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I'm looking to get pointed in the right direction with regards to research on a particular (Stochastic) Integer Linear Programming case. I've been looking into stochastic, chance-constrained, and robust programming -- but none of these, at least I don't think, fit my particular case very well.

Suppose I wanted to solve for a set of deterministic constraints given a maximization of a function of the sum of random variables whose distributions are known. I'm essentially trying to build on this question.

The optimization function would have something to this form, a combination of a random variable and decision variable.

$max \sum_{i}\delta _{i}A_i$

The optimal result, in this case, would be the standard integer linear programming case over the expectation of each random variable. Simply optimize for the highest combined expectation.

However, I'd like to take this one step further and generate a set of results. These results would build on the optimal result -- weighted by some term that is proportional to the probability of that result truly being optimal for a set of observations. That is, in the scenario where my expectation-driven optimal result doesn't yield high value, could one of the alternatives provide a higher value? One can think of each result being orthogonal to one another.

Does this type of problem already fit a defined model? I was hoping that stochastic programming would fit the bill, but it doesn't seem to be the case as I am only concerned with one step evaluation -- this is only evaluated once.

I'm assuming the random variables are independent, but I'd like to eventually throw away this assumption.

jab
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    Could you be a bit more precise about how you want your model to work, or maybe provide a small example, or both? For instance: “maximization of a function of the sum of random variables” — you don’t say anything about decision variables so this objective function would be a constant. Also “proportional to the expectation of that result truly being optimal” — what is the expectation of a result being optimal? Do you mean the probability that it is optimal? – LarrySnyder610 May 17 '19 at 22:01
  • @LarrySnyder610 Probability that it is optimal -- proportional considering there would be a very low chance that the choices would be optimal, given the nature of random variables. – jab May 18 '19 at 05:02
  • @LarrySnyder610 Also, I'll edit and include an example function, but this is really a generalization. Optimization function would be a linear combination of random variables weighted by integer decision variables. – jab May 18 '19 at 05:04
  • Do you have any second-stage variables, i.e., decision variables that are allowed to depend on the scenario? Also, I still think your answer is vague -- maybe that's just because you're still trying to figure out exactly how you want your model to work -- but I can't understand, for example, what you mean by results that are weighted based on the probability of being optimal for a given scenario. – LarrySnyder610 May 20 '19 at 13:28

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