具有增量补偿的鲁棒优化
Robust optimization with incremental recourse
摘要：
在本文中，我们考虑一种自适应方法来解决不确定成本参数的优化问题。在这里，决策者选择一个初始决策，观察不确定的成本参数的实现，然后被允许修改初始决策。我们对待的不确定性使用的框架中的不确定参数位于一个给定的集合的鲁棒优化。决策者进行优化，以便根据最坏情况分析制定最佳成本保证。追索权决策是“增量的”;也就是说，决策者被允许改变一个小的固定金额的初始解决方案。我们将由此产生的问题称为鲁棒增量问题。我们研究了几个优化问题的鲁棒增量变量。我们表明，一个线性规划的鲁棒增量对应本身是一个线性规划，如果不确定性集是多面体。因此，它在多项式时间内是可解的。对于离散不确定性集，我们建立了鲁棒增量线性规划的NP-困难性。我们证明了当费用从多面体不确定集合中选取时，鲁棒增量最短路问题是NP完全的，即使在初始路径上只增加一条新弧的情况下，我们也讨论了鲁棒增量最短路问题和鲁棒增量最小生成树问题的几种特殊情况的复杂性.
英文摘要：
In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem.
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