structural investigation of piecewise linearized flow problems · compatibility graph as before,...

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

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]

n∑i=1

qi , f (q) =

n∑i=1

f (Bi)zi + (qi − Bi)f (Bi+1)− f (Bi)

Bi+1 − Bi

Main application: piecewise linear approximation/relaxation

→ solve MINLPs by means of MIP-techniques[e.g. Geißler, Martin, Morsi, Schewe, 2012]

n∑i=1

qi , f (q) =

n∑i=1

Bi+1 − Bi

Main application: piecewise linear approximation/relaxation→ solve MINLPs by means of MIP-techniques[e.g. Geißler, Martin, Morsi, Schewe, 2012]

n∑i=1

qi , f (q) =

n∑i=1

Bi+1 − Bi

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)

Problem setting

Outline

Introduction

Structural Results

Empirical Results

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.

d = 0a b

a1 [1,2]

a2 [1,4]

a3 [3,5]

b1 [1,2]

b2 [3,4]

b3 [3,6]

d = 0a b

a1 [1,2]

a2 [1,4]

a3 [3,5]

b1 [1,2]

b2 [3,4]

b3 [3,6]

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.

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.

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= ∅.

Proof outline

to show:

• GC is bipartite

Proof outline

to show:

1. GC is a perfect graph• GC is bipartite

Proof outline

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= ∅.

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.

d = 0 d = 0 d = 0e1 e2 e3 ek...

Proof outline

LetP = {z ∈ [0,1]|V (GC)| | z(C) = 1 ∀ partitions C, z(S) ≤ 1 ∀ stable sets S ⊆ V (GC)}

to show:

• GC is bipartite

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= ∅.

Proof outline

LetP = {z ∈ [0,1]|V (GC)| | z(C) = 1 ∀ partitions C, z(S) ≤ 1 ∀ stable sets S ⊆ V (GC)}

to show:

• 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

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.

Perfect graphs

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):

Theorem

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.

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.

Proof outline

LetP = {z ∈ [0,1]|V (GC)| | z(C) = 1 ∀ partitions C, z(V ) ≤ z(NC2

(V )),V ⊆ C1 6= C2}

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

Sketch of the proof

a3 [3,4]

[2,6]!?

b1 [1,1]

b2 [1,2]

c4 [5,6]

• 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 ...

Sketch of the proof

a3 [3,4]

[2,6]!?

b1 [1,1]

b2 [1,2]

c4 [5,6]

• 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}))...

Sketch of the proof

a3 [3,4]

[2,6]!?

b1 [1,1]

b2 [1,2]

c4 [5,6]

z(c4) ≤ z(NA({c4}))...

• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.

• Proof by induction ...

Sketch of the proof

a3 [3,4]

a4 [2,6]!?

b1 [1,1]

b2 [1,2]

c4 [5,6]

z(c4) ≤ z(NA({c4}))...

• ... if {b1,b2,a3} ∪NA({c4}) is actually a stable set.

• Proof by induction ...

Sketch of the proof

a3 [3,4]

[2,6]!?

b1 [1,1]

b2 [1,2]

c4 [5,6]

z(c4) ≤ z(NA({c4}))...

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) satisfies all stable set inequalities with

variables from only two partitions,but does not lie in P as za1 + zb1 + zc1 ≤ 1 is violated.

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.

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.

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

∑a ∈ outflows

laja, (the symmetric case is analogous)

We have the valid cover inequality∑a ∈ inflows

∑a ∈ outflows

zaja≤ d − 1

It can be strengthened to∑a ∈ inflows

jk∑i=1

zai +∑

a ∈ outflows

na∑i=jk

zai ≤ d − 1

a ∈ inflows

∑a ∈ outflows

zaja≤ d − 1

jk∑i=1

zai +∑

a ∈ outflows

na∑i=jk

zai ≤ d − 1

a ∈ inflows

∑a ∈ outflows

zaja≤ d − 1

jk∑i=1

zai +∑

a ∈ outflows

na∑i=jk

zai ≤ d − 1

Outline

Introduction

Structural Results

Empirical Results

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)

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

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)

degree (FORKCUT), called at every 50th branch&bound node

• Separation of both types of cutting planes (CUT)

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

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

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 ∞

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

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

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!

Summary

structural investigation of piecewise linearized flow problems · compatibility graph as before,...

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