Robustness and duality in linear programming

In this paper, we consider a linear program in which the right hand sides of theconstraints are uncertain and inaccurate. This uncertainty is represented byintervals, that is to say that each right hand side can take any value in itsinterval regardless of other constraints. The problem is then to determine arobust solution, which is satisfactory for all possible coefficient values.Classical criteria, such as the worst case and the maximum regret, are appliedto define different robust versions of the initial linear program. Morerecently, Bertsimas and Sim have proposed a new model that generalizes the worstcase criterion. The subject of this paper is to establish the relationshipsbetween linear programs with uncertain right hand sides and linear programs withuncertain objective function coefficients using the classical duality theory. Weshow that the transfer of the uncertainty from the right hand sides to theobjective function coefficients is possible by establishing new dualityrelations. When the right hand sides are approximated by intervals, we alsopropose an extension of the Bertsimas and Sim's model and we show that themaximum regret criterion is equivalent to the worst case criterion.