Theory, computations and applications of structured Mixed-Integer Programs

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Mixed-Integer Programming (MIP) is an extremely important tool in quantitative decision making within modern corporations. Its applications include air transportation, telecommunications, logistics and manufacturing. The long-term goal of my research is to improve solution methods for MIPs, by providing better theoretical understanding of its properties, improving algorithmic ideas and ultimately allowing one to solve real-world optimization problems at a larger scale. This proposal advances this goal by studying particular structures that appear in some applications. Two research streams will be addressed. The first stream deals with considering sparsity in MIPs, particularly in cutting planes. Sparsity is a measure of the number of nonzero components, and it is an important factor that allows several algorithms to become faster. It is a prominent feature in several benchmark MIPs that came from practical applications - in a large set of such benchmark MIPs, the average sparsity is close to 1%. Cutting planes are one of the most important components in the practical solution of MIPs and, among all cutting planes, split cuts are the ones with greater impact. In a recent work, we have investigated split cuts obtained from sparse splits and show, through computational experiments, that this subclass of split cuts is very important. Therefore, this first stream will address theoretical and computational questions that arise based on such a sparsity parameter, with the ultimate goal to improve the performance of MIP algorithms. The second stream is dedicated to studying the structure obtained in the context of MIP formulations for vehicle routing (VRP), particularly under uncertainty. The VRP is an important optimization problem that seeks to find the minimum cost way to route a fleet of vehicles to satisfy a set of customer demands in such a way that the capacity of each vehicle is respected. It finds applications in logistics and transportation. One limiting assumption in the VRP is that all customer demands are known in advance, which is not always the case. Thus, recently, renewed research interest has been devoted to studying variants of the problem where demands are uncertain (i.e. stochastic VRP). Thus, this second stream aims to significantly advance the state-of-the-art in solving the stochastic VRP. Of particular interest are data-driven approaches, i.e. approaches that optimize using only historical data as input. The proposal will investigate new algorithms for existing versions of the problem and new paradigms for modeling more realistic versions of it. Ultimately, the development of such approaches will significantly increase the applicability of VRP models in real-world applications. HQP trained as a result of this proposal will gain valuable skills in quantitative decision making, acquiring both theoretical and implementation knowledge, which is important in both academia and industry.