In this work we investigate the min-max-min robust optimization problem for binary problems with uncertain cost-vectors. 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 is 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 cases, i.e. k lies between one and the dimension of the problem. We approach this open question and provide an efficient algorithm which achieves problem-specific additive and multiplicative approximation guarantees for the cases where k is close to and where k is a fraction of the dimension. The derived bounds can be used to show that the min-max-min robust problem is solvable in oracle-polynomial time under certain conditions even if k is smaller than the dimension. We show that the derived approximation guarantees can be extended to the k-adaptability problem. As a consequence we can provide better bounds on the number of required second-stage policies to achieve a certain approximation guarantee for the exact two-stage robust problem. Additionally we can show that these bounds are also promising for recoverable robust optimization. Finally we incorporate our efficient approximation algorithm into a branch & bound method to solve the min-max-min problem for arbitrary values of k. The experiments show that the performance of the branch & bound method scales well with the number of solutions, confirming our theoretical insights. Thus we are able to solve instances where k is of intermediate size efficiently.

翻译：本文研究具有不确定成本向量的二元问题中的最小-最大-最小鲁棒优化问题。该方法的核心思想是计算一个包含k个可行解的集合，使得在每种可能的情景下，当采用集合中最优解时，该集合达到最坏情况下的最优性。已知当k至少等于问题的维数时，最小-最大-最小鲁棒问题可高效求解；而当k较小时，该问题在理论上和计算上均具有难度。尽管这两种极端情况在文献中已有充分研究，但对于中间情形（即k介于1与问题维数之间）却鲜有探讨。本文针对这一开放问题，提出了一种高效算法，在k接近维数或k为维数的某一分数时，能够实现问题特定的加性及乘性近似保证。所得边界可用于证明：即便k小于维数，在特定条件下，最小-最大-最小鲁棒问题仍可在神谕多项式时间内求解。我们进一步证明该近似保证可推广至k-适应性问题。由此，我们可为精确两阶段鲁棒问题提供更优的第二阶段策略数量上界，以实现特定近似保证。此外，这些边界对可恢复鲁棒优化同样具有应用前景。最后，我们将所提高效近似算法嵌入分支定界方法中，以求解任意k值下的最小-最大-最小问题。实验表明，分支定界方法的性能随解集规模扩展良好，验证了理论分析的有效性。因此，我们能够高效求解k处于中间规模的实例。