Robust Solutions for Geographic Resource Allocation Problems

Abstract

This thesis describes different ways to use robust optimization concepts and techniques in problems that arise in geographic resource allocation. Many problems in geographic resource allocation deal with uncertainty just like many other domains. Some of them, like the k-centers problem, are naturally defined in a {\em minimax} fashion, and some others can be treated under uncertainty where we seek robust solutions. Most geographic resource allocation problems can be settled in one or more of the categories, such as location problems, segmentation (partitioning) problems, assignment problems, routing problems, and backbone network design problems. In all such problems there are parameters that can be unknown in practice. It is sensible to define an uncertainty set for the unknown parameter based on some crude knowledge about that unknown parameter and then to treat the uncertainty like some deterministic variability of the values of the parameter, followed by ultimately solving the problem as that parameter is another variable in a higher dimension. For such problems in geographic resource allocation, we take a robust approach to tackle the uncertainty. Depending on the problem and also geometry of the uncertainty set, the robust optimization model can be tractable or difficult to solve. We deal with both cases in this thesis where we combine elements from computational geometry, geometric probability theory, vector space optimization, and topology, to either solve the problem to optimality or develop fast algorithms to settle with an approximation solution. We also present a divide and conquer type of approach using geometric partitioning to solve robust optimization problems. In a generic robust optimization problem, if the uncertainty set is an infinite set (which is the case in most practical situations), then we will have an infinite or semi-infinite dimensional optimization problem since the model will have infinite number of constraints. We describe the drawbacks of the current approaches to solving such problems and their inability to obtain reasonable solutions for some special but common and practical cases, like clustered data, and then we show that our approach makes these problems easy to solve.