Inner approximation algorithm for solving linear multiobjective optimization problems
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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 constituents: one-dimensional linear optimization one one hand, and a combinatorial part closely related to vertex numeration on the other. Their separation provides a deeper insight into Benson's algorithm, and points toward a dual approach different from the one defined by Ehrgott, Löhne and Shan. Two skeletal algorithms are defined and instantiated with different separation oracles to yield a sequential convex hull algorithm, a new version of Benson's algorithm with the theoretically best possible iteration count, and the new algorithm. The new algorithm has several advantages. First, the corresponding one-dimensional 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 20 or more objectives the implementation shows a more than 10-fold increase in efficiency compared to Bensolve -- due to the reduced number of iterations and the improved combinatorial handling.
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