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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/operations-research-theory
课程评论：没有评论
课程名称：运筹学（3）：理论 课程概述：运筹学（OR）是一个应用数学和工程方法研究商业与管理、经济学、计算机科学、土木工程、电气工程等领域优化问题的领域。该系列课程分为三个部分，本课程重点介绍确定性优化技术，这是运筹学的主要内容作为系列的第三部分，我们将研究线性规划、整数规划和非线性规划的数学性质，并介绍这些理论性质的应用，以及它们如何帮助我们发展更好的数学程序求解方法。 课程大纲： 1. **课程概览**：在第一节课中，介绍课程及数学性质的重要性，学习使用矩阵方法进行单纯形法的运算。熟悉矩阵将帮助我们理解后续课程内容。 2. **对偶性**：本周学习线性规划对偶性的理论与应用，介绍原始-对偶对所拥有的性质，包括弱对偶性、强对偶性、互补松弛性，以及如何根据原始最优解构建对偶最优解。此外，介绍线性规划对偶性的一个重要应用：使用影子价格确定线性程序中的最关键约束。 3. **敏感性分析与对偶单纯形法**：在过去两周学习单纯形法和对偶性基础上，本节课讨论对偶单纯形法，并应用于敏感性分析中的一个重要问题：评估带有新约束的线性规划模型。 4. **网络流**：本节课介绍网络流模型，广泛应用于运输、物流、库存、项目管理等决策问题。首先介绍最小成本网络流（MCNF）模型，并展示其如何成为众多著名模型（如指派、运输、转运、最大流和最短路径）的推广，证明MCNF的特殊性质——完全单调性，以及其如何将线性规划和整数规划联系起来。 5. **凸分析**：作为本课程的最后一节，介绍NEC台湾的案例，该公司提供IT和网络解决方案，包括云计算、人工智能、物联网等。由于维护所有服务中心的成本过高，他们计划重新安排中心位置并重新分配每个中心的员工数量。包括解决NEC台湾所面临的设施选址问题的算法。 6. **拉格朗日对偶性与KKT条件**：本周学习带约束的非线性程序，引入两个主要工具，拉格朗日松弛法和KKT条件，用于求解带约束的非线性程序，同时展示线性规划对偶性是拉格朗日对偶性的特例。 7. **案例研究**：本周介绍两个通过应用所介绍的数学性质构建的知名模型。首先，将简单线性回归问题公式化为非线性程序，并推导出封闭式回归公式。其次，从对偶性角度介绍支持向量机，这是一种最著名的分类模型。 8. **课程总结与未来学习方向**：在最后一周，回顾我们所介绍的主题，给出总结性言论，并提出进一步学习的方向，以便于进阶研究。 此课程旨在提升学生对运筹学理论及其在实际中的应用能力，为未来的研究奠定基础。
Name:Course Overview
Description:In the first lecture, after introducing the course and the importance of mathematical properties, we study the matrix way to run the simplex method. Being more familiar with matrices will help us understand further lectures.
Name:Duality
Description:In this week, we study the theory and applications of linear programming duality. We introduce the properties possessed by primal-dual pairs, including weak duality, strong duality, complementary slackness, and how to construct a dual optimal solution given a primal optimal one. We also introduce one important application of linear programming duality: Using shadow prices to determine the most critical constraint in a linear program.
Name:Sensitivity Analysis and Dual Simplex Method
Description:In the past two weeks, we study the simplex method and the duality. On top of them, the dual simplex method is discussed in this lecture. We apply it to one important issue in sensitivity analysis: evaluating a linear programming model with a new constraint. A linear programming model with a new variable is also discussed.
Name:Network Flow
Description:In this lecture, we introduce network flow models, which are widely used for making decision regarding transportation, logistics, inventory, project management, etc. We first introduce the minimum cost network flow (MCNF) model and show hot it is the generalization of many famous models, including assignment, transportation, transshipment, maximum flow, and shortest path. We also prove a very special property of MCNF, total unimodularity, and how it connects linear programming and integer programming.
Name:Convex Analysis
Description:As the last lesson of this course, we introduce a case of NEC Taiwan, which provides IT and network solutions including cloud computing, AI, IoT etc. Since maintaining all its service hubs is too costly, they plan to rearrange the locations of the hubs and reallocate the number of employees in each hub. An algorithm is included to solve the facility location problem faced by NEC Taiwan.
Name:Lagrangian Duality and the KKT condition
Description:In this week, we study nonlinear programs with constraints. We introduce two major tools, Lagrangian relaxation and the KKT condition, for solving constrained nonlinear programs. We also see how linear programming duality is a special case of Lagrangian duality.
Name:Case Study
Description:In this week, we introduce two well-known models constructed by applying the mathematical properties we have introduced. First, we formulate a simple linear regression problem as a nonlinear program and derive the closed-form regression formula. Second, we introduce support-vector machine, one of the most famous classification model, from the perspective of duality.
Name:Course Summary and Future Learning Directions
Description:In the final week, we review the topics we have introduced and give some concluding remarks. We also provide some learning directions for advanced studies.
Operations Research (OR) is a field in which people use mathematical and engineering methods to study optimization problems in Business and Management, Economics, Computer Science, Civil Engineering, Electrical Engineering, etc. The series of courses consists of three parts, we focus on deterministic optimization techniques, which is a major part of the field of OR. As the third part of the series, we study mathematical properties of linear programs, integer programs, and nonlinear programs. We also introduce applications of these theoretical properties: How they help us develop better ways to solve mathematical programs.
