Optimization with interval data: new problems, algorithms, and applications.

The parameters of real-world optimization problems are often uncertain due to the failure of exact estimation of data entries. Throughout the years, several approaches have been developed to cope with uncertainty in the input parameters of optimization problems, such as robust optimization, stochastic optimization, fuzzy programming, parametric programming, and interval optimization. Each of these approaches tackles the uncertainty in the input data with different assumptions on the source of uncertainty and imposes different requirements on the returned solutions. In this dissertation, the approach we take is that of interval optimization, and more specifically, interval linear programming. The two main problems we consider in this context are, considering all realizations of the interval data, the problems of finding the range of the optimal values and determining the set of all possible optimal solutions. While the theoretical aspects of these problems are well-studied, the algorithmic aspects and the engineering implications have not been explored. In this dissertation, we partially fill these voids. In the first part of the dissertation, we present and test three heuristics to find bounds on the worst optimal value of the equality-constrained interval linear program, which is known to be an intractable problem. In the second part of the dissertation, motivated by a real-case problem in the healthcare context, we define and analyze a new problem, the outcome range problem, in interval linear programming. The solution to the problem would help decision-makers quantify unintended/further consequences of optimal decisions made under uncertainty. Basically, the problem finds the range of an extra function of interest (different from the objective function) over all possible optimal solutions of an interval linear program. We analyze the problem from the theoretical and algorithmic perspectives. We evaluate the performance of our algorithms on simulated problem instances and on a real-world healthcare application. In the third part of the dissertation, we extend our analysis of the outcome range problem, exploring different theoretical properties and designing several new solution algorithms. We also test our solution approaches on different datasets, highlighting the strengths and weaknesses of each approach. Finally, in the last part of the dissertation, we discuss an application of interval optimization in the sensor location problem in the traffic management context. Particularly, we propose an optimization approach to handle the measurement errors in the full link flow observability problem. We show the applicability of our approach on several real-world traffic networks