Maximize minimum optimization problem

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Chris B
Chris B am 29 Jul. 2014
Kommentiert: Chris B am 31 Jul. 2014
Hello! I would like to ask for your help concerning the following optimization problem as I am trying to get familiar with such kind of optimization...
Can someone help me about how to write down my objective function and the first inequality constraint (in order to include them in the intlinprog solver)? I mainly don't know how to describe the 'ε' as the obective function... The rest of the constraints can be easily formed in matrices.
Thank you in advance!
Regards,
Chris
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Matt J
Matt J am 29 Jul. 2014
Looks like bintprog would be sufficient, since x is binary. No need for intlinprog.

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Alan Weiss
Alan Weiss am 29 Jul. 2014

It seems to me that your variable $\epsilon$ is unneeded. You simply want to find

 max(sum_t sum_i sum_j rates(i,j) x(t,i,j))

while still satisfying your other constraints. (This maximum, taken over the x(t,i,j), is the value of $\epsilon$.) This is the same as minimizing the negative of that sum. So your objective function vector has components rates(i,j) for the (t,i,j) entry.

Alan Weiss

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Matt J
Matt J am 30 Jul. 2014
Yes, binary variables happen all the time. That's why you have solvers like intlinprog and bintprog to enforce it.
Chris B
Chris B am 30 Jul. 2014
Bearbeitet: Chris B am 30 Jul. 2014
But in this case, we don't know how to express the objective f for using it as an input to the intlinprog for example (this epsilon confuses me I could say).. That's my main concern and sorry for asking 'easy- to answer' questions.. I just want to be sure that I understand it properly!
edit : if it is easy for you, can you post just an example of how to write it? Not analytically, in few words if you don't have any problem!

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Matt J
Matt J am 30 Jul. 2014
Bearbeitet: Matt J am 30 Jul. 2014
But in this case, we don't know how to express the objective f for using it as an input to the intlinprog (this epsilon confuses me I could say)
The objective and constraints need to be posed in terms of the concatenated vector of knowns p=[x(:);epsilon] very similar to what we did in your earlier thread,
http://www.mathworks.com/matlabcentral/answers/140614-maximize-d2d-rate-fmincon-problem
So, f(i) would be 0 for all i corresponding to the x(i) while f(end) corresponding to the epsilon would equal -1.
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Matt J
Matt J am 31 Jul. 2014
and then I have to decide for the corresponding values to the b column vector. It should be valued with '-epsilon' but this is also the asked parameter that we want to maximize...
No, b should be a vector of zeros. Rearrange your constraint inequalities to be of the form
epsilon - sum_i sum_j rates(i,j) x(t,i,j) <=0
This is a system of the form
A*[x(:);epsilon]<=0
for some appropriately chosen matrix A.
Chris B
Chris B am 31 Jul. 2014
It worked for me! Thank you so much.
Chris

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