Using approximate gradients in developing an interactive interior primal–dual multiobjective linear programming algorithm

We present a new interactive multiobjective linear programming algorithm that is based on one variant of Karmarkar's algorithm known as the path-following primal–dual algorithm. The modification of this single-objective linear programming algorithm to the multiobjective case is done by deriving an approximate gradient to the implicitly-known utility function. By interacting with the decision maker, locally-relevant preference information are elicited and the approximated gradient can therefore be continuously updated. The interior step direction is then generated by projecting the approximate gradient and taking an interior step from the current iterate to the new one along this projection.