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Structural Investigationof Piecewise Linearized Flow Problems
Frauke Liers, Maximilian MerkertFAU Erlangen-NürnbergAussois, January 8, 2015
Outline
Introduction
Structural Results
Empirical Results
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 2
Outline
Introduction
Structural Results
Empirical Results
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 3
MotivationNetwork flow problem, flow values are restricted to lie in certain intervals
Main application: piecewise linear approximation/relaxation→ solve MINLPs by means of MIP-techniques[e.g. Geißler, Martin, Morsi, Schewe, 2012]
q
∆p
q =
n∑i=1
qi , f (q) =
n∑i=1
f (Bi)zi + (qi − Bi)f (Bi+1)− f (Bi)
Bi+1 − Bi
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 4
MotivationNetwork flow problem, flow values are restricted to lie in certain intervals
Main application: piecewise linear approximation/relaxation
→ solve MINLPs by means of MIP-techniques[e.g. Geißler, Martin, Morsi, Schewe, 2012]
q
∆p
q =
n∑i=1
qi , f (q) =
n∑i=1
f (Bi)zi + (qi − Bi)f (Bi+1)− f (Bi)
Bi+1 − Bi
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 4
MotivationNetwork flow problem, flow values are restricted to lie in certain intervals
Main application: piecewise linear approximation/relaxation→ solve MINLPs by means of MIP-techniques[e.g. Geißler, Martin, Morsi, Schewe, 2012]
q
∆p
q =
n∑i=1
qi , f (q) =
n∑i=1
f (Bi)zi + (qi − Bi)f (Bi+1)− f (Bi)
Bi+1 − Bi
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 4
Problem setting
min f (z)s.t. lizi ≤ qi ≤ uizi ∀i (flow value in predefined interval)
Mq = d (demands)∑i∈Ia
zi = 1 ∀arcs a (exactly one interval active per arc)(q, z) ∈ Rm × {0,1}n
Formulation is locally ideal for one network arc.[Vielma, Ahmed, Nemhauser, 2009]
Question: How can the formulation be strengthened for multiplearcs/simple subnetworks?
Consider the convex hull of the projection onto the integer variables (=: P)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 5
Problem setting
min f (z)s.t. lizi ≤ qi ≤ uizi ∀i (flow value in predefined interval)
Mq = d (demands)∑i∈Ia
zi = 1 ∀arcs a (exactly one interval active per arc)(q, z) ∈ Rm × {0,1}n
Formulation is locally ideal for one network arc.[Vielma, Ahmed, Nemhauser, 2009]
Question: How can the formulation be strengthened for multiplearcs/simple subnetworks?
Consider the convex hull of the projection onto the integer variables (=: P)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 5
Problem setting
min f (z)s.t. lizi ≤ qi ≤ uizi ∀i (flow value in predefined interval)
Mq = d (demands)∑i∈Ia
zi = 1 ∀arcs a (exactly one interval active per arc)(q, z) ∈ Rm × {0,1}n
Formulation is locally ideal for one network arc.[Vielma, Ahmed, Nemhauser, 2009]
Question: How can the formulation be strengthened for multiplearcs/simple subnetworks?
Consider the convex hull of the projection onto the integer variables (=: P)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 5
Problem setting
min f (z)s.t. lizi ≤ qi ≤ uizi ∀i (flow value in predefined interval)
Mq = d (demands)∑i∈Ia
zi = 1 ∀arcs a (exactly one interval active per arc)(q, z) ∈ Rm × {0,1}n
Formulation is locally ideal for one network arc.[Vielma, Ahmed, Nemhauser, 2009]
Question: How can the formulation be strengthened for multiplearcs/simple subnetworks?
Consider the convex hull of the projection onto the integer variables (=: P)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 5
Outline
Introduction
Structural Results
Empirical Results
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 6
Two adjacent arcs at a degree-2 node
d = 0a b
Construct compatibility graph:
a1 [1,2]
a2 [1,4]
a3 [3,5]
b1 [1,2]
b2 [3,4]
b3 [3,6]
→ integral point in P = edge (2-clique) in the compatibility graph.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 7
Two adjacent arcs at a degree-2 node
d = 0a b
Construct compatibility graph:
a1 [1,2]
a2 [1,4]
a3 [3,5]
b1 [1,2]
b2 [3,4]
b3 [3,6]
→ integral point in P = edge (2-clique) in the compatibility graph.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 7
Two adjacent arcs at a degree-2 node
d = 0a b
Construct compatibility graph:
a1 [1,2]
a2 [1,4]
a3 [3,5]
b1 [1,2]
b2 [3,4]
b3 [3,6]
→ integral point in P = edge (2-clique) in the compatibility graph.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 7
Two adjacent arcs at a degree-2 node
A first class of cutting planes
Let V be a set of vertices of the compatibility graph GC that belong to the samenetwork arc. Then the inequality
z(V ) ≤ z(N(V )) (1)
is valid for P, where N(V ) denotes the set of neighbors of the set V .
Theorem (Complete description for paths of lengths 2)
The inequalities of type (1) (stable set constraints of GC) together with the trivialinequalities and the clique equations form a complete description of P.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 8
Two adjacent arcs at a degree-2 node
A first class of cutting planes
Let V be a set of vertices of the compatibility graph GC that belong to the samenetwork arc. Then the inequality
z(V ) ≤ z(N(V )) (1)
is valid for P, where N(V ) denotes the set of neighbors of the set V .
Theorem (Complete description for paths of lengths 2)
The inequalities of type (1) (stable set constraints of GC) together with the trivialinequalities and the clique equations form a complete description of P.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 8
Proof outline
Let P = {z ∈ [0,1]n+m | z(A) = 1, z(B) = 1, z(V ) ≤ z(N(V )) for all V ⊆ V (GC)}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph
• GC is bipartite
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅)
• All inequalities needed for a complete description of PCLIQUE are implied by theconstraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = 2 is valid for PCLIQUE and PCLIQUE|z(V (GC))=2 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 9
Proof outline
Let P = {z ∈ [0,1]n+m | z(A) = 1, z(B) = 1, z(V ) ≤ z(N(V )) for all V ⊆ V (GC)}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph
• GC is bipartite
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅)
• All inequalities needed for a complete description of PCLIQUE are implied by theconstraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = 2 is valid for PCLIQUE and PCLIQUE|z(V (GC))=2 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 9
Proof outline
Let P = {z ∈ [0,1]n+m | z(A) = 1, z(B) = 1, z(V ) ≤ z(N(V )) for all V ⊆ V (GC)}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph• GC is bipartite
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅)
• All inequalities needed for a complete description of PCLIQUE are implied by theconstraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = 2 is valid for PCLIQUE and PCLIQUE|z(V (GC))=2 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 9
Proof outline
Let P = {z ∈ [0,1]n+m | z(A) = 1, z(B) = 1, z(V ) ≤ z(N(V )) for all V ⊆ V (GC)}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph• GC is bipartite
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅)• All inequalities needed for a complete description of PCLIQUE are implied by the
constraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = 2 is valid for PCLIQUE and PCLIQUE|z(V (GC))=2 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 9
Paths of network arcs
d = 0 d = 0 d = 0e1 e2 e3 ek...
Compatibility graph as before, including compatibility edges for non-adjacentnetwork arcs
Observation: the interval property guarantees that GC detects all variableconflicts!
Theorem (Complete description for paths of network arcs)
The stable set constraints of GC together with the trivial inequalities and theclique equations form a complete description of P.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 10
Paths of network arcs
d = 0 d = 0 d = 0e1 e2 e3 ek...
Compatibility graph as before, including compatibility edges for non-adjacentnetwork arcs
Observation: the interval property guarantees that GC detects all variableconflicts!
Theorem (Complete description for paths of network arcs)
The stable set constraints of GC together with the trivial inequalities and theclique equations form a complete description of P.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 10
Paths of network arcs
d = 0 d = 0 d = 0e1 e2 e3 ek...
Compatibility graph as before, including compatibility edges for non-adjacentnetwork arcs
Observation: the interval property guarantees that GC detects all variableconflicts!
Theorem (Complete description for paths of network arcs)
The stable set constraints of GC together with the trivial inequalities and theclique equations form a complete description of P.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 10
Proof outline
LetP = {z ∈ [0,1]|V (GC)| | z(C) = 1 ∀ partitions C, z(S) ≤ 1 ∀ stable sets S ⊆ V (GC)}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph
• GC is bipartite
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅)
X• All inequalities needed for a complete description of PCLIQUE are implied by the
constraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = k is valid for PCLIQUE and PCLIQUE|z(V (GC))=k 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 11
Proof outline
LetP = {z ∈ [0,1]|V (GC)| | z(C) = 1 ∀ partitions C, z(S) ≤ 1 ∀ stable sets S ⊆ V (GC)}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph
• GC is bipartite
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅) X• All inequalities needed for a complete description of PCLIQUE are implied by the
constraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = k is valid for PCLIQUE and PCLIQUE|z(V (GC))=k 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 11
Perfect graphs
Examples of well-known classes of perfect graphs:bipartite graphs, line graphs of bipartite graphs, chordal graphs.
Theorem (Chudnovsky, Robertson, Seymour, Thomas, 2006)
A graph is perfect if and only if it has neither odd holes nor odd antiholes(complements of odd holes) of size ≥ 5 as induced subgraphs.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 12
Perfect graphs
Examples of well-known classes of perfect graphs:bipartite graphs, line graphs of bipartite graphs, chordal graphs.
Theorem (Chudnovsky, Robertson, Seymour, Thomas, 2006)
A graph is perfect if and only if it has neither odd holes nor odd antiholes(complements of odd holes) of size ≥ 5 as induced subgraphs.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 12
Perfect graphs
Examples of well-known classes of perfect graphs:bipartite graphs, line graphs of bipartite graphs, chordal graphs.
Theorem (Chudnovsky, Robertson, Seymour, Thomas, 2006)
A graph is perfect if and only if it has neither odd holes nor odd antiholes(complements of odd holes) of size ≥ 5 as induced subgraphs.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 12
A class of perfect graphs
Definition (Partition-chordal graphs)
A graph G is called partition-chordal (with partition order k)⇔ it has ak -partition and a set A ⊆ {(u, v) | u 6= v belong to the same partition} (fill edges)of edges such that adding all edges in A yields a chordal graph.
Theorem
A graph that is partition-chordal is also perfect.
Proof: use characterization via forbidden induced subgraphs(here only for the case of odd cycles):
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 13
A class of perfect graphs
Definition (Partition-chordal graphs)
A graph G is called partition-chordal (with partition order k)⇔ it has ak -partition and a set A ⊆ {(u, v) | u 6= v belong to the same partition} (fill edges)of edges such that adding all edges in A yields a chordal graph.
Theorem
A graph that is partition-chordal is also perfect.
Proof: use characterization via forbidden induced subgraphs(here only for the case of odd cycles):
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 13
A class of perfect graphs
Definition (Partition-chordal graphs)
A graph G is called partition-chordal (with partition order k)⇔ it has ak -partition and a set A ⊆ {(u, v) | u 6= v belong to the same partition} (fill edges)of edges such that adding all edges in A yields a chordal graph.
Theorem
A graph that is partition-chordal is also perfect.
Proof: use characterization via forbidden induced subgraphs(here only for the case of odd cycles):
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 13
A class of perfect graphs
Definition (Partition-chordal graphs)
A graph G is called partition-chordal (with partition order k)⇔ it has ak -partition and a set A ⊆ {(u, v) | u 6= v belong to the same partition} (fill edges)of edges such that adding all edges in A yields a chordal graph.
Theorem
A graph that is partition-chordal is also perfect.
Proof: use characterization via forbidden induced subgraphs(here only for the case of odd cycles):
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 13
A class of perfect graphs
Definition (Partition-chordal graphs)
A graph G is called partition-chordal (with partition order k)⇔ it has ak -partition and a set A ⊆ {(u, v) | u 6= v belong to the same partition} (fill edges)of edges such that adding all edges in A yields a chordal graph.
Theorem
A graph that is partition-chordal is also perfect.
Proof: use characterization via forbidden induced subgraphs(here only for the case of odd cycles):
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 13
Interval configurations: a special case
d = 0 d = 0 d = 0e1 e2 e3 ek...
How strong are inequalities that use variables from only two partitions (networkarcs) each?
Theorem
If for each ordered pair (I1, I2) of intervals belonging to the same partition I1\I2 isconnected, then the inequalities of type
z(V ) ≤ z(NC2(V )),V ⊆ C1 6= C2,
together with the trivial inequalities and the clique equations form a completedescription of P, where V ⊆ V (GC); C1,C2 partitions.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 14
Interval configurations: a special case
d = 0 d = 0 d = 0e1 e2 e3 ek...
How strong are inequalities that use variables from only two partitions (networkarcs) each?
Theorem
If for each ordered pair (I1, I2) of intervals belonging to the same partition I1\I2 isconnected, then the inequalities of type
z(V ) ≤ z(NC2(V )),V ⊆ C1 6= C2,
together with the trivial inequalities and the clique equations form a completedescription of P, where V ⊆ V (GC); C1,C2 partitions.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 14
Proof outline
LetP = {z ∈ [0,1]|V (GC)| | z(C) = 1 ∀ partitions C, z(V ) ≤ z(NC2
(V )),V ⊆ C1 6= C2}
Idea: use knowledge on a complete description of the clique polytope on perfectgraphs: PCLIQUE = {x ∈ RV (G) | 0 ≤ xi ≤ 1 ∀i ,
∑vi∈S xi ≤ 1 ∀ stable sets S}
= conv({x ∈ RV (G) | x is a clique vector}).
to show:
1. GC is a perfect graph X• GC is partition-chordal
2. P is a face of the clique polytope PCLIQUE (if P 6= ∅)• all inequalities needed for a complete description of PCLIQUE are implied by the
constraints of P ⇒ P ⊆ PCLIQUE.• z(V (GC)) = k is valid for PCLIQUE and PCLIQUE|z(V (GC))=k 6= ∅.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 15
Sketch of the proof
a1
a2
a3 [3,4]
a4
[2,6]!?
b1 [1,1]
b2 [1,2]
b3
b4
c1
c2
c3
c4 [5,6]
d1
d2
d3
d4
• Consider the stable set S = {b1,b2,a3, c4}
• The stable-set inequality of S is implied by that of {b1,b2,a3} ∪NA({c4}) asz(c4) ≤ z(NA({c4}))...
• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.• Proof by induction ...
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 16
Sketch of the proof
a1
a2
a3 [3,4]
a4
[2,6]!?
b1 [1,1]
b2 [1,2]
b3
b4
c1
c2
c3
c4 [5,6]
d1
d2
d3
d4
• Consider the stable set S = {b1,b2,a3, c4}• The stable-set inequality of S is implied by that of {b1,b2,a3} ∪NA({c4}) as
z(c4) ≤ z(NA({c4}))...
• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.• Proof by induction ...
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 16
Sketch of the proof
a1
a2
a3 [3,4]
a4
[2,6]!?
b1 [1,1]
b2 [1,2]
b3
b4
c1
c2
c3
c4 [5,6]
d1
d2
d3
d4
• Consider the stable set S = {b1,b2,a3, c4}• The stable-set inequality of S is implied by that of {b1,b2,a3} ∪NA({c4}) as
z(c4) ≤ z(NA({c4}))...
• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.
• Proof by induction ...
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 16
Sketch of the proof
a1
a2
a3 [3,4]
a4 [2,6]!?
b1 [1,1]
b2 [1,2]
b3
b4
c1
c2
c3
c4 [5,6]
d1
d2
d3
d4
• Consider the stable set S = {b1,b2,a3, c4}• The stable-set inequality of S is implied by that of {b1,b2,a3} ∪NA({c4}) as
z(c4) ≤ z(NA({c4}))...
• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.
• Proof by induction ...
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 16
Sketch of the proof
a1
a2
a3 [3,4]
a4
[2,6]!?
b1 [1,1]
b2 [1,2]
b3
b4
c1
c2
c3
c4 [5,6]
d1
d2
d3
d4
• Consider the stable set S = {b1,b2,a3, c4}• The stable-set inequality of S is implied by that of {b1,b2,a3} ∪NA({c4}) as
z(c4) ≤ z(NA({c4}))...
• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.• Proof by induction ...
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 16
An example
a1 [1,2]
a2 [1,6]
b1 [3,4]
b2 [1,6]
c1 [5,6]
c2 [1,6]
(za1, za2, zb1, zb2, zc1, zc2) = (12,
12,
12,
12,
12,
12) satisfies all stable set inequalities with
variables from only two partitions,but does not lie in P as za1 + zb1 + zc1 ≤ 1 is violated.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 17
Linearizations based on the incremental methodModeling a network arc with the δ-method: [Markowitz, Manne]given an interval [l ,u] and breakpoints B1 = l ,B2, ...,Bn,Bn+1 = u
q = B1z1 +
n∑i=1
(Bi+1 − Bi)δi
zi ≥ δi , i = 1, ...n and δi ≥ zi+1, i = 1, ...,n − 1, δn ≥ 0 (filling condition)
Results can be transferred, P and Pδ are isomorphic under the linear bijectionT : zδ 7→ z with zi := ziδ − zi+1δ
, i = 1, . . . ,n − 1; zn := znδwith inverse
T−1 : zi 7→∑n
j=i zi .
Theorem (Version for the incremental method)
The valid inequalities of typezai ≥ zbi
together with the trivial inequalities and the filling condition form a completedescription of Pδ.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 18
Linearizations based on the incremental methodModeling a network arc with the δ-method: [Markowitz, Manne]given an interval [l ,u] and breakpoints B1 = l ,B2, ...,Bn,Bn+1 = u
q = B1z1 +
n∑i=1
(Bi+1 − Bi)δi
zi ≥ δi , i = 1, ...n and δi ≥ zi+1, i = 1, ...,n − 1, δn ≥ 0 (filling condition)
Results can be transferred, P and Pδ are isomorphic under the linear bijectionT : zδ 7→ z with zi := ziδ − zi+1δ
, i = 1, . . . ,n − 1; zn := znδwith inverse
T−1 : zi 7→∑n
j=i zi .
Theorem (Version for the incremental method)
The valid inequalities of typezai ≥ zbi
together with the trivial inequalities and the filling condition form a completedescription of Pδ.
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 18
Forks: a simple class of cutting planes
Multiple in- and outflows at a network node of degree dsuppose an integral vector z with zai = 1 on arc a⇔ i = ja is infeasible since∑
a ∈ inflows
uaja<
∑a ∈ outflows
laja, (the symmetric case is analogous)
We have the valid cover inequality∑a ∈ inflows
zaja+
∑a ∈ outflows
zaja≤ d − 1
It can be strengthened to∑a ∈ inflows
jk∑i=1
zai +∑
a ∈ outflows
na∑i=jk
zai ≤ d − 1
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 19
Forks: a simple class of cutting planes
Multiple in- and outflows at a network node of degree dsuppose an integral vector z with zai = 1 on arc a⇔ i = ja is infeasible since∑
a ∈ inflows
uaja<
∑a ∈ outflows
laja, (the symmetric case is analogous)
We have the valid cover inequality∑a ∈ inflows
zaja+
∑a ∈ outflows
zaja≤ d − 1
It can be strengthened to∑a ∈ inflows
jk∑i=1
zai +∑
a ∈ outflows
na∑i=jk
zai ≤ d − 1
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 19
Forks: a simple class of cutting planes
Multiple in- and outflows at a network node of degree dsuppose an integral vector z with zai = 1 on arc a⇔ i = ja is infeasible since∑
a ∈ inflows
uaja<
∑a ∈ outflows
laja, (the symmetric case is analogous)
We have the valid cover inequality∑a ∈ inflows
zaja+
∑a ∈ outflows
zaja≤ d − 1
It can be strengthened to∑a ∈ inflows
jk∑i=1
zai +∑
a ∈ outflows
na∑i=jk
zai ≤ d − 1
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 19
Outline
Introduction
Structural Results
Empirical Results
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 20
The algorithms/implementations
Comparison of 4 methods:
• Gurobi with standard parameter settings (MIP)• Exact separation of cutting planes from the complete description for paths of
network arcs (PATHCUT), called at every 50th branch&bound node• Heuristic separation of a simple class of cutting planes at nodes of arbitrary
degree (FORKCUT), called at every 50th branch&bound node• Separation of both types of cutting planes (CUT)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 21
The algorithms/implementations
Comparison of 4 methods:• Gurobi with standard parameter settings (MIP)
• Exact separation of cutting planes from the complete description for paths ofnetwork arcs (PATHCUT), called at every 50th branch&bound node• Heuristic separation of a simple class of cutting planes at nodes of arbitrary
degree (FORKCUT), called at every 50th branch&bound node• Separation of both types of cutting planes (CUT)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 21
The algorithms/implementations
Comparison of 4 methods:• Gurobi with standard parameter settings (MIP)• Exact separation of cutting planes from the complete description for paths of
network arcs (PATHCUT), called at every 50th branch&bound node
• Heuristic separation of a simple class of cutting planes at nodes of arbitrarydegree (FORKCUT), called at every 50th branch&bound node• Separation of both types of cutting planes (CUT)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 21
The algorithms/implementations
Comparison of 4 methods:• Gurobi with standard parameter settings (MIP)• Exact separation of cutting planes from the complete description for paths of
network arcs (PATHCUT), called at every 50th branch&bound node• Heuristic separation of a simple class of cutting planes at nodes of arbitrary
degree (FORKCUT), called at every 50th branch&bound node
• Separation of both types of cutting planes (CUT)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 21
The algorithms/implementations
Comparison of 4 methods:• Gurobi with standard parameter settings (MIP)• Exact separation of cutting planes from the complete description for paths of
network arcs (PATHCUT), called at every 50th branch&bound node• Heuristic separation of a simple class of cutting planes at nodes of arbitrary
degree (FORKCUT), called at every 50th branch&bound node• Separation of both types of cutting planes (CUT)
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 21
Setting for the computationsinstances:• Random graphs based on a preferential attachment model• Node demands, objective function coefficients randomly generated• Intervals constructed by random subdivision of a large flow interval• 5 instances per configuration• Test set I: network sizes of 50-100 nodes,10 intervals per arc• Test set II: 100 network nodes, varying number of intervals• Test set III: realistic gas network topology with 152 nodes, varying number of
intervals
Implementation with Gurobi 5.6.3, C++-interfaceComputations on queuing cluster koebes (in Cologne)• In total 720 CPU cores, 4,5 TB central memory• Each core has 2 CPUs with 10 cores each (3,0
GHz), 128 GB RAM• Each Job is assigned 4 cores• Time limit: 10 hours
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 22
Test set I (varying number of nodes) - runtimes
|V| MIP FORKCUT PATHCUT CUTsolved CPU[s] solved CPU[s] solved CPU[s] solved CPU[s]
50 5 4.33 5 4.70 5 4.38 5 4.9060 5 9.72 5 11.80 5 8.71 5 7.3970 5 23.03 5 20.12 5 20.40 5 20.3980 5 20.11 5 21.20 5 18.65 5 17.3190 5 47.08 5 45.85 5 36.93 5 38.84
100 5 89.73 5 65.88 5 84.09 5 59.17110 5 170.34 5 144.85 5 127.06 5 104.02120 5 167.37 5 113.98 5 115.19 5 74.48130 5 485.31 5 330.24 5 538.29 5 281.52140 5 916.05 5 707.98 5 720.72 5 418.94150 5 1 076.37 5 616.40 5 565.13 5 416.12
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 23
Test set II (varying number of intervals) - runtimes
# intervals MIP FORKCUT PATHCUT CUTper arc solved CPU[s] solved CPU[s] solved CPU[s] solved CPU[s]
3 5 0.18 5 0.28 5 0.26 5 0.335 5 3.90 5 3.99 5 3.37 5 3.567 5 27.05 5 21.18 5 26.78 5 18.97
10 5 70.90 5 69.21 5 63.60 5 54.2715 5 1 028.48 5 862.44 5 573.18 5 367.9020 4 6 586.35 5 6 851.56 5 6 831.81 5 5 376.4025 4 14 866.45 4 5 480.13 4 6 085.32 5 8 389.7330 2 48 910.73 3 6 392.94 3 10 733.53 5 8 899.5850 1 65 283.80 1 4 424.47 1 6 983.83 2 16 539.8770 0 ∞ 0 ∞ 0 ∞ 1 85 237.93
100 0 ∞ 0 ∞ 0 ∞ 0 ∞
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 24
Test set III (gas network topology) - runtimes
# intervals MIP FORKCUT PATHCUT CUTper arc solved CPU[s] solved CPU[s] solved CPU[s] solved CPU[s]
3 5 0.06 5 0.09 5 0.12 5 0.135 5 0.28 5 0.36 5 0.40 5 0.467 5 1.37 5 1.63 5 1.81 5 1.76
10 5 3.99 5 4.95 5 5.12 5 5.8715 5 33.69 5 39.25 5 45.29 5 36.4520 5 447.82 5 189.33 5 143.47 5 148.8625 5 948.00 5 379.26 5 371.83 5 239.0430 5 5 507.18 5 559.79 5 646.48 5 600.7250 1 110 578.15 5 3 817.25 5 9 298.63 5 3 504.3570 0 ∞ 5 10 176.50 5 28 236.29 5 12 521.38
100 0 ∞ 5 21 512.62 2 58 528.93 5 39 990.03
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 25
Test set III - Branch&Bound nodes
# intervals MIP FORKCUT PATHCUT CUTper arc
3 21 16 15 155 177 113 105 917 1 261 1 032 918 781
10 2 072 1 823 1 598 1 49015 14 261 8 886 9 378 5 35120 168 550 31 434 21 987 15 22425 262 863 48 709 46 117 17 99630 1 344 796 51 087 66 624 33 514
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 26
Summary
Theory:• Complete description for paths of arbitrary length• A class of perfect graphs• Simple cutting planes for stars• Transferable to a modeling according to the δ-method
Computations:• Implementation of separating routines for Gurobi• On random instances reduction of branch&bound nodes and improvement in
runtime, in particular for many intervals per arc
Thank you for your attention!
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 27
Summary
Theory:• Complete description for paths of arbitrary length• A class of perfect graphs• Simple cutting planes for stars• Transferable to a modeling according to the δ-method
Computations:• Implementation of separating routines for Gurobi• On random instances reduction of branch&bound nodes and improvement in
runtime, in particular for many intervals per arc
Thank you for your attention!
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 27
Summary
Theory:• Complete description for paths of arbitrary length• A class of perfect graphs• Simple cutting planes for stars• Transferable to a modeling according to the δ-method
Computations:• Implementation of separating routines for Gurobi• On random instances reduction of branch&bound nodes and improvement in
runtime, in particular for many intervals per arc
Thank you for your attention!
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 27
Summary
Theory:• Complete description for paths of arbitrary length• A class of perfect graphs• Simple cutting planes for stars• Transferable to a modeling according to the δ-method
Computations:• Implementation of separating routines for Gurobi• On random instances reduction of branch&bound nodes and improvement in
runtime, in particular for many intervals per arc
Thank you for your attention!
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 27
Summary
Theory:• Complete description for paths of arbitrary length• A class of perfect graphs• Simple cutting planes for stars• Transferable to a modeling according to the δ-method
Computations:• Implementation of separating routines for Gurobi• On random instances reduction of branch&bound nodes and improvement in
runtime, in particular for many intervals per arc
Thank you for your attention!
Maximilian Merkert · FAU Erlangen-Nürnberg · Structural Investigation of Piecewise Linearized Flow Problems Aussois, January 8, 2015 27