In this work we investigate the min-max-min robust optimization problem applied to combinatorial problems with uncertain cost-vectors which are contained in a convex uncertainty set. The idea of the approach is to calculate a set of k feasible solutions which are worst-case optimal if in each possible scenario the best of the k solutions would be implemented. It is known that the min-max-min robust problem can be solved efficiently if k is at least the dimension of the problem, while it is theoretically and computationally hard if k is small. While both cases are well studied in the literature nothing is known about the intermediate case, namely if k is smaller than but close to the dimension of the problem. We approach this open question and show that for a selection of combinatorial problems the min-max-min problem can be solved exactly and approximately in polynomial time if some problem specific values are fixed. Furthermore we approach a second open question and present the first implementable algorithm with pseudopolynomial runtime for the case that k is at least the dimension of the problem. The algorithm is based on a projected subgradient method where the projection problem is solved by the classical Frank-Wolfe algorithm. Additionally we derive a branch & bound method to solve the min-max-min problem for arbitrary values of k and perform tests on knapsack and shortest path instances. The experiments show that despite its theoretical impact the projected subgradient method cannot compete with an already existing method. On the other hand the performance of the branch & bound method scales very well with the number of solutions. Thus we are able to solve instances where k is above some small threshold very efficiently.

中文翻译：

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最小-最大-最小稳健组合优化的新复杂度结果和算法

在这项工作中，我们研究了最小-最大-最小稳健优化问题，该优化问题适用于包含在凸不确定性集中的不确定成本向量的组合问题。该方法的思想是计算一组 k 个可行的解决方案，如果在每种可能的情况下都将实现 k 个解决方案中的最佳解决方案，则这些解决方案是最坏情况下的最优解。众所周知，如果 k 至少是问题的维度，则可以有效地解决最小-最大-最小稳健问题，而如果 k 很小，则理论上和计算上都很困难。虽然文献中对这两种情况都进行了深入研究，但对中间情况一无所知，即 k 是否小于但接近问题的维度。我们处理这个开放问题，并表明如果某些特定问题的值是固定的，那么对于选择的组合问题，可以在多项式时间内精确且近似地解决最小-最大-最小问题。此外，我们处理第二个开放性问题，并针对 k 至少是问题的维度的情况提出了第一个具有伪多项式运行时间的可实现算法。该算法基于投影次梯度方法，其中投影问题由经典的 Frank-Wolfe 算法解决。此外，我们导出了一种分支定界方法来解决任意 k 值的最小-最大-最小问题，并对背包和最短路径实例进行测试。实验表明，尽管具有理论影响，投影次梯度方法无法与现有方法竞争。另一方面，分支定界方法的性能与解决方案的数量成正比。因此，我们能够非常有效地解决 k 高于某个小阈值的情况。