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Linear regression on an usual domain, hyperplane, sphere or simplex.

The problem I have in mind is as follows:

I try to reverse-engineer the recipe for the coca cola beverage. The response, Y, is how close my recipe is to the actual formula, based on a number of tastings performed by a number of different people, according to a design of experiment plan. Indeed, it's quite similar to a clinical trial where a mix of atoms or chemical radicals (each combination producing a unique molecule) is tested to optimize a drug. The independent variables are binary, each one representing an ingredient: salt, water, corn syrup etc.

Read more at http://bit.ly/18PDU7e

Can someone provide guidance on how best to proceed? 

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