Before revision:  The budget perturbations are set based on the dual variables of the additional physical constraints such that the KKT conditions of the newly defined social and individual optimization problems are equivalent. This establishes that our mechanism sets market clearing pricessuch that each agent obtains their maximum possible utilities subject to their constraints .
After revision:  The budget perturbations are set based on the dual variables of the additional physical constraints such that the KKT conditions of the perturbed social problem establishes equilibrium prices .