Using efficient anchoring points for generating search directions in interior multiobjective linear programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along 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 acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.