Robust And Adaptive Optimization

Robust and Adaptive Optimization

Robust and Adaptive Optimization

Optimization is one of the most interesting and well-studied domains in Mathematics andComputer Science. It has attracted the interest of researcher communities from diverse backgrounds for centuries. Some of the optimization problems are harder than others, because ofthe uncertainties involved in them due to the structure of the problem, or due to the uncertainty in the input data. The uncertainties can be passive or they can be induced activelyby an adversary. Many of those problems are NP-Hard. Scheduling problems and spanningtree problems have also attracted the researchers for a long time.In this work, we have developed robust and adaptive algorithms for some problems from theabove-mentioned domains. We have provided polynomial-time algorithms for tractable problems. As we investigated many NP-Hard problems, either we have provided polynomial-timealgorithms for those problems with special structures, or we have designed approximation algorithms and heuristics for the general problems. We have reported experimental results andthe outcomes of comparative studies between different schemes to evaluate those heuristicsand approximation algorithms.

Advances in Robust and Adaptive Optimization

Optimization in the presence of uncertainty is at the heart of operations research. There are many approaches to modeling the nature of this uncertainty, but this thesis focuses on developing new algorithms, software, and insights for an approach that has risen in popularity over the last 15 years: robust optimization (RO), and its extension to decision making across time, adaptive optimization (AO). In the first chapter, we perform a computational study of two approaches for solving RO problems: "reformulation" and "cutting planes". Our results provide useful evidence for what types of problems each method excels in. In the second chapter, we present and analyze a new algorithm for multistage AO problems with both integer and continuous recourse decisions. The algorithm operates by iteratively partitioning the problem's uncertainty set, using the approximate solution at each iteration. We show that it quickly produces high-quality solutions. In the third chapter, we propose an AO approach to a general version of the process flexibility design problem, whereby we must decide which factories produce which products. We demonstrate significant savings for the price of flexibility versus simple but popular designs in the literature. In the fourth chapter, we describe computationally practical methods for solving problems with "relative" RO objective functions. We use combinations of absolute and relative worst-case objective functions to find "Pareto-efficient" solutions that combine aspects of both. We demonstrate through three in-depth case studies that these solutions are intuitive and perform well in simulation. In the fifth chapter, we describe JuMPeR, a software package for modeling RO and AO problems that builds on the JuMP modeling language. It supports many features including automatic reformulation, cutting plane generation, linear decision rules, and general data-driven uncertainty sets.