[PPT] - Projected Chv atal-Gomory cuts for Mixed Integer Linear Programs PowerPoint Presentation

SLIDE 1

Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs Pierre Bonami

CMU, USA

Gerard Cornu´ ejols

CMU, USA and LIF Marseille, France

Sanjeeb Dash

IBM T.J. Watson, USA

Matteo Fischetti

University of Padova, Italy

Andrea Lodi

University of Bologna, Italy alodi@deis.unibo.it Aussois X, January 9, 2006

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs

SLIDE 2

Notation and background

Consider an Integer Linear Program (ILP) of the form:

min{cTx : Ax ≤ b, x ≥ 0 integer}

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 1

SLIDE 3

Notation and background

Consider an Integer Linear Program (ILP) of the form:

min{cTx : Ax ≤ b, x ≥ 0 integer} and two associated polyhedra: P := {x ∈ I Rn

+ : Ax ≤ b}

PI := conv{x ∈ Zn

+ : Ax ≤ b} = conv(P ∩ Zn)

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 1

SLIDE 4

Notation and background

Consider an Integer Linear Program (ILP) of the form:

min{cTx : Ax ≤ b, x ≥ 0 integer} and two associated polyhedra: P := {x ∈ I Rn

+ : Ax ≤ b}

PI := conv{x ∈ Zn

+ : Ax ≤ b} = conv(P ∩ Zn)

A Chv´

atal-Gomory (CG) cut is a valid inequality for PI of the form:

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 1

SLIDE 5

Notation and background

Consider an Integer Linear Program (ILP) of the form:

min{cTx : Ax ≤ b, x ≥ 0 integer} and two associated polyhedra: P := {x ∈ I Rn

+ : Ax ≤ b}

PI := conv{x ∈ Zn

+ : Ax ≤ b} = conv(P ∩ Zn)

A Chv´

atal-Gomory (CG) cut is a valid inequality for PI of the form: ⌊uTA⌋x ≤ ⌊uTb⌋ where u ∈ Rm

+ is called the CG multiplier vector, and ⌊·⌋ denotes lower integer part.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 1

SLIDE 6

Notation and background

Consider an Integer Linear Program (ILP) of the form:

min{cTx : Ax ≤ b, x ≥ 0 integer} and two associated polyhedra: P := {x ∈ I Rn

+ : Ax ≤ b}

PI := conv{x ∈ Zn

+ : Ax ≤ b} = conv(P ∩ Zn)

A Chv´

atal-Gomory (CG) cut is a valid inequality for PI of the form: ⌊uTA⌋x ≤ ⌊uTb⌋ where u ∈ Rm

+ is called the CG multiplier vector, and ⌊·⌋ denotes lower integer part.

The first Chv´

atal closure of P is defined as: P1 := {x ≥ 0 : Ax ≤ b, ⌊uTA⌋x ≤ ⌊uTb⌋ for all u ∈ I Rm

+}

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 1

SLIDE 7

Notation and background

Consider an Integer Linear Program (ILP) of the form:

min{cTx : Ax ≤ b, x ≥ 0 integer} and two associated polyhedra: P := {x ∈ I Rn

+ : Ax ≤ b}

PI := conv{x ∈ Zn

+ : Ax ≤ b} = conv(P ∩ Zn)

A Chv´

atal-Gomory (CG) cut is a valid inequality for PI of the form: ⌊uTA⌋x ≤ ⌊uTb⌋ where u ∈ Rm

+ is called the CG multiplier vector, and ⌊·⌋ denotes lower integer part.

The first Chv´

atal closure of P is defined as: P1 := {x ≥ 0 : Ax ≤ b, ⌊uTA⌋x ≤ ⌊uTb⌋ for all u ∈ I Rm

+}

P1 is indeed a polyhedron, i.e., a finite number of CG cuts suffice to define it.

[Chv´ atal 1973]

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 1

SLIDE 8

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 9

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
Chv´

atal-Gomory separation problem (CG-SEP) is NP-hard.

[Eisenbrand 1999]

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 10

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
Chv´

atal-Gomory separation problem (CG-SEP) is NP-hard.

[Eisenbrand 1999]

Recently Fischetti & Lodi have shown that:

Optimizing over P 1 is possible in practice via an MIP model (MIPping).

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 11

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
Chv´

atal-Gomory separation problem (CG-SEP) is NP-hard.

[Eisenbrand 1999]

Recently Fischetti & Lodi have shown that:

Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of PI in practice.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 12

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
Chv´

atal-Gomory separation problem (CG-SEP) is NP-hard.

[Eisenbrand 1999]

Recently Fischetti & Lodi have shown that:

Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of PI in practice. CG cuts in the first closure have a nice numerical behavior and stability.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 13

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
Chv´

atal-Gomory separation problem (CG-SEP) is NP-hard.

[Eisenbrand 1999]

Recently Fischetti & Lodi have shown that:

Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of PI in practice. CG cuts in the first closure have a nice numerical behavior and stability.

Thus, the natural question is:

What does it happen in the Mixed IP case?

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 14

Notation and background (cont.d)

Clearly, PI ⊆ P1 ⊆ P .
Chv´

atal-Gomory separation problem (CG-SEP) is NP-hard.

[Eisenbrand 1999]

Recently Fischetti & Lodi have shown that:

Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of PI in practice. CG cuts in the first closure have a nice numerical behavior and stability.

Thus, the natural question is:

What does it happen in the Mixed IP case?

Of course, the natural answer would be using Gomory Mixed Integer cuts (GMI) (also known as

MIR cuts and split cuts) but their separation is much more involved than CG separation: nobody knows a MIP model for GMI yet!

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 2

SLIDE 15

Projected Chv´ atal-Gomory cuts

Our first order of business is to extend the classical definition of Chv´

atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 3

SLIDE 16

Projected Chv´ atal-Gomory cuts

Our first order of business is to extend the classical definition of Chv´

atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP.

We then consider the MIP:

min{cTx + f Ty : Ax + Cy ≤ b, x ≥ 0, x integral, y ≥ 0}

with the two associated polyhedra:

P (x, y) := {(x, y) ∈ Rn

+ × Rr + : Ax + Cy ≤ b}

PI(x, y) := conv({(x, y) ∈ P (x, y) : x integral})

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 3

SLIDE 17

Projected Chv´ atal-Gomory cuts

Our first order of business is to extend the classical definition of Chv´

atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP.

We then consider the MIP:

min{cTx + f Ty : Ax + Cy ≤ b, x ≥ 0, x integral, y ≥ 0}

with the two associated polyhedra:

P (x, y) := {(x, y) ∈ Rn

+ × Rr + : Ax + Cy ≤ b}

PI(x, y) := conv({(x, y) ∈ P (x, y) : x integral})

and we project P (x, y) onto the space of x variables as:

P (x) := {x ∈ Rn

+ : there exists y ∈ Rr + s.t. Ax + Cy ≤ b}

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 3

SLIDE 18

Projected Chv´ atal-Gomory cuts

Our first order of business is to extend the classical definition of Chv´

atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP.

We then consider the MIP:

min{cTx + f Ty : Ax + Cy ≤ b, x ≥ 0, x integral, y ≥ 0}

with the two associated polyhedra:

P (x, y) := {(x, y) ∈ Rn

+ × Rr + : Ax + Cy ≤ b}

PI(x, y) := conv({(x, y) ∈ P (x, y) : x integral})

and we project P (x, y) onto the space of x variables as:

P (x) := {x ∈ Rn

+ : there exists y ∈ Rr + s.t. Ax + Cy ≤ b}

= {x ∈ Rn

+ : ukA ≤ ukb, k = 1, . . . , K}

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 3

SLIDE 19

Projected Chv´ atal-Gomory cuts

Our first order of business is to extend the classical definition of Chv´

atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP.

We then consider the MIP:

min{cTx + f Ty : Ax + Cy ≤ b, x ≥ 0, x integral, y ≥ 0}

with the two associated polyhedra:

P (x, y) := {(x, y) ∈ Rn

+ × Rr + : Ax + Cy ≤ b}

PI(x, y) := conv({(x, y) ∈ P (x, y) : x integral})

and we project P (x, y) onto the space of x variables as:

P (x) := {x ∈ Rn

+ : there exists y ∈ Rr + s.t. Ax + Cy ≤ b}

= {x ∈ Rn

+ : ukA ≤ ukb, k = 1, . . . , K}

=: {x ∈ Rn

+ : ¯

Ax ≤ ¯ b} where u1, . . . , uK are the (finitely many) extreme rays of the projection cone {u ∈ Rm

+ : uTC ≥ 0T}.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 3

SLIDE 20

Projected Chv´ atal-Gomory cuts (cont.d)

We then define a projected Chv´

atal-Gomory (pro-CG) cut as a CG cut derived from the system ¯ Ax ≤ ¯ b, x ≥ 0, i.e., an inequality of the form ⌊wT ¯ A⌋x ≤ ⌊wT¯ b⌋ for some w ≥ 0.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 4

SLIDE 21

Projected Chv´ atal-Gomory cuts (cont.d)

We then define a projected Chv´

atal-Gomory (pro-CG) cut as a CG cut derived from the system ¯ Ax ≤ ¯ b, x ≥ 0, i.e., an inequality of the form ⌊wT ¯ A⌋x ≤ ⌊wT¯ b⌋ for some w ≥ 0.

More directly any pro-CG can be defined as an inequality of the form:

⌊uTA⌋x ≤ ⌊uTb⌋ for any u ≥ 0 such that uTC ≥ 0T

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 4

SLIDE 22

Projected Chv´ atal-Gomory cuts (cont.d)

We then define a projected Chv´

atal-Gomory (pro-CG) cut as a CG cut derived from the system ¯ Ax ≤ ¯ b, x ≥ 0, i.e., an inequality of the form ⌊wT ¯ A⌋x ≤ ⌊wT¯ b⌋ for some w ≥ 0.

More directly any pro-CG can be defined as an inequality of the form:

⌊uTA⌋x ≤ ⌊uTb⌋ for any u ≥ 0 such that uTC ≥ 0T

As such, its associated separation problem can be modeled as:

max αTx∗ − α0 (1) αj ≤ uTAj, for j = 1, . . . , n (2) 0 ≤ uTCj, for j = 1, . . . , r (3) α0 + 1 − ǫ ≥ uTb (4) ui ≥ 0, for i = 1, . . . , m (5) αj integer, for j = 0, . . . , n. (6)

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 4

SLIDE 23

On the strength of pro-CG cuts

Given the above definition of pro-CG cuts it is straightforward to extend the definition of

Chv´ atal-Gomory closure for the MIP case.

We will denote as P 1(x, y) the intersection of P (x, y) with all pro-CG cuts viewed as

inequalities αTx + 0Ty ≤ α0 in Rn × Rr.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 5

SLIDE 24

On the strength of pro-CG cuts

Given the above definition of pro-CG cuts it is straightforward to extend the definition of

Chv´ atal-Gomory closure for the MIP case.

We will denote as P 1(x, y) the intersection of P (x, y) with all pro-CG cuts viewed as

inequalities αTx + 0Ty ≤ α0 in Rn × Rr.

For any π ∈ Zn and π0 ∈ Z, the disjunction πTx ≤ π0 or πTx ≥ π0 + 1 is valid for MIP. In
ther words, PI(x, y) ⊆ conv(Π0 ∪ Π1) where

Π0 := P (x, y) ∩ {(x, y) : πTx ≤ π0} Π1 := P (x, y) ∩ {(x, y) : πTx ≥ π0 + 1} A valid inequality for conv(Π0 ∪ Π1) is called a split cut.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 5

SLIDE 25

On the strength of pro-CG cuts

Given the above definition of pro-CG cuts it is straightforward to extend the definition of

Chv´ atal-Gomory closure for the MIP case.

We will denote as P 1(x, y) the intersection of P (x, y) with all pro-CG cuts viewed as

inequalities αTx + 0Ty ≤ α0 in Rn × Rr.

For any π ∈ Zn and π0 ∈ Z, the disjunction πTx ≤ π0 or πTx ≥ π0 + 1 is valid for MIP. In
ther words, PI(x, y) ⊆ conv(Π0 ∪ Π1) where

Π0 := P (x, y) ∩ {(x, y) : πTx ≤ π0} Π1 := P (x, y) ∩ {(x, y) : πTx ≥ π0 + 1} A valid inequality for conv(Π0 ∪ Π1) is called a split cut.

A somehow expected result is:

Theorem 1. Let S(x, y) denote the intersection of P (x, y) with all the split cuts where

ne of the sets Π0, Π1 defined aboveis empty. Then P 1(x, y) = S(x, y).
A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 5

SLIDE 26

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 27

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

Observe that the pro-CG cut x ≤ 1 cuts off the vertex (3/2, 0), but there is no pro-CG cut

which cuts off the non-integral vertex (3/4, 3/4).

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 28

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

Observe that the pro-CG cut x ≤ 1 cuts off the vertex (3/2, 0), but there is no pro-CG cut

which cuts off the non-integral vertex (3/4, 3/4).

Thus, if the objective is to maximize x, pro-CG cuts help, and P 1(x) = PI(x).
A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 29

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

Observe that the pro-CG cut x ≤ 1 cuts off the vertex (3/2, 0), but there is no pro-CG cut

which cuts off the non-integral vertex (3/4, 3/4).

Thus, if the objective is to maximize x, pro-CG cuts help, and P 1(x) = PI(x).

On the other hand, if the objective is to maximize y, pro-CG cuts do not help.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 30

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

Observe that the pro-CG cut x ≤ 1 cuts off the vertex (3/2, 0), but there is no pro-CG cut

which cuts off the non-integral vertex (3/4, 3/4).

Thus, if the objective is to maximize x, pro-CG cuts help, and P 1(x) = PI(x).

On the other hand, if the objective is to maximize y, pro-CG cuts do not help.

More generally, suppose that the projection of the optimum of the MIP relaxation P (x, y)

belongs to the first Chv´ atal closure P 1(x). In this case, no pro-CG cut can cut off that point, although there might possibly be a huge gap between the MIP and its relaxation.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 31

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

Observe that the pro-CG cut x ≤ 1 cuts off the vertex (3/2, 0), but there is no pro-CG cut

which cuts off the non-integral vertex (3/4, 3/4).

Thus, if the objective is to maximize x, pro-CG cuts help, and P 1(x) = PI(x).

On the other hand, if the objective is to maximize y, pro-CG cuts do not help.

More generally, suppose that the projection of the optimum of the MIP relaxation P (x, y)

belongs to the first Chv´ atal closure P 1(x). In this case, no pro-CG cut can cut off that point, although there might possibly be a huge gap between the MIP and its relaxation.

On the other hand, pro-CG cuts are well suited to handle those MIPs where the continuous

variables are only used to model some feasibility condition, possibly by using big-M coefficients, but are not present in the objective function.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 32

On the strength of pro-CG cuts (cont.d)

Consider the following simple example in two variables x and y:

P (x, y) := {x + y ≤ 3/2, y ≤ x, x, y ≥ 0}

Observe that the pro-CG cut x ≤ 1 cuts off the vertex (3/2, 0), but there is no pro-CG cut

which cuts off the non-integral vertex (3/4, 3/4).

Thus, if the objective is to maximize x, pro-CG cuts help, and P 1(x) = PI(x).

On the other hand, if the objective is to maximize y, pro-CG cuts do not help.

More generally, suppose that the projection of the optimum of the MIP relaxation P (x, y)

belongs to the first Chv´ atal closure P 1(x). In this case, no pro-CG cut can cut off that point, although there might possibly be a huge gap between the MIP and its relaxation.

On the other hand, pro-CG cuts are well suited to handle those MIPs where the continuous

variables are only used to model some feasibility condition, possibly by using big-M coefficients, but are not present in the objective function.

More precisely, it is not difficult to prove that:

Theorem 2. MIPs where the continuous variables do not appear in the objective function can be optimized to proven optimality by using only pro-CG cuts (in an iterative way of course).

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 6

SLIDE 33

How tight is the pro-CG closure for MIPLIB instances?

Instances from MIPLIB 3.0 and MIPLIB 2003, time limit of 20 minutes,
(% gap closed) = 100 − 100
pt value(PI)−opt value(P 1)
pt value(PI)−opt value(P ) .
A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 7

SLIDE 34

How tight is the pro-CG closure for MIPLIB instances?

Instances from MIPLIB 3.0 and MIPLIB 2003, time limit of 20 minutes,
(% gap closed) = 100 − 100
pt value(PI)−opt value(P 1)
pt value(PI)−opt value(P ) .

pro-CG CPU % gap instance n r rc # iter # cuts time closed bell3a 71 62 46 70 241 65.3 48.10 bell5 58 46 32 36 126 4.4 91.73 egout 55 86 55 35 168 6.8 81.77 fixnet6 378 500 416 34 83 42.9 67.51 khb05250 24 1,326 1,249 5 13 3.5 4.70 noswot 100 28 39 118 68.0 — rentacar 55 9,502 177 7 15 5.1 0.00 set1ch 240 472 232 29 89 34.2 51.41 vpm1 168 210 27 53 14.9 100.00 vpm2 168 210 89 275 1,021.9 62.86 mas74 150 1 1 1 0.0 0.00 mas76 150 1 1 1 0.0 0.00 misc06 112 1,696 1 1 0.0 0.00 mod011 96 10,862 7,489 1 0.4 0.00 modglob 98 324 324 1 0.0 0.00 pk1 55 31 1 1 0.0 0.00 rgn 100 80 80 1 0.6 0.00

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 7

SLIDE 35

How tight is the pro-CG closure for MIPLIB instances? (cont.d)

pro-CG CPU % gap instance n r rc # iter # cuts time closed 10teams 1,800 225 225 455 2,001 1,200.0 ≥ 57.14 arki001 538 850 1 62 215 1,200.0 ≥ 28.04 blend2 264 89 363 1,032 1,200.0 ≥ 36.40 dcmulti 75 473 473 46 132 1,200.0 ≥ 47.25 fiber 1,254 44 289 1,556 1,200.0 ≥ 4.83 flugpl 11 7 7 3 2 1,200.0 ≥ 19.19 gen 150 720 432 171 427 1,200.0 ≥ 86.60 gesa2 408 816 624 383 1,660 1,200.0 ≥ 94.84 gesa2 o 720 504 312 76 306 1,200.0 ≥ 94.93 gesa3 384 768 528 138 381 1,200.0 ≥ 58.96 gesa3 o 672 480 264 49 193 1,200.0 ≥ 64.53 mkc 5,323 2 87 267 1,200.0 ≥ 1.27 misc03 159 1 1 303 852 1,200.0 ≥ 34.92 misc07 259 1 1 331 889 1,200.0 ≥ 3.86 pp08a 64 176 112 7 8 1,200.0 ≥ 4.32 qiu 48 792 264 7 8 1,200.0 ≥ 10.71 qnet1 1,417 124 124 214 715 1,200.0 ≥ 7.32 qnet1 o 1,417 124 124 318 1,340 1,200.0 ≥ 8.61 rout 315 241 1 459 1,715 1,200.0 ≥ 0.03 swath 6,724 81 1 354 1,222 1,200.0 ≥ 7.68

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 8

SLIDE 36

Is this a good result?

pro-CG cuts have the advantage of being separated solving a “simple” MIP model and they

seem to be effective (as already shown for CG cuts).

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 9

SLIDE 37

Is this a good result?

pro-CG cuts have the advantage of being separated solving a “simple” MIP model and they

seem to be effective (as already shown for CG cuts).

In order to additionally test their behavior we tested their effect when used in conjunction with
ther split cuts which are “easy” to separate.
A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 9

SLIDE 38

Is this a good result?

pro-CG cuts have the advantage of being separated solving a “simple” MIP model and they

seem to be effective (as already shown for CG cuts).

In order to additionally test their behavior we tested their effect when used in conjunction with
ther split cuts which are “easy” to separate.
More precisely, we applied one round of GMI and one round of MIR separated from the tableau
f the initial continuous relaxation and we optimized over the lift-and-project closure (Bonami &

Minoux) before starting separating pro-CG cuts (only using the initial constraint set Ax ≤ b).

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 9

SLIDE 39

Is this a good result?

pro-CG cuts have the advantage of being separated solving a “simple” MIP model and they

seem to be effective (as already shown for CG cuts).

In order to additionally test their behavior we tested their effect when used in conjunction with
ther split cuts which are “easy” to separate.
More precisely, we applied one round of GMI and one round of MIR separated from the tableau
f the initial continuous relaxation and we optimized over the lift-and-project closure (Bonami &

Minoux) before starting separating pro-CG cuts (only using the initial constraint set Ax ≤ b).

% gap closed GMI GMI +MIR +MIR +L&P +L&P instance +pro-CG bell3a 64.02 91.68 egout 93.85 100.00 fixnet6 86.01 92.33 set1ch 40.17 69.27 flugpl 11.74 ≥ 41.75 gesa2 o 49.27 ≥ 99.27

In other words, pro-CG cuts seem to be diverse wrt other cuts, thus increasing the arsenal of a

cutting plane algorithm.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 9

SLIDE 40

Is this a good result? (cont.d)

A final experiment to assert pro-CG’s effectiveness is a comparison with the split closure itself.
A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 10

SLIDE 41

Is this a good result? (cont.d)

A final experiment to assert pro-CG’s effectiveness is a comparison with the split closure itself.
A possible way of attacking the problem is looking at the separation of MIR cuts

(Oktay G¨ unl¨ uk, Sanjeeb Dash & Andrea Lodi): min cT

+y∗ + ˆ

aTx∗− ˆ d(⌈d⌉ − ¯ aTx∗) s.t. ˆ a + ¯ a ≥ λTA, ¯ ai ∈ Z, ˆ ai ∈ (0, 1) c+ ≥ λTC, c+ ≥ 0 d ≤ λTb

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 10

SLIDE 42

Is this a good result? (cont.d)

A final experiment to assert pro-CG’s effectiveness is a comparison with the split closure itself.
A possible way of attacking the problem is looking at the separation of MIR cuts

(Oktay G¨ unl¨ uk, Sanjeeb Dash & Andrea Lodi): min cT

+y∗ + ˆ

aTx∗− ˆ d(⌈d⌉ − ¯ aTx∗) s.t. ˆ a + ¯ a ≥ λTA, ¯ ai ∈ Z, ˆ ai ∈ (0, 1) c+ ≥ λTC, c+ ≥ 0 d ≤ λTb

and approximate it:

Approximate ˆ d as ˆ d = k

i=1 ǫiπi where ǫi = 1 2i, πi ∈ {0, 1}

For any violated MIR cut, if ∆ = ⌈d⌉ − ¯ aTx∗, then 0 < ∆ < 1

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 10

SLIDE 43

An Exact MIR separation model

Lemma 1. The multipliers λi corresponding to an equation without continuous variables can

be assumed to lie in (0, 1) in an optimal solution to MIR-sep.

Lemma 2. The multipliers λi can be assumed to lie in (−mδ, mδ) in an optimal solution to

MIR-sep, where m is the number of rows in P , and δ is the maximum value of sub-determinants of [A, C]. Corollary 1. The MIR closure of P is a polyhedron. Corollary 2. ˆ d can be assumed to have finite precision. Corollary 3. MIR-sep can be solved as an MIP.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 11

SLIDE 44

Is this a good result? (cont.d)

Of course the model is larger and more difficult to solve (binary variables for the approximation
f d), thus, although theoretically dominating, the tradeoff of using such a model must be

analyzed.

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 12

SLIDE 45

Is this a good result? (cont.d)

Of course the model is larger and more difficult to solve (binary variables for the approximation
f d), thus, although theoretically dominating, the tradeoff of using such a model must be

analyzed.

% gap closed instance pro-CG MIR bell3a 40.10 ≥ 76.06 blend2 ≥ 36.40 ≥ 33.28 dcmulti ≥ 47.25 ≥ 79.75 egout ≥ 81.77 ≥ 77.19 fixnet6 67.51 ≥ 21.62 flugpl ≥ 19.19 ≥ 99.79 khb05250 4.70 99.98 pp08a ≥ 4.32 ≥ 95.35 misc06 0.00 ≥ 98.69 set1ch 51.41 ≥ 10.76 rgn 0.00 ≥ 97.09 vpm1 100.00 ≥ 49.89

A. Lodi, Projected Chv´

atal-Gomory cuts for Mixed Integer Linear Programs 12