Anchoring points and cones of opportunities in interior multiobjective linear programming

This paper presents a modification of one variant of Karmarkar’s interior-point linear programming algorithm to Multiobjective Linear Programming (MOLP) problems. We show that by taking the variant known as the affine-scaling primal algorithm, generating locally-relevant scaling coefficients and applying them to the projected gradients produced by it, we can define what we refer to as anchoring points that then define cones in which we search for an optimal solution through interaction with the decision maker. Currently existing MOLP algorithms are simplex-based and make their progress toward the optimal solution by following the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an optimal solution may prove less sensitive to problem size. We refer to the class of MOLP algorithms resulting from this variant as Affine-Scaling Interior Multiobjective Linear Programming (ASIMOLP) algorithms.