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Operations Research (3): Theory

Operations Research (3): Theory

This course is part of Operations Research Specialization

Instructor: 孔令傑 (Ling-Chieh Kung)

12,160 already enrolled

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8 modules

Gain insight into a topic and learn the fundamentals.

97 reviews

Advanced level

Recommended experience

1 week to complete

at 10 hours a week

Flexible schedule

Learn at your own pace

8 modules

Gain insight into a topic and learn the fundamentals.

97 reviews

Advanced level

Recommended experience

1 week to complete

at 10 hours a week

Flexible schedule

Learn at your own pace

What you'll learn

Understand the theoretical properties of linear programs, integer programs, and nonlinear programs.
Apply the mathematical properties to reduce the complexity of real-world problems or to solve it.

Skills you'll gain

Tools you'll learn

Classification Algorithms

Details to know

Shareable certificate

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Assessments

8 assignments

Taught in English

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This course is part of the Operations Research Specialization

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There are 8 modules in this course

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.

Module details

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.

What's included

5 videos1 reading1 assignment

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.

What's included

14 videos1 assignment

14 videosTotal 120 minutes

2-0: Opening.5 minutes
2-1: Introduction.2 minutes
2-2: Primal-dual pairs – The first example.9 minutes
2-3: Primal-dual pairs – More examples.8 minutes
2-4: Primal-dual pairs – General rule.12 minutes
2-5: Weak duality and sufficiency of optimality.8 minutes
2-6: Dual optimal solution and strong duality.11 minutes
2-7: An example for the theorems.5 minutes
2-8: Complementary slackness.13 minutes
2-9: Motivating examples for shadow prices.9 minutes
2-10: Shadow prices.9 minutes
2-11: Shadow prices and duality.12 minutes
2-12: Computers – Gurobi and Python for shadow prices.14 minutes
2-13: Closing remarks.4 minutes

1 assignmentTotal 20 minutes

Quiz for Week 220 minutes

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.

What's included

8 videos1 assignment

8 videosTotal 52 minutes

3-0: Opening.5 minutes
3-1: Introduction.1 minute
3-2: New variable – Motivation.8 minutes
3-3: New variable – Solution.9 minutes
3-4: New constraint – Motivation.8 minutes
3-5: Dual simplex – Idea.6 minutes
3-6: Dual simplex – Example and remark.12 minutes
3-7: Closing remarks.3 minutes

1 assignmentTotal 20 minutes

Quiz for Week 320 minutes

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.

What's included

11 videos1 assignment

11 videosTotal 89 minutes

4-0: Opening.5 minutes
4-1: Introduction.5 minutes
4-2: MCNF problems.12 minutes
4-3 LP formulation for MCNF9 minutes
4-4: Total unimodularity.13 minutes
4-5: MCNF and total unimodularity.7 minutes
4-6: Transportation problems.7 minutes
4-7: Assignment and transshipment problems.7 minutes
4-8: Shortest path and maximum flow problems.12 minutes
4-9: Computers – Gurobi and Python for network flow.8 minutes
4-10: Closing remarks.3 minutes

1 assignmentTotal 20 minutes

Quiz for Week 420 minutes

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.

What's included

13 videos1 assignment

13 videosTotal 125 minutes

5-0: Opening.5 minutes
5-1: Motivating examples.7 minutes
5-2: Convex sets and functions.13 minutes
5-3: Global optimality and extreme point.11 minutes
5-4: Convex programming.13 minutes
5-5: Convexity of twice differentiable functions.7 minutes
5-6: Example – EOQ12 minutes
5-7: Second-order derivatives.8 minutes
5-8: Positive semi-definiteness.14 minutes
5-9: Analytically solving multi-variate NLPs.6 minutes
5-10: Example – Two-product pricing.11 minutes
5-11: Computers – Implementation of gradient descent.13 minutes
5-12: Closing remarks.4 minutes

1 assignmentTotal 20 minutes

Quiz for Week 520 minutes

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.

What's included

15 videos1 assignment

15 videosTotal 120 minutes

6-0: Opening.5 minutes
6-1: Motivation.8 minutes
6-2: Lagrange relaxation.8 minutes
6-3: An example of Lagrange relaxation.4 minutes
6-4: Weak duality of Lagrange relaxation.6 minutes
6-5: The KKT condition.10 minutes
6-6: Visualizing the KKT condition.12 minutes
6-7: Example 1 of applying the KKT condition.11 minutes
6-8: Example 2 of applying the KKT condition.14 minutes
6-9: The KKT condition in general.5 minutes
6-10: More about Lagrange duality.3 minutes
6-11: Convexity and strong duality of Lagrange relaxation.9 minutes
6-12: An example of Lagrange duality.9 minutes
6-13: Lagrange duality vs. LP duality.11 minutes
6-14: Closing remarks.5 minutes

1 assignmentTotal 20 minutes

Quiz for Week 620 minutes

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.

What's included

14 videos1 assignment

14 videosTotal 90 minutes

7-0: Opening.6 minutes
7-1: Introduction.3 minutes
7-2: Simple linear regression.6 minutes
7-3: Solving the simple linear regression problem.8 minutes
7-4: Remarks and other regression models.9 minutes
7-5: Support vector machine.8 minutes
7-6: Formulating the SVM model.10 minutes
7-7: Simplifying the objective function.5 minutes
7-8: SVM for imperfect separation.6 minutes
7-9: Dualization for the SVM problem (1).7 minutes
7-10: Dualization for the SVM problem (2).6 minutes
7-11: Convexity of the dual program.5 minutes
7-12: Final remarks.4 minutes
7-13: Closing remarks.5 minutes

1 assignmentTotal 20 minutes

Quiz for Week 720 minutes

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.