Computational Studies in Discrete Optimization

离散优化用于解决实际问题，涉及从可能性领域中选择最佳方案;它在经济的几乎每个部门都有广泛的应用。从电信到超大规模集成电路的许多行业都完全依赖离散优化来指导他们完成复杂的设计过程。也许最重要的一类离散优化问题是混合整数规划（MIP）模型。这些模型具有最小化或最大化线性函数的形式，该线性函数受到线性不等式约束，其中一些或所有变量需要采用整数值。如果不要求变量是整数，则模型是线性规划（LP）问题。积分变量使得MIP模型比相应的LP模型更难求解，但是正是这种积分性成分使得MIP模型能够捕获本质上离散的决策。离散优化的巨大成功在很大程度上归功于乔治丹齐格著名的单纯形算法，该算法被设计用于求解LP模型;该算法在2000年被评为世纪十大算法之一。在离散优化中，单纯形算法与称为切割平面法的技术相结合，用于改进离散问题的LP近似。切割平面的使用是由Dantzig，Fulkerson和约翰逊在1954年在他们的旅行商问题（TSP）的工作中开创的;多年来，人们提出了许多变化，切割平面现在是商业MIP求解器和许多类离散优化问题的最成功攻击中的重要组成部分。我们提出了一系列旨在扩展离散优化范围的研究混合整数规划这项工作有两个主要目标。首先，我们将寻求开发工具，将允许更大的类的LP模型的解决方案与单纯形法，利用并行计算平台。其次，我们将研究MIP和TSP模型，以开发更好的技术，将切割平面方法应用于大规模模型，再次关注并行计算平台的使用。预期的成果是大大改善的工具和软件的LP和MIP模型的解决方案中出现的工业应用，以及广泛使用的协和代码的旅行商problem. A的解决方案的改进版本的建议项目的一个重要组成部分是培训研究生的先进技术在大规模离散优化模型的实际解决方案。参与该项目的学生将非常适合目前在工业界和学术界这一领域提供的许多职位。

Discrete optimization is used to solve practical problems that involve choosing the best alternative from a field of possibilities; it has broad applications in nearly every segment of the economy. Many industries, ranging from telecommunications to VLSI, are fully dependent on discrete optimization to guide them through complex design procedures.Perhaps the most important class of discrete optimization problems are the mixed-integer-programming (MIP) models. These models have the form of minimizing or maximizing a linear function subject to linear inequality constraints, where some or all of the variables are required to take on integer values. If no variables are required to be integer, then the model is a linear-programming (LP) problem. The integral variables make MIP models much more difficult to solve than the corresponding LP models, but it is this integrality component that allows MIP models to capture decisions that are discrete in nature.The great success of discrete optimization is due in large part to the power of George Dantzig's famous simplex algorithm, designed to solve LP models; this algorithm was named one of the Top Ten Algorithms of the Century in the year 2000. In discrete optimization, the simplex algorithm is combined with a technique known as the cutting-plane method for improving the LP approximations of discrete problems. The use of cutting planes was pioneered by Dantzig, Fulkerson, and Johnson in 1954 in their work on the traveling salesman problem (TSP); many variations have been proposed over the years and cutting planes are now an essential ingredient in commercial MIP solvers and in the most successful attacks on many classes of discrete optimization problems.We propose a line of research aimed at extending the reach of discrete optimization and mixed-integer programming. The work has two main thrusts. First, we will seek to develop tools that will allow for the solution of much larger classes of LP models with the simplex method, utilizing parallel computing platforms. Secondly, we will study MIP and TSP models to develop better techniques for applying the cutting-plane method to large-scale models, again focusing on the use of parallel computing platforms. The expected outcomes are greatly improved tools and software for the solution of LP and MIP models arising in industrial applications, as well as an improved version of the widely-used Concorde code for the solution of the traveling salesman problem.An important component of the proposed project is the training of graduate students in advanced techniques in the practical solution of large-scale discrete optimization models. Students involved in the project will be well-suited for the many positions that are currently available in this area in both industry and academia.