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# Robust Optimization and Ellipsoid Uncertainty

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Userlevel 1
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Dear All,

I have been trying to use ellipsoid uncertainty in a simple problem that I have.

The problem is an assortment optimization. Then, I have the Demand Mean for each products Dem(p) and the Covariance matrix or the variance Var(p,p1) for each par of products.

The objective is to select the products what will maximize the expected demand Dem(p) at the same time, penalizes products with high variance. The uncertainty set will be a ellipsoid uncertainty.

I would like to create the eelipsoid uncertainty using the uncertainty region like:

Demand(p).level - sum[p1 , Var(p,p1) * …………

There is a example in the book on page 341. but it’s not clear for me.

Could you help me to model the ellipsoid uncertainty using the uncertainty property?

Thank you

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Best answer by Marcel Hunting 18 October 2021, 17:59

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Userlevel 4
+6

Thanks Luis,

It is not entirely clear to us what question you are asking and how you think we can help you. Would it be possible to elaborate on your question a bit further? Many thanks!

Userlevel 1
+2

Dear

I included all details about my question on microsoft word. I also send the aimms file with the model.

I’m creating an assortment model and I would like to use the ellipsoid uncertianty.

I k ow that I can use the function Ellispoid from aimms. However, I would like to use the uncertainty region, and the uncertianty constraint to build the ellipsoid.

I think that I was able to create the model with ellipsoid. I described the process that I used to build the model and I would like to check if there is something missing.

Thank you

Userlevel 5
+4

Hi @luiswilbert , I am not sure whether the model that you defined as ‘robust model’ in your document matches the robust optimization model that you defined in the AIMMS project.

In your robust model (in the document) you have defined a new objective which also penalizes products with high variance. Are expected demand (i.e., DemandMean) and covariance uncertain, or should they be considered fixed or given?

In your AIMMS project you have defined DemandMean as an uncertain parameter which depends on another uncertain parameter (z), and the relationship of both these uncertain parameters depends on the (fixed) parameter Covariance. In your robust optimization model in AIMMS you are telling AIMMS to find the best solution considering all possible realizations of the uncertain parameter DemandMean(a). All possible realizations of DemandMean(a) are restricted by its Uncertainty definition and (indirectly) by the ellipsoidal uncertainty constraint. I don’t think that this robust optimization model will penalize products with high covariance. The objective function in your AIMMS model does not contain a term for the covariance.

Userlevel 1
+2

AmDear

When I move from the nominal model to the robust model, I have the following model with ellipsoid (in this case is a minimization problem).

In my case, the model is a maximization. Then the second part of the last equation will be -p||Px||2.

When I’m building this equation,

I’m trying to replicate this function

IS it corret?

Sorry for this basic question

Userlevel 5
+4

Hi @luiswilbert , Thanks for the additional information on how the robust counterpart can be reformulated. In that case your AIMMS model seems to be correct.

Userlevel 1
+2

Thank you so much