Robust optimisation in Maths Programming

[stukbv]

I am a bit confused as to how to formulate the robust optimisation counterpart for the following problem,

Homework Statement

Consider the random linear constraint Ʃ_j ( ~a_ijx_j ) ≤ b_i, where ~a_ij's are the random parameters,
Assume ~a_ij belongs to the uncertainty interval [a_ij-a_ij*, a_ij + a_ij*] for all j=1...n, and in addition
Ʃ_j|~a_ij-a_ij| ≤r for all j=1...n

Formulate the robust counterpart for this random constraint.


2. The attempt at a solution

All I can think to do is ;

Objective; minc^Tx

Constraint; Ʃ_j~a_ijx_j ≤b_i for all ~a_i in Ui where Ui = {~a_i:|~a_ij-a_ij|≤a_ij*}

 

[KataKoniK]

This is a good start, but there are a few things you can do to make it more robust and efficient. Firstly, you can include the uncertainty interval in the constraint instead of creating a separate set Ui. This will ensure that the constraint is satisfied for all possible values of ~aij within the interval.

Secondly, you can add a term to the objective function that penalizes the deviation from the nominal values of ~aij. This will ensure that the solution is not overly sensitive to small changes in the random parameters.

Lastly, you can add a term to the objective function that penalizes the variability of the random parameters. This will encourage the solution to be robust to a wider range of variations in the parameters.

Putting it all together, the robust counterpart for this random constraint would be:

Objective: min cTx + λ∑j|~aij-aij| + μ∑j|~aij-aij*|

Constraint: Ʃj(~aijxj) ≤ bi for all ~aij in [aij-aij*, aij+aij*] and Ʃj|~aij-aij| ≤ r

Where λ and μ are tuning parameters that control the trade-off between robustness and optimality.

I hope this helps clarify the formulation for you. Let me know if you have any further questions.