Compromise Solutions in Robust Optimisation

A new preprint is out, on “Compromise Solutions for Robust Combinatorial Optimization with Variable-Sized Uncertainty”. It uses a similar setting as our previous paper on variable-sized uncertainty, by assuming that only the shape of the uncertainty set is known, but not its size. But instead of finding a set of candidate robust solutions, we consider the problem of finding a single solution, that performs well over all possible uncertainty set sizes.

For min-max robustness, this problem can be solved quite efficiently by reformulating it as a single min-max problem for an uncertainty set of specific size. For min-max regret, however, things may get complicated, as the regret of a fixed solutions is a piecewise-linear function in the size of the uncertainty. We present general solution algorithms for this case, and consider the computational complexity for some classic combinatorial problems.

Here is the abstract:

In classic robust optimization, it is assumed that a set of possible parameter realizations, the uncertainty set, is modeled in a previous step and part of the input. As recent work has shown, finding the most suitable uncertainty set is in itself already a difficult task. We consider robust problems where the uncertainty set is not completely defined. Only the shape is known, but not its size. Such a setting is known as variable-sized uncertainty.
In this work we present an approach how to find a single robust solution, that performs well on average over all possible uncertainty set sizes. We demonstrate that this approach can be solved efficiently for min-max robust optimization, but is more involved in the case of min-max regret, where positive and negative complexity results for the selection problem, the minimum spanning tree problem, and the shortest path problem are provided. We introduce an iterative solution procedure, and evaluate its performance in an experimental comparison.

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