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

Hi Will,

the short answer is: your uncertainty set is too large.

First, recall that a robust constraint must be satisfied for all values of the uncertain parameter within the uncertainty set. If u is an uncertain parameter between 0 and 1 and x is a variable, then for the robust constraint

(1+u) x <= 5

to be satisfied for all values of u, it must be x <= 5 / max{(1+u): u in [0,1]} = 5/2. This creates an upper bound for x and allows us to drop the robust constraint (but retain the bound of 1/2 on x). Note that the problem at the denominator attains its optimum for u=1.

The robust constraint

(1+u) x >= 1

is instead equivalent to a new lower bound on x, because x >= 1 / min{(1+u): u in [0,1]} = 1, because x must be >= 1 for the robust constraint to be satisfied for any value of u (note that this time the optimal value of u is 0 instead). The crucial point here is that if both constraints are in the problem, they can both be dropped but x has a bound interval of [1, 5/2], even though these limits correspond each to a different value of the uncertain parameter: one for each of the two robust constraints. It is a defining feature for the robust optimization module that two robust constraints correspond to different values of the uncertain parameters that they use.

For all t, your model uses the same uncertain parameter P23UNCERTAIN(t) over four constraints. For the first two:

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

P23MODL2 := sum(t in VARIABLES1) (P23COEFF(t)-P23UNCERTAIN(t))*OPTS(t) >= P23TARGETL

the coefficient of OPTS(t) can be low or high depending on P23UNCERTAIN(t). The data file has P23PERCENT(t) positive for all t but P23COEFF(t) sometimes positive, sometimes negative. All uncertain parameters are unbounded from below, therefore all of these uncertain parameters have an interval of [-inf, a(i)]. This implies that even one such constraint makes the problem infeasible: just make all a(i) a sufficiently large negative number for all coefficients of OPTS(t) to be small positive numbers, and the P23TARGETL will never be attained.

A similar conclusion can be drawn for the other two constraints, where a(i) just need to be made negative so that the coefficients on OPTS(t) are negative and hence OPTS(t) obtains a negative upper bound, incompatible with its lower bound of 0.

This could be mitigated by adding a lower bound to all P23UNCERTAIN(t), but a quick test shows that the problem remains infeasible even with bounds as tight as follows:

P23UNCERTAIN(t) <= P23PERCENT(t) * P23COEFF(t)

P23UNCERTAIN(t) >= P23PERCENT(t) * P23COEFF(t) - 0.001

I am not sure of the meaning of such parameters so I don't know if such small value would make sense. If it is indeed very small, consider defining an uncertainty set where all uncertains have both a lower and upper bound AND, for example, a weighted sum of them is bounded from above, or perhaps more suited for this example, the "cardinality" robustness constraint.