Robust LP with Right-Handside Uncertainty, Duality and Applications

The various robust linear programming models investigated so far in the literature essentially appear to be based either on what is referred to as ’rowwise’ uncertainty models or on ’columnwise’ uncertainty models (these basically assume that the rows resp: the columns of the constraint matrix are subject to changes within a well specified uncertainty set). In this paper, we discuss a special case of columnwise uncertainty namely the subclass of robust LP models with uncertainty limited to the right handside only. (this subclass does not appear to have been significantly investigated so far). In this context we introduce the concept of ’ two-stage robust LP model’ as opposed to the standard case (which might be referred to as ’single-stage robust LP model’) and we address the question of whether LP duality can be used to convert a LP problem with RHS-uncertainty into a robust LP problem with uncertainty on the objective function. We show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The result∗University Paris-6, France Email : [email protected] ing expressions of the objective function to be optimized in both cases, appear to be clearly distinct. Moreover, from a complexity pointof-view, one appears to be efficiently solvable ( it reduces to a convex optimization problem) whereas the other, as a nonconvex optimization problem, is expected to be computationally difficult in the general case. As an application of the 2-stage robust LP model introduced here, we next investigate the robust PERT scheduling problem, considering two possible natural ways of specifying the uncertainty set for the task durations: the case where the uncertainty set is a scaled ball with respect to the L∞ norm; the case where the uncertainty set is a scaled Hamming ball of bounded radius (which, though leading to a quite different model, bears some resemblance to the well-known Bertsimas-Sim approach to robustness). We show that in both cases, the resulting robust optimization problem can be efficiently solved in polynomial time.