Duality, Robustness, and 2-stage robust LP decision models. Application to Robust PERT Scheduling

Michel Minoux 1
1 DECISION
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : 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). We first address the question of whether LP duality can be used to convert a robust LP problem with, say, rowwise uncertainty, into a robust LP with columnwise uncertainty. We provide simple examples supporting the general statement that the dual of a given robust linear programming model is NOT equivalent to the robust version of the dual (assuming of course that uncertainty for the rows - resp: the columns - of the primal, is specified in exactly the same way as uncertainty for the corresponding columns - resp: rows - of the dual). Next, we further investigate this issue of duality in the context of robustness by considering 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 show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The resulting expressions of the objective function to be optimized in both cases, though bearing some formal resemblance, appear to be clearly distinct. Moreover, from a complexity point-of-view, one appears to be effi- ciently 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.
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Michel Minoux. Duality, Robustness, and 2-stage robust LP decision models. Application to Robust PERT Scheduling. 2007. ⟨hal-00180529⟩

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