Probust constraints: A new concept do deal with uncertainty in constraints

What is it about?

Many decisions have to be taken with uncertain data. For rational decisions that are modelled by optimization problems, this means that in the constraints often random variables occur. A well establishted method to deal with this uncertainty that takes into account information about the probability distribution of the uncertain data is to use probabilistic constraints (chance constraints) that require that the probability that the inequality constraint is satisfied is greater than or equal to an a priori given minimal probability. If the constraints are supposed to be satisfied for each moment in a whole time interval, it makes sense to require that the probability that the constraint is satisfied throughout the time is greater than the a priori bound rather than to require only that at each moment in the time interval the chance constraints hold.

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Photo by Martin Adams on Unsplash

Photo by Martin Adams on Unsplash

Why is it important?

Uncertain data occur in different classes: For some data experience from the past yields excellent approximations for the corresponding probability distribution, whereas for other types of data at most an uncertainty set (for example defined by lower and upper bounds) containing the data is available. The *probust constraints* provide a tool that allows to treat optimization problems where both types of data occur simultaneously. As a driving example, the operation of gas transport networks is considered where the nominations can be modelled as a random variable with a known distribution function within the interval of capacities. The system operator has the problem of maximization of theses capacities that can be modelled with probust constraints.

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