Robust Optimisation: Equality vs Inequality constraint (Fixed vs Uncertains)

Hi Will,

The box uncertainty set you have defined is a sensible approach, but there are a few reasons why you get an infeasible problem.

First, the lower and upper bound should be used within the same constraint, as otherwise the problem becomes “too robust”, i.e. too strong constraints on the mpvars for the solution to be feasible for all uncertain assignments. Rather than writing two constraints with a different sign on the uncertain parameter and only an upper bound on the uncertain itself, I’d rather write one robust constraint and set bounds on the uncertain parameters, i.e. the following uncertainty set:

P23UNCERTAIN(t) <=  P23PERCENT(t) * abs(P23COEFF(t))
P23UNCERTAIN(t) >= - P23PERCENT(t) * abs(P23COEFF(t))

(note the abs() in order to make P23UNCERTAIN(t) vary between a small negative and a small positive) and the following single robust constraint (which replaces the first two with P23TARGETL):

P23MODL1 := sum(t in VARIABLES1) (P23COEFF(t) + P23UNCERTAIN(t))*OPTS(t) >= P23TARGETL

The second pair of constraints would be replaced by a similar single robust constraint. This reflects your box-constrained uncertainty set: the coefficient of OPTS(t) can vary between P23COEFF(t) * (1 – P23PERCENT(t)) and P23COEFF(t) * (1 + P23PERCENT(t)).

I haven’t tried this approach, so I don’t know if it results in an infeasible problem, which anyway depends on your data. The bottom line is that your initial model is infeasible because the robust constraints depend on a very large uncertainty set with due to loose bounds on the uncertain parameters. The above constraints should achieve the robustness you had in mind: the uncertainty set is now a bounded box, but this doesn’t mean the problem is going to be feasible.

Second, the size of the uncertainty set determines how robust the solution will be. The larger the uncertainty set, the more stringent constraints the solution will have to satisfy in order to achieve robustness; this can lead to worse objective function values (because the set of feasible OPTS(t) is more restricted) or even to infeasibility.

Box uncertainty sets have the peculiarity that the uncertain parameters realized at an optimal solution of the robust problem are a vertex of the box, i.e. they are all at their lower or upper bound. This is a rather unlikely practical situation where ALL parameters are far from their expected value. One could imagine only a few such parameters are off. In your case you could also imagine a confidence ellipsoid containing all uncertain realizations, something of the form

sum(t in VARIABLES1) ((P23UNCERTAIN(t) / (abs(P23COEFF(t)) * P23PERCENT(t)))**2) <= 1

which is an ellipsoid contained in the aforementioned box (this would hence replace the upper/lower bound constraints on P23UNCERTAIN(t)) and that prevents all uncertain parameters from being at an extreme of the box (obviously, only use it if this makes sense for your application).

(Note: in Mosel I believe the ** should be replaced by a caret, but the caret i not accepted in text by this forum platform.)

An alternative to the above ellipsoid that limits the (weighted) l-2 distance is an uncertainty set that limits the l-1 distance, and it’s the cardinality() uncertainty set mentioned in the manual.

Hope this helps.