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I have 3 variables where 2 are predictors (positive integers) and the response variable is a real number.

To give an example:

a b c
1 2 5
1 5 7

Because this dataset is unique to my problem, I don't have an equation that can explain the relationship among a, b, c.

However, when I plot these as a surface, there is a nonlinear relationship.

How do I go about to find the maximum value of c while minimizing a, b. I know this sounds like a numerical optimization problem but without an equation, I won't be able to solve it.

Edit: can machine learning help here? Large amount of data can be obtained.

Patrick
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    You'll need some kind of equation to express exactly what you want to optimize. Suppose you have {A, B, C} = {0, 0, 10}. Is that preferable to {1, 1, 50}, which maximizes C but fails to minimize A and B? What about {0, 10, 100}? We have no way of knowing which is "best". This is an optimization problem, but you can't solve it without defining what to optimize. – Nuclear Hoagie Jul 28 '21 at 20:59
  • Will I need one equation to tie a, c together and another one for b, c, or is it one equation for all the variables? How can I generate these equations? – Patrick Jul 28 '21 at 21:09
  • Are you asking about creating a surrogate surface of c in terms of a,b? Because that's easily done using e.g. Gaussian processes; here's a bibliography https://stats.stackexchange.com/a/207025/22311. But finding the maximum of c is generally different than minimizing a,b. In order to answer this, you'll need to be able to write down the tradeoff that you're willing to accept between larger c and smaller a,b. – Sycorax Jul 29 '21 at 20:23
  • In your example, c=7 is the largest value, but a+b for the second row is also larger than a+b for the first row. So is the correct answer row 2 (because c is largest) or is the correct answer row 1 (because a + b is smallest)? And more importantly, why is the chosen row the correct answer? These are the questions you'll need to answer to be able to solve this problem. – Sycorax Aug 03 '21 at 19:53

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