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[Help-glpk] Problem with absolute value

From: Alessandro Saccoia
Subject: [Help-glpk] Problem with absolute value
Date: Mon, 14 Jan 2013 00:43:06 +0100

Hi list,
I am new here, and not that experienced with mathematical methods such as 
Linear Programming. I am trying to solve a problem (related to audio alignment) 
for which, on stack exchange, I have been told to use LP. Please check my 
original question:

I have then studied it the whole afternoon, and it opened me a world, this is 
so useful! But the absolute value in my problem is giving me a big headache and 
I don't fully understand the suggestion I got.

If you have read the formulation of the problem in the link above, here follows 
the modeling I am trying to plug into gplk: please try to follow my solution. 
As this is the first time I try gplk I am going to use numbers, then of course 
my program will use runtime data.

I want to find the alignment S from the following points:
Q0 = 3
Q1 = 5
Q2 = 8
Q3 = 12

with reference points

R0 = 2
R1 = 5
R2 = 7
R3 = 11

where the segments Si-Si+1 can't move more than 50% of the original length
1 <= S1 - S0 <= 3               (original length Q0-Q1 = 2)
1.5 <= S2 - S1 <= 4.5  (original length Q1-Q2 = 3)
2 <= S3 - S2 <= 6               (original length Q2-Q3 = 4)

The problem is a minimization problem with 

min: |r0 - s0| + |r1 - s1| + |r2 - s2| + |r3 - s3|

for each one of the absolute values i we need introduce one variable Zi and two 

Zi >= Ri - Si
Zi >= -Ri + Si

(this is what I have understood by the answer on SE, and to get it I have also 
read the page : paragraph 
"minimization and sign is positive or maximization and sign is negative")

To create the matrix A I use just my relationships over the segment lengths: 
since these are inequalities, I introduce for each "i" two new variables called 
SLi (stretch low) and SHi (stretch high), and so I can remove the >= and <= by 
adding constraints.

for each i from 0 to 2:

Lower bound of the length of the segment
Si+1 - Si >= Qi - alpha * (Qi+1 - Qi)
Si+1 - Si = SLi (Stretch lower)
Introduces constraint SLi >=  Qi - alpha * (Qi+1 - Qi) 

Upper bound of the length of the segment
Si+1 - Si <= Qi +  beta * (Qi+1 - Qi)
Si+1 - Si <= SHi
Introduces constraint SHi <= Qi +  beta * (Qi+1 - Qi)  

Since all the Q values are known, the constraint on SLi and SHi sems very 

Now my problem arrives: for what I got, each one of the new variables just 
introduced creates a row,
while each of the original unknowns is a column.

S0  S1  S2  S3    Variable
-1  1   0   0   0       Sl0
-1  1   0   0   0       Sh0
0   -1  1   0   0       Sl1
0   -1  1   0   0       Sh1
0   0   -1  1   0       Sl2
0   0   -1  1   0       Sh2

At this point I am stuck: I have an objective function made of the Zs 
introduced before, and I know the constraints on them. But those Zs do not show 
up in the A matrix, so… stuck, and I don't know if any of the above is correct. 
Thanks in advance for your precious help.


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