Discrete Optimization

Go to Course: https://www.coursera.org/learn/discrete-optimization

Introduction

### Course Review: Discrete Optimization on Coursera If you've ever found joy (or frustration) in solving Sudokus, then the "Discrete Optimization" course on Coursera might just be your next favorite challenge. This course, cleverly designed for people interested in the mathematical modeling and algorithmic techniques that underpin complex search problems, is not just for math enthusiasts—it's a crucial tool for anyone from data scientists to operations managers. In this review, we'll explore the details of the course content and share a recommendation for potential learners. #### Course Overview The Discrete Optimization course covers a wide array of topics related to optimization techniques, particularly focusing on discrete problems. It enlightens participants on how modern optimization is applied in various real-world scenarios, such as scheduling aircraft, managing production lines, and coordinating transportation logistics. #### Course Structure and Content **1. Welcome** The course kicks off with a comprehensive introduction, providing learners with essential insights into the course philosophy and structure. It sets clear expectations about the assignment-based grading and offers tips on how to navigate the course successfully. **2. Knapsack Problem** Diving straight into the heart of optimization, the course utilizes the knapsack problem as a foundational case study. It breaks down how to formalize optimization problems and introduces critical techniques such as dynamic programming and branch and bound. **3. Constraint Programming** One of the standout features of this course is the in-depth exploration of constraint programming. By teaching students to express optimization problems at a granular level, it cultivates an understanding of how constraints can streamline complex searches. **4. Local Search** This segment sheds light on the intuitive local search technique. By guiding students through concepts like neighborhood and connectivity, it prepares them for more advanced techniques such as tabu search and simulated annealing, which are essential for large and complex optimization scenarios. **5. Linear Programming** The course delves into linear programming—an indispensable tool in the optimization toolkit. The lectures cover fundamental principles, including the simplex algorithm and duality, instilling a robust understanding of this technique's importance. **6. Mixed Integer Programming** Transitioning to mixed-integer programming, learners discover how introducing integer variables complicates problems while expanding the scope of applications. The course equips participants with necessary modeling techniques and advanced solution strategies. **7. Advanced Topics (Part I & II)** In the advanced sections, students are introduced to cutting-edge concepts such as large neighborhood search and column generation. These advanced techniques provide practical applications and deepen the learner's grasp on real-world optimization challenges. #### Recommendations The "Discrete Optimization" course on Coursera is highly recommended for anyone seeking to enhance their analytical and problem-solving skills. Here are some reasons why: - **Practical Applications**: The course is steeped in real-world applications, making it instantly relatable for professionals in fields ranging from logistics to computer science. - **Comprehensive Curriculum**: It covers both foundational concepts and advanced methodologies, making it suitable for beginners and seasoned practitioners alike. - **Flexible Learning**: The online format allows learners to progress at their own pace, accommodating various schedules and learning styles. - **Expert Instruction**: Given that courses on Coursera are developed in collaboration with top universities and professionals, you can trust the quality of the education you're receiving. #### Conclusion In a world increasingly driven by data and efficiency, the "Discrete Optimization" course on Coursera stands out as a vital resource for those eager to deepen their understanding of optimization techniques. Whether you're solving complex logistical problems or looking to boost your computational skills, this course prepares you to tackle modern challenges with confidence. Don't miss the opportunity to take your problem-solving skills to the next level—sign up today and start your journey into the enlightening world of discrete optimization!

Syllabus

Welcome

These lectures and readings give you an introduction to this course: its philosophy, organization, and load. They also tell you how the assignments are a significant part of the class. This week covers the common input/output organization of the assignments, how they are graded, and how to succeed in this class.

These lectures introduce optimization problems and some optimization techniques through the knapsack problem, one of the most well-known problem in the field. It discusses how to formalize and model optimization problems using knapsack as an example. It then reviews how to apply dynamic programming and branch and bound to the knapsack problem, providing intuition behind these two fundamental optimization techniques. The concept of relaxation and search are also discussed.

Constraint programming is an optimization technique that emerged from the field of artificial intelligence. It is characterized by two key ideas: To express the optimization problem at a high level to reveal its structure and to use constraints to reduce the search space by removing, from the variable domains, values that cannot appear in solutions. These lectures cover constraint programming in detail, describing the language of constraint programming, its underlying computational paradigm and how it can be applied in practice.

Local search is probably the oldest and most intuitive optimization technique. It consists in starting from a solution and improving it by performing (typically) local perturbations (often called moves). Local search has evolved substantially in the last decades with a lot of attention being devoted on which moves to explore. These lectures explore the theory and practice of local search, from the concept of neighborhood and connectivity to meta-heuristics such as tabu search and simulated annealing.

Linear programming has been, and remains, a workhorse of optimization. It consists in optimizing a linear objective subject to linear constraints, admits efficient algorithmic solutions, and is often an important building block for other optimization techniques. These lectures review fundamental concepts in linear programming, including the infamous simplex algorithm, simplex tableau, and duality. .

Mixed Integer Programming generalizes linear programming by allowing integer variables, which dramatically changes the complexity of the problems but also broadens the potential applications significantly. These lectures review how to model problems in mixed-integer programming and how to solve mixed-integer programs using branch and bound. Advanced techniques such as cutting planes and polyhedral cuts are also covered.

These lectures cover some more advanced concepts in optimization. They introduce constraint-programming techniques for scheduling and routing.

These lectures continues to cover some more advanced concepts in optimization. They introduce large neighborhood search, which often combines constraint programming and local search, and column generation which decomposes an optimization model into a master and pricing problem, using more complex variables.