Efficient gradient calculation of the Pareto optimal curve in multicriteria optimization

An original method is presented to evaluate the gradient information of the Pareto optimal curve so as to give a more rigorous and precise description of its characteristics. By introducing the novel concept of the subgradient in the study of the Pareto optimal curve, it is revealed that when the weighting method is used, the insensitivity of certain Pareto optima with respect to the weighting coefficients is due to the nondifferentiability at these points. The subgradient bounds can be readily calculated by linear programming. Numerical examples will be dealt with to demonstrate this method. Some seemingly strange published results can thus be interpreted by this method.