Mathematical Sciences: Theoretical and Algorithmic Studies in Combinatorial Problem-Solving

数学规划研究的是受侧约束的多变量函数的优化问题。组合优化解决了同样的问题，但由于包含不可分割的活动（即必须采用整数值的变量）而带来了额外的复杂性。在纯组合问题中，所有变量必须是整数。自第二次世界大战后不久，数学规划思想被成功地用于制定和解决许多复杂的规划问题，其应用涵盖了从定量社会科学到工程科学的整个范围。以数字方式解决数学规划问题的能力的提高，带来了将不可分割的活动合并到规划模型中的更大需求。本研究旨在扩展组合问题解决的计算前沿，并实现对组合优化问题的更好的数学理解。

Mathematical programming is concerned with the optimization of functions in many variables subject to side constraints. Combinatorial optimization addresses the same problem with an additional complication brought about by the inclusion of indivisible activities, i.e., by variables that must assume integer values. In purely combinatorial problems, all variables must be integer. Since its inception shortly after World War II, mathematical programming ideas have been successfully utilized to formulate and to solve many complex planning problems, with applications covering the entire range from the quantitative social sciences to engineering science. The increased ability to numerically solve mathematical programming problems has brought about a greater demand to incorporate indivisible activities into the planning models. This research aims to extend the computational frontiers of combinatorial problem-solving as well as to achieve a better mathematical understanding of combinatorial optimization problems.