Operations Research (3): Theory

Go to Course: https://www.coursera.org/learn/operations-research-theory

Introduction

### Course Review: Operations Research (3): Theory on Coursera #### Overview of the Course Operations Research (OR) is an essential field that combines mathematical and engineering methods to tackle optimization problems across various disciplines, including business, management, economics, computer science, and engineering. The "Operations Research (3): Theory" course is the third installment in a comprehensive series, focusing specifically on deterministic optimization techniques. This course delves into the mathematical properties of linear, integer, and nonlinear programs, making it an invaluable resource for both new and seasoned professionals looking to enhance their decision-making and problem-solving skills. #### Course Structure and Syllabus The course is structured to facilitate a gradual understanding of complex topics, starting from foundational concepts and advancing to intricate models. Here’s a detailed breakdown of the syllabus: 1. **Course Overview**: The initial lecture introduces the fundamental importance of mathematical properties, setting the stage for the matrix approach to the simplex method. Gaining familiarity with matrices is crucial for comprehending further lectures. 2. **Duality**: This week dives into the theory and applications of linear programming duality. Key concepts include weak duality, strong duality, and complementary slackness, alongside practical applications like using shadow prices to identify critical constraints in a linear program. 3. **Sensitivity Analysis and Dual Simplex Method**: Building on the simplex method and duality, this section explores the dual simplex method and how it applies to sensitivity analysis, particularly evaluating the impact of new constraints and variables in linear programming models. 4. **Network Flow**: Focused on practical applications, this lecture introduces network flow models used in transportation, logistics, and project management. The minimum cost network flow (MCNF) model serves as a generalization for various well-known models, illustrating the connection between linear and integer programming. 5. **Convex Analysis**: Using a real-world case study from NEC Taiwan, participants learn about facility location problems—a common challenge faced by businesses. The course presents an algorithm to optimally reallocate resources and costs associated with service hubs. 6. **Lagrangian Duality and KKT Conditions**: This critical week covers nonlinear programs with constraints, introducing essential tools like Lagrangian relaxation and the Karush-Kuhn-Tucker (KKT) conditions. Participants see how linear programming duality is encapsulated within Lagrangian duality concepts. 7. **Case Study**: Here, students apply the mathematical principles learned to tangible problems. They tackle a linear regression problem framed as a nonlinear program, deriving the closed-form regression formula, and explore support vector machines from a duality perspective. 8. **Course Summary and Future Learning Directions**: The final week wraps up the course with a comprehensive review of the topics discussed and suggests avenues for further study, ensuring students are prepared for advanced learning opportunities in Operations Research. #### Course Experience and Recommendations This course does an exceptional job of blending theory with practical application. The use of real-world examples, especially the case study about NEC Taiwan, brings relevance and context to the mathematical concepts discussed. Additionally, the progression from basic principles to complex applications ensures that learners of varying expertise can follow along without feeling overwhelmed. The course's interactive elements—exercises, quizzes, and discussion forums—further enrich the learning experience. Participants are also encouraged to collaborate, which enhances understanding through peer interaction and knowledge sharing. ### Recommendation I highly recommend the "Operations Research (3): Theory" course on Coursera for anyone interested in developing a solid foundation in optimization. It’s well-suited for students pursuing degrees in engineering, mathematics, economics, and business, as well as professionals seeking to enhance their analytical skills in a practical context. By completing this course, you will not only master critical concepts in deterministic optimization but also gain valuable insights into how these techniques can be applied to solve significant real-world challenges. Whether you’re looking to improve your career prospects or simply expand your intellectual horizons, this course is a worthwhile investment in your education.

Syllabus

Course Overview

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.

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.

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.

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.

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.

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.

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.

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.