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 components: 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. 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, Lohne and Shao, 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 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.

中文翻译：

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求解线性多目标优化问题的内逼近算法

Benson 的外逼近算法及其变体是解决线性多目标优化问题最常用的方法。这些算法有两个相互交织的组成部分：一方面是一维线性优化，另一方面是与顶点计数密切相关的组合部分。它们的分离提供了对 Benson 算法的更深入的了解，并指向了一种双重方法。定义了两种侧重于组合部分的骨架算法。使用不同的单目标优化问题——称为预言机调用——会产生不同的算法，例如顺序凸包算法、具有理论上最佳迭代次数的 Benson 算法的另一个版本、Ehrgott、Lohne 和 Shao 的对偶算法，以及新的算法。新算法有几个优点。首先，对应的一维优化问题使用原始约束，没有添加任何额外的变量或约束。其次，它的迭代次数符合理论上最好的次数。作为一个对偶算法，它是顺序的：在每次迭代中它产生一个极值解，因此当找到一个满意的解时可以中止。可以随时从多个方向“探测”或“扫描”帕累托前沿，而不会对效率产生不利影响。最后，它非常适合处理约束之间存在许多线性相关性的高度退化问题。