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 oracle-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.

翻译：在这项工作中,我们调查了用于对具有不确定成本-矢量的组合问题(其成本-矢量的不确定性包含在康韦克斯的不确定因素中)的最小最大优化问题。 方法的构想是计算一套最坏的k可行的解决方案, 如果在每种可能的假设中, 最优的 k 解决方案将会得到实施。 众所周知, 如果 k 至少是问题的层面, 最小最大- 最小优化问题可以有效解决, 而 k 则在理论上和计算上很难解决。 虽然在文献中已经很好地研究了关于中间案例的未知问题, 即 k 是否小于问题的范围, 但接近于问题的业绩层面。 我们处理这个开放式问题, 并显示, 对于组合问题的选择问题, 最小- 最小- 最小- 最小- 最小- 最小- 最小- 最小- 最小- 最小- 最小- 最小- 的 水平- 测试方法, 其当前 K- 的直径- 直径, 其直径- 的直径- 直方- 直方- k- 直方- 直方- 直方- 直方- 直方- 直方- 解- 直方- 直方- 解- 解- 解- 解- k- 解- 解- 解- 解- k- 方法, 直方- 方法显示- 直方- 直方- 直方- 直方- 直方- 法- 直方- 直方- 直方- 直方- 直方- 直方- 直方- 直方- 方- 方- 法- 方- 法- 方- 直- 直- 法- 直- 直- 直- 法- 法- 方- 法- 法- 直- 直- 法- 法- 直- 直- 直- 直- 直- 直- 直- 直- 直- 方- 方- 直- 直- 直- 直- 法- 直- 直- 方- 直- 直- 方- 方- 方- 方- 直- 方- 直