Math Models of OR: Modeling Piecewise Linear Functionseaton.math.rpi.edu/faculty/Mitchell/courses/matp4700/notesMATP47… · Mitchell Modeling Piecewise Linear Functions 9 / 18. Modeling

Math Models of OR:

Modeling Piecewise Linear Functions

John E. Mitchell

Department of Mathematical Sciences

RPI, Troy, NY 12180 USA

November 2018

Mitchell Modeling Piecewise Linear Functions 1 / 18

Introduction

Outline

1 Introduction

2 Modeling the first line segment

3 Modeling the second line segment

4 Modeling the third line segment

5 Modeling the piecewise linear function


Introduction

Piecewise linear functions

Piecewise linear functions can be modeled using variables that satisfy

what is known as a special order set (SOS) constraint of type 2, which

in turn can be modeled using binary variables.

Consider the following example of a continuous piecewise linear

function:

z =

8<

:

3 + 4x for 0 x 2

15 � 2x for 2 x 3

6 + x for 3 x 7.

The variable x is restricted to lie between 0 and 7.

Note that the function is nonconvex.


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linesegment i t above c u r v e .

Ei.p÷÷÷÷÷÷÷÷:÷÷÷÷.m i n f(x) Equivalent

m i n 2

s - t . 2 3 6 - 2t o : 2 7 , 4 - x

OEXEG z ? - 5 t 2 x

r u i n 2 2 hs i ? z s , 6 -3x .

2 ? 4 - × 6 ): : : : : µ¥,i f

Introduction

Our function

x

z

0 2 3 7

3

9

11

13

(0,3)

(2,11)

(3,9)

(7,13)

z =

8<

:

3 + 4x for 0 x 2

15 � 2x for 2 x 3

6 + x for 3 x 7.


io

n o nc o n v e x :

liec segment

i s below c u r v e .

Modeling the first line segment

Outline

1 Introduction







The first line segment

For 0 x 2, we have the linear function z = 3 + 4x .

Note that in this range, the points (x , z) on the line segment are

convex combinations of the two endpoints of the line segment, (0, 3)and (2, 11).

If we give a weight w1 to the point (0, 3) and a weight w2 to the point

(2, 11) then any point on the line segment can be expressed as

✓xz

◆= w1

✓0

3

◆+ w2

✓2

11

◆,

so x = 0w1 + 2w2

and z = 3w1 + 11w2.

The weights w1 and w2 must be nonnegative and they must add up to

one for this to be a convex combination.



The first line segment graphically

x

z

0 2 3 7

3

9

11

13

(0,3)

(2,11)

x = w1 + 2w2, z = 3w1 + 11w2


4=0,w e l

41=491*41WI,W L ?O

=w , t w >= /.

'u,= I ,w e 0

Modeling the second line segment

Outline

1 Introduction







The second line segment

For 2 x 3, we have the linear function z = 15 � 2x .





✓xz

◆= w2

✓2

11

◆+ w3

✓3

9

◆,

so x = 2w2 + 3w3

and z = 11w2 + 9w3.





The second line segment graphically

x

z

0 2 3 7

3

9

11

13

(2,11)

(3,9)

x = 2w2 + 3w3, z = 11w2 + 9w3


(ft . K i l t u s(9)W ,i was7 0 , o ,#w,s /

Modeling the third line segment

Outline

1 Introduction







The third line segment

For 3 x 7, we have the linear function z = 6 + x .





✓xz

◆= w3

✓3

9

◆+ w4

✓7

13

◆,

so x = 3w3 + 7w4

and z = 9w3 + 13w4.





The third line segment graphically

x

z

0 2 3 7

3

9

11

13

(3,9)

(7,13)

x = 3w3 + 7w4, z = 9w3 + 13w4


(1)= ↳ ({It-4lb)U s ,4470 , w ,-104= /

Modeling the piecewise linear function

Outline

1 Introduction







Putting it all together

We can combine the three convex combination models for the three

line segments to give a model for the piecewise linear function.

A point on the piecewise linear function must be on one of the line

segments, so it is a convex combination of two of the four points (0, 3),(2, 11), (3, 9) and (7, 13).

One tricky part is that the two points must be adjacent.



Some constraints

We require (x , z) to be a convex combination of the four breakpoints.

So we need

✓xz

◆= w1

✓0

3

◆+ w2

✓2

11

◆+ w3

✓3

9

◆+ w4

✓7

13

◆.

We can express these two conditions as well as the requirement that

we have a convex combination using the following linear constraints in

the six variables x , z,w1,w2,w3,w4:

0w1 + 2w2 + 3w3 + 7w4 = x3w1 + 11w2 + 9w3 + 13w4 = zw1 + w2 + w3 + w4 = 1

The variables w1,w2,w3,w4 must also be nonnegative.


V.¥÷÷÷*i÷:÷÷÷÷i.•

¥

w ,i wait, ( x4 = w ,⇒

:

2)= (3£)

Onlast linesegment,s oshould

have : 2 = 6e x = 9 4


SOS-2 constraints

The calculated value of z is not accurate if, for example, w1 = 0.5 and

w4 = 0.5. This would give x = 3.5 and z = 8, whereas we should have

z = 6 + 3.5 = 9.5.

The restriction we need to impose is that at most two of the wi can be

nonzero, and the two nonzero wi must be adjacent.

This is known as a SOS-2 constraint, that is, a special order set of type

2 constraint.



SOS-2 using binary variables

The SOS-2 restriction can be modeled using binary variables yi ,

i = 1, . . . , 4:

wi yi i = 1, . . . , 4P4

i=1yi 2

y1 + y3 1

y1 + y4 1

y2 + y4 1.

The summation constraint allows at most two of the yi to be equal to

one, and hence at most two components of wi can be positive.

The last three constraints prevent non-adjacent points being chosen.

For example, the constraint y1 + y4 1 forces y1 = 0 and/or y4 = 0;

in turn, the first constraint then forces either w1 = 0 and/or w4 = 0.


Documents

Math Models of OR: Modeling Piecewise Linear Functionseaton.math.rpi.edu/faculty/Mitchell/courses/matp4700/notesMATP47… · Mitchell Modeling Piecewise Linear Functions 9 / 18. Modeling