Structure and Interpretation of Dual-Feasible Functions · The superadditive duality theory of integer linear optimization appears in several concrete forms including: Dual-feasible

Structure and Interpretationof Dual-Feasible Functions

Matthias KoppeJiawei Wang

Department of Mathematics,University of California, Davis

September 13, 2017

Matthias Koppe , Jiawei Wang Dual-Feasible Functions September 13, 2017 1 / 23

Superadditive Duality in Integer Programs

The superadditive duality theory of integer linear optimization appears inseveral concrete forms including:

Dual-feasible functions

Cut-generating functions

Superadditive Duality in IP

Dual-feasible functions ([1] Alves et al, 2016):

Superadditive and nondecreasing DFFs havebeen proved to be very useful for the efficientcomputation of lower bounds for many integerprograms and combinatorial optimizationproblems, including bin-packing and cuttingstock problems.

DFFs can be used to generate valid inequalitiesfor integer programs which have knapsack typeinequalities.

Superadditive Duality in IP

Gomory–Johnson cut-generating functions:

Gomory and Johnson ([3], 1972) first studied so-called infinite groupproblem and found certain facet-defining inequalities.

The fundamental Gomory (fractional) cuts, which are crucial for theefficiency of today’s most powerful codes for solving integer programs,have an interpretation in terms of superadditive duality.

Amitabh Basu, Robert Hildebrand, Matthias Koppe ([2], 2014)presented a survey on the infinite group problem in the moderncontext of cut generating functions. The survey introduced aninteractive companion program [4], implemented in the open-sourcecomputer algebra package Sage, which tests extremality for a givenfunction and provides an updated compendium of known extremefunctions.

DFF examples

φBJ,1(x ;C ≥ 1) =bCxc+ max(0, {Cx}−{C}1−{C} )

bCc(1)

Figure: Dual-Feasible Function (1) φBJ,1 by Burdett and Johnson for parametervalues C ∈ { 3

2 ,73 ,

134 }. {a} is the fractional part of a.

Gomory–Johnson cut-generating functions example

The simplest extreme Gomory–Johnson function is so-called GomoryMixed Integer Cut:

π(x) = gmic(x ; f ) =

{1f x if 0 ≤ x ≤ f

1f−1 (x − 1) if f < x ≤ 1

where f is the fractional part of b. Since π is Z-periodic, we just define πin [0, 1].

Figure: Gomory Mixed Integer Cut (2) for parameter values f ∈ { 13 ,

Our work

Relate Dual-feasible functions to Gomory–Johnson cut-generatingfunctions.

Use computer-based tools to search for new extreme Dual-feasiblefunctions.

Definitions

Definition

A function φ : [0, 1]→ [0, 1] is called a (valid) classical Dual-FeasibleFunction, if for any finite index set I of nonnegative real numbersxi ∈ [0, 1], it holds that,∑

i∈Ixi ≤ 1⇒

∑i∈I

φ(xi ) ≤ 1

Definition

A function φ : R→ R is called a (valid) general Dual-Feasible Function,if for any finite index set I of real numbers xi ∈ R, it holds that,∑

i∈Ixi ≤ 1⇒

∑i∈I

φ(xi ) ≤ 1

Definitions

Definition

A function φ : [0, 1]→ [0, 1] is called a (valid) classical Dual-FeasibleFunction, if for any finite index set I of nonnegative real numbersxi ∈ [0, 1], it holds that,∑

i∈Ixi ≤ 1⇒

∑i∈I

φ(xi ) ≤ 1

Definition

A function φ : R→ R is called a (valid) general Dual-Feasible Function,if for any finite index set I of real numbers xi ∈ R, it holds that,∑

i∈Ixi ≤ 1⇒

∑i∈I

φ(xi ) ≤ 1

DFF examples

φBJ,1(x ;C ≥ 1) =bCxc+ max(0, {Cx}−{C}1−{C} )

bCc(3)

Figure: Dual-Feasible Function (3) φBJ,1 by Burdett and Johnson for parametervalues C ∈ { 3

2 ,73 ,

134 }. {a} is the fractional part of a. It is both a classical and

general DFF, which is the quasiperiodic extension of the classical DFF.

Two Main Applications

Provide fast lower bounds for several types of combinatorialoptimization problems including bin-packing and cutting stockproblems.

Generate valid inequalities for integer programs which have knapsacktype inequalities.

Applications

Provide fast bounds

Consider a bin packing problem: there are 10 item A with weight 0.3and 20 item B with weight 0.4. Use a minimum number of bins topack all items such that the weight of each bin is at most 1.

Three packing patterns: 2× B, 2× A + 1× B, 3× A.

Formulation:minimize x1 + x2 + x3

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ Z+

Applications

Provide fast boundsConsider a bin packing problem: there are 10 item A with weight 0.3and 20 item B with weight 0.4. Use a minimum number of bins topack all items such that the weight of each bin is at most 1.

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ Z+

Applications

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ Z+

Applications

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ Z+

Applications

We consider the LP relaxation:

minimize x1 + x2 + x3

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ R+

(φ(0.3), φ(0.4)) is a feasible solution to the dual problem of the LPrelaxation for any Dual-Feasible Function φ:

maximize 10u1 + 20u2

subject to 0u1 + 2u2 ≤ 1

2u1 + 1u2 ≤ 1

3u1 + 0u2 ≤ 1

u1, u2 ∈ R+

Then d10φ(0.3) + 20φ(0.4)e is a lower bound of the IP problem.

Applications

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ R+

2u1 + 1u2 ≤ 1

3u1 + 0u2 ≤ 1

u1, u2 ∈ R+

Applications

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ R+

2u1 + 1u2 ≤ 1

3u1 + 0u2 ≤ 1

u1, u2 ∈ R+

Applications

subject to 0x1 + 2x2 + 3x3 ≥ 10

2x1 + x2 + 0x3 ≥ 20

x1, x2, x3 ∈ R+

2u1 + 1u2 ≤ 1

3u1 + 0u2 ≤ 1

u1, u2 ∈ R+

Then d10φ(0.3) + 20φ(0.4)e is a lower bound of the IP problem.Matthias Koppe , Jiawei Wang Dual-Feasible Functions September 13, 2017 12 / 23

Applications

Generate valid inequalites

Theorem ([1] by Alves et. al.)

If φ is a general Dual-Feasible Function andS = {x ∈ Zn

+ :∑n

i=1 aixi ≤ 1}. Then∑n

i=1 φ(ai )xi ≤ 1 is a validinequality.

Applications

Generate valid inequalites

If φ is a general Dual-Feasible Function andS = {x ∈ Zn

+ :∑n

i=1 aixi ≤ 1}. Then∑n

i=1 φ(ai )xi ≤ 1 is a validinequality.

Hierarchy

Definition

A classical/general DFF φ is called maximal, if there is no other DFF φ′

such that φ′ ≥ φ

Definition

A maximal classical/general DFF φ is called extreme, if it can not bewritten as a convex combination of two other DFFs.

The hierarchy indicates the strength of the corresponding valid inequalitiesand lower bounds.

Hierarchy

Definition

Hierarchy

Definition

Maximality

Definition

A function φ is superadditive if φ(x) + φ(y) ≤ φ(x + y). A function φ issubadditive if φ(x) + φ(y) ≥ φ(x + y).

Characterization of maximal classcial DFFs:

A function φ : [0, 1]→ [0, 1] is a classical maximal DFF if and only if thefollowing conditions hold:

(i) φ is superadditive.

(ii) φ is symmetric in the sense φ(x) + φ(1− x) = 1

(iii) φ(0) = 0

Maximality

Definition

A function φ is superadditive if φ(x) + φ(y) ≤ φ(x + y). A function φ issubadditive if φ(x) + φ(y) ≥ φ(x + y).

Characterization of maximal classcial DFFs:

A function φ : [0, 1]→ [0, 1] is a classical maximal DFF if and only if thefollowing conditions hold:

(ii) φ is symmetric in the sense φ(x) + φ(1− x) = 1

(iii) φ(0) = 0

Maximality of general DFFs

The characterization of minimal cut-generating functions in theYıldız–Cornuejols model [5] can be easily adapted to give a fullcharacterization of maximal general DFFs, which is missing in [1].

Theorem

(NEW) A function φ : R→ R is a general maximal DFF if and only if thefollowing conditions hold:

(ii) φ is symmetric in the sense φ(x) = inf{ 1k (1− φ(1− kx)) : k ∈ Z++}

(iii) φ(0) = 0

(iv) φ(x) ≥ 0 for all x ≥ 0

DFF examples

φBJ,1(x ;C ) =bCxc+ max(0, {Cx}−{C}1−{C} )

bCc(4)

Figure: Dual-Feasible Function (4) φBJ,1 for parameter values C ∈ { 32 ,

134 }.

φBJ,1(x ;C ) : [0, 1]→ [0, 1] is a continuous 2-slope classical maximal DFFfor all parameter value C ≥ 1 and extreme for C ≥ 2 or C = 1.φBJ,1(x ;C ) : R→ R is an extreme general DFF for C ≥ 1.

Gomory–Johnson cut-generating functions

Consider the single-row Gomory–Johnson model, which takes the followingform:

x +∑r∈R

r × y(r) = b, b /∈ Z, b > 0 (5)

x ∈ Z, y : R→ Z+, and y has finite support.

Let π : R→ R to be a nonnegative function. Then π is a validGomory–Johnson function if

∑r∈R π(r)y(r) ≥ 1 holds for any feasible

solution (x , y). The valid inequality cuts off the non-integral solution(x,y)=(b,0).π is minimal (non-dominating) if and only if π is Z-periodic, π(0) = 0, πis subadditive and π(x) + π(b − x) = 1 for all x ∈ R.

x +∑r∈R

r × y(r) = b, b /∈ Z, b > 0 (5)

solution (x , y). The valid inequality cuts off the non-integral solution(x,y)=(b,0).

π is minimal (non-dominating) if and only if π is Z-periodic, π(0) = 0, πis subadditive and π(x) + π(b − x) = 1 for all x ∈ R.

x +∑r∈R

r × y(r) = b, b /∈ Z, b > 0 (5)

solution (x , y). The valid inequality cuts off the non-integral solution(x,y)=(b,0).π is minimal (non-dominating) if and only if π is Z-periodic, π(0) = 0, πis subadditive and π(x) + π(b − x) = 1 for all x ∈ R.

Relation to Gomory–Johnson cut-generating functions

Theorem

(NEW) Let π be a minimal piecewise linear Gomory–Johnson functioncorresponding to a row of the form (5) with the right hand side b. Assumeπ is continuous at 0 from the right. Then there exists δ > 0, such that forall 0 < λ < δ, φλ : R→ R, defined by φλ(x) = bx−λπ(bx)

b−λ is a maximalgeneral DFF and φλ|[0,1] is a maximal classical DFF.

Such maximal generalDFF φλ has the following properties:

(i) π has k different slopes if and only if φλ has k different slopes.

(ii) φλ( 1b )x +

∑r∈R φλ( r

b )y(r) ≤ 1 is a valid inequality for (5).Moreover, φλ cuts off the solution (x , y) = (b, 0).

(iii) The general DFF φλ is extreme if π is also continuous with only 2slope values where its positive slope s satisfies sb > 1 and λ = 1

s .The classical DFF φλ|[0,1] is extreme if π and λ satisfy the previousconditions and b > 3.

Theorem

b−λ is a maximalgeneral DFF and φλ|[0,1] is a maximal classical DFF. Such maximal generalDFF φλ has the following properties:

(ii) φλ( 1b )x +

∑r∈R φλ( r

Theorem

(ii) φλ( 1b )x +

∑r∈R φλ( r

Theorem

(ii) φλ( 1b )x +

∑r∈R φλ( r

Example of the Conversion

π = gmic functions can be converted to φBJ,1 functions.

Figure: Gomory Mixed Integer Cut π = gmic(x ; f = 13 ) can be converted to

Dual-Feasible Function φBJ,1(x ;C = 73 ) using b = 7

3 and λ = 13 .

Computer-based search

Results:

(1) automate maximality test and extremality test of classical DFFs.

(2) find piecewise linear extreme classical DFFs with rationalbreakpoints, which have fixed common denominator q ∈ N.

The strategy is to discretize the interval [0, 1] and define discretefunctions on 1

qZ ∩ [0, 1]. After adding the inequalities fromcharacterization of maximality, the space of functions becomes aconvex polytope with finite dimensions. Extreme points of thepolytope can be found by vertex enumeration tools.

Candidates for extreme DFFs φ are obtained by interpolating valueson 1

qZ ∩ [0, 1] from each extreme point (discrete function). Then weuse our extremality test to filter out the non-extreme functions.

Codes are available at [4]:https://github.com/mkoeppe/cutgeneratingfunctionology

Results:

Computer-base search

Table: Search for extreme DFFs and efficiency of vertex enumeration codes

Polytope Φcontinuous (q) Running times (s)

q dim inequalities vertices extreme DFFs PPL Normaliz

original minimized total 2-slope from G–J conversion

2 0 4 3 1 1 0 1 0.00006 0.002

3 1 5 5 2 1 0 1 0.00009 0.006

5 2 9 7 3 2 1 2 0.00014 0.007

7 3 15 10 5 3 2 3 0.0002 0.007

9 4 23 14 9 3 2 3 0.0004 0.008

11 5 33 18 14 7 6 7 0.0006 0.010

13 6 45 23 25 8 7 8 0.001 0.012

15 7 59 29 66 14 10 14 0.003 0.018

17 8 75 35 94 22 17 22 0.005 0.025

19 9 93 42 221 32 24 32 0.010 0.042

21 10 113 50 677 55 30 48 0.036 0.105

23 11 135 58 1360 105 52 94 0.110 0.226

25 12 159 67 3898 189 74 139 0.526 0.725

27 13 185 77 12279 291 94 206 5.1 2.991

29 14 213 87 28877 626 143 377 41 9.285

31 15 243 98 91761 1208 212 634 595 35.461

Future Work

Discover new parametric family of classical DFFs and recognizeextreme functions found by computer-based search.

Implement general DFFs, which have unbounded domain R, and theirmaximality and extremality test.

Relate DFFs to other cut-generating functions.

C. Alves, F. Clautiaux, J. V. de Carvalho, and J. Rietz, Dual-feasiblefunctions for integer programming and combinatorial optimization:Basics, extensions and applications, EURO Advanced Tutorials onOperational Research, Springer, 2016,doi:10.1007/978-3-319-27604-5, ISBN 978-3-319-27602-1.

A. Basu, R. Hildebrand, and M. Koppe, Light on the infinite grouprelaxation, version 1 as posted to arXiv.org, eprint arXiv:1410.8584v1[math.OC], October 2014.

R. E. Gomory and E. L. Johnson, Some continuous functions relatedto corner polyhedra, Math. Programming 3 (1972), 23–85.

C. Y. Hong, M. Koppe, and Y. Zhou, Sage code for thegomory-johnson infinite group problem,https://github.com/mkoeppe/cutgeneratingfunctionology,(Version 1.0).

S. Yıldız and G. Cornuejols, Cut-generating functions for integervariables, Mathematics of Operations Research 41 (2016), no. 4,1381–1403, doi:10.1287/moor.2016.0781.

Structure and Interpretation of Dual-Feasible Functions · The superadditive duality theory of integer linear optimization appears in several concrete forms including: Dual-feasible

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