Computational Studies in Discrete Optimization

离散优化用于解决涉及从一系列可能性中选择最佳替代方案的实际问题；它在经济的几乎每一个领域都有广泛的应用。许多行业，从电信到VLSI，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受线性不等约束的线性函数的形式，其中某些或全部变量被要求取整数值。如果不要求变量为整数，则该模型是线性规划(LP)问题。整数变量使得MIP模型比相应的LP模型更难求解，但正是这种完整性组件使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George Dantzig为求解LP模型而设计的著名单纯形算法的强大功能；该算法在2000年被评为世纪十大算法之一。在离散优化中，单纯形算法与一种称为割平面法的技术相结合，以改进离散问题的Lp逼近。割面的使用最早是由Dantzig，Fulkerson和Johnson于1954年在研究旅行商问题(TSP)时提出的；多年来，人们已经提出了许多变体，现在割面已经成为商业MIP解算器和对许多类离散优化问题最成功的攻击的重要组成部分。**我们提出了一系列旨在扩大离散优化和混合整数规划的研究范围。这项工作有两个主要推动力。首先，我们将寻求开发工具，利用并行计算平台，用单纯形法解决更大类别的线性规划模型。其次，我们将研究MIP和TSP模型，以开发更好的技术，将割平面法应用于大型模型，再次侧重于并行计算平台的使用。预期的结果是极大地改进了解决工业应用中出现的LP和MIP模型的工具和软件，以及用于解决旅行商问题的广泛使用的协和代码的改进版本。**拟议项目的一个重要组成部分是对研究生进行大规模离散优化模型实用解的高级技术培训。参与该项目的学生将非常适合目前这一领域的许多工业界和学术界的职位。

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.