I have a very basic, but important question regarding the Ben-Tal, Robust Counterpart (RC), methodology.
What is the mathematical justification, for solving a dual version of the semi-infinite RC, instead of a just Worst-Case formulation.
I know it doesn't not sound clear. Let me say it more simple, assume we have an uncertain parameter, a.
we know this parameter varies between 1 and 1.1
I wanna know, what is the superiority of defining a box uncertainty set for a, over just putting it 1.1 instead of a, in the constraint.
Thank you in advance.
Hi, Aliriza
1. "Robust" in RO basically means robust feasibility, i.e., the solution of RO is feasible for all the possible realization of uncertain parameters. However, the solution from the deterministic problem by simply taking the worst case parameters (e.g., put your parameter to 1.1) does not necessarily have this feasibility.
2. Sometimes, it is not trivial at all to know what is the "worst case" of your uncertain parameters if your objective function is not linearly related to the uncertain parameters. Imaging the case that your "worst case" is defined as the "maximum regret" of your decision problem.
3. The solution from RO is quite conservative in the sense that RO gives you a solution that is both feasible no matter what the constraints turn out to be, and optimal for the worst-case objective function. For some problems, the adjustable robust optimization (ARO) a remedy to this conservatism.
Yiping Fang , thank you very much Dr. Fang. Your answer is very helpful for me to get a better sense of RO.
I have read, at least once, entire RO book from Ben-Tal, and related papers from him and Bertsimas, but this tiny, yet always repeated fact I neglected.
But still there is one subtlety here,
for implanting RO, we treat each constraint, separately ('Constraint-Wise').
How can after putting the worst-case value, assuming that this constraint has only one parameter, be more prone to infeasiblity, compared to RO in which we put the uncertain parameter |a|
If your parameter is only right hand side and you can easily figure out its worst-case, then directly repalce "1" with "1.1" or "0.9" is equivalent to using a box set and formulate Robust Couterpart.
The beauty of robust counterpart lies in that the uncertain parameters are somehow related to each other and they have little chance to reach their worst-case simultanuously (budget set, or ellipsoidal set). Under such situation, the robust optimization is less conservative than let all parameters reach their worst case values simultanuously.
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