| Mathematical optimization, or mathematical programming, has been studied for several decades. Researchers are constantly searching for optimization techniques which allow one to determine the ideal course of action in extremely complex situations. This line of scientific inquiry motivates the primary focus of this dissertation --- nontraditional optimization problems having either multiple objective functions or parametric input. Utilizing multiple objective functions allows one to account for the fact that the decision process in many real-life problems in engineering, business, and management is often driven by several conflicting criteria such as cost, performance, reliability, safety, and productivity. Additionally, incorporating parametric input allows one to account for uncertainty in models' data, which can arise for a number of reasons, including a changing availability of resources, estimation or measurement errors, or implementation errors caused by storing data in a fixed precision format. However, when a decision problem has either parametric input or multiple objectives, one cannot hope to find a single, satisfactory solution. Thus, in this work we develop techniques which can be used to determine sets of desirable solutions. The two main problems we consider in this work are the biobjective mixed integer linear program (BOMILP) and the multiparametric linear complementarity problem (mpLCP).;BOMILPs are optimization problems in which two linear objectives are optimized over a polyhedron while restricting some of the decision variables to be integer. We present a new data structure in the form of a modified binary tree that can be used to store the solution set of BOMILP. Empirical evidence is provided showing that this structure is able to store these solution sets more efficiently than other data structures that are typically used for this purpose. We also develop a branch-and-bound (BB) procedure that can be used to compute the solution set of BOMILP. Computational experiments are conducted in order to compare the performance of the new BB procedure with current state-of-the-art methods for determining the solution set of BOMILP. The results provide strong evidence of the utility of the proposed BB method.;We also present new procedures for solving two variants of the mpLCP. Each of these procedures consists of two phases. In the first phase an initial feasible solution to mpLCP which satisfies certain criteria is determined. This contribution alone is significant because the question of how such an initial solution could be generated was previously unanswered. In the second phase the set of feasible parameters is partitioned into regions such that the solution of the mpLCP, as a function of the parameters, is invariant over each region. For the first variant of mpLCP, the worst-case complexity of the presented procedure matches that of current state-of-the-art methods for nondegenerate problems and is lower than that of current state-of-the-art methods for degenerate problems. Additionally, computational results show that the proposed procedure significantly outperforms current state-of-the-art methods in practice. The second variant of mpLCP we consider was previously unsolved. In order to develop a solution strategy, we first study the structure of the problem in detail. This study relies on the integration of several key concepts from algebraic geometry and topology into the field of operations research. Using these tools we build the theoretical foundation necessary to solve the mpLCP and propose a strategy for doing so. Experimental results indicate that the presented solution method also performs well in practice. |
