Inner approximation algorithm for solving linear multiobjective optimization problems

Abstract

Benson’s outer approximation algorithm and its variants are the most frequently used methods for solving linear multiobjective optimization problems. These algorithms have two intertwined parts: single-objective linear optimization on one hand, and a combinatorial part closely related to vertex enumeration on the other. Their separation provides a deeper insight into Benson’s algorithm, and points toward a dual approach. Two skeletal algorithms are defined which focus on the combinatorial part. Using different single-objective optimization problems – called oracle calls – yield different algorithms, such as a sequential convex hull algorithm, another version of Benson’s algorithm with the theoretically best possible iteration count, the dual algorithm of Ehrgott, L ̈ ohne and Shao [7], and the new algorithm. The new algorithm has several advantages. First, the corresponding single-objective optimization problem uses the original constraints without adding any extra variables or constraints. Second, its iteration count meets the theoretically best possible one. As a dual algorithm, it is sequential: in each iteration it produces an extremal solution, thus can be aborted when a satisfactory solution is found. The Pareto front can be “probed” or “scanned” from several directions at any moment without adversely affecting the efficiency. Finally, it is well suited to handle highly degenerate problems where there are many linear dependencies among the constraints. On problems with ten or more objectives the implementation shows a significant increase in efficiency compared to Bensolve – due to the reduced number of iterations and the improved combinatorial handling.