXτ=1
xkτ ≥ Dk
1T ∀k , t
xkt ≤ Dk
tT ykt ∀k , t
xkt ≥ 0 ∀k , t
ykt ∈ {0, 1} ∀k , t}
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Numerical experimentsMulti Item Lot Sizing – small bucket
[R] ≡ min{Xktu
cktu zk
tu :
Xk
TXu=t
zktu ≤ 1 ∀t,
Xu≥t
z0kt = 1 ∀k,
z0kt +
Xu<t
zku,t−1 =
Xu≥t
zktu ∀k, t = 1, . . . T ,
TXt=1
zkt,T = 1 ∀k,
zktu ∈ {0, 1} ∀k, u, t}
[M] ≡ min{X
k,g∈Gk(ck xg + f k yq ) λk
g :X
k,g∈Gkyg
t λkg ≤ 1 ∀t,
Xg∈Gk
λkg = 1 ∀k, λ ∈ IN|G|×K}
(K , T ) It[R] Sp[R] T[R] It[M] Sp[M] T[M](5,40) 65 109 2 109 540 9
(10,80) 116 341 25 166 1643 151
Iteration decrease: factor 1.4; time decrease: factor 6.
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations
Numerical experimentsConclusion
1 Column generation for an extended formulation isto be considered when:
The extended formulation← decomposition principle,SP solutions can be recombined into alternative ones.
2 There are computational evidence in the literature thatthis can be a competitive approach. There are also studieswhere it could have been used (and wasn’t).
3 The approach can be interpreted a stabilization methodfor column generation:
disaggregation helps,related to the use of exchange vectors.
4 It can be applied to an approximation of an extendedformulation [Van Vyve & Wolsey MP06] (see bin packing)
R. Sadykov and F. Vanderbeck Column Generation for Extended Formulations