
From:  Jeffrey Kantor 
Subject:  Re: [Helpglpk] Problem with absolute value 
Date:  Mon, 14 Jan 2013 07:04:34 0500 
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:
http://math.stackexchange.com/questions/276492/stretchingofasetofnumberstoaligntoareference
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 SiSi+1 can't move more than 50% of the original length
1 <= S1  S0 <= 3 (original length Q0Q1 = 2)
1.5 <= S2  S1 <= 4.5 (original length Q1Q2 = 3)
2 <= S3  S2 <= 6 (original length Q2Q3 = 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 contraints
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 http://lpsolve.sourceforge.net/5.1/absolute.htm : 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 simple.
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.
Alessandro
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