Operations Research (3): Theory

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孔令傑 (Ling-Chieh Kung)

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Operations Research (3): Theory

National Taiwan University

About this Course

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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.

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Shareable Certificate

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Advanced Level

This course is appropriate for students with experience Calculus, Linear Algebra, and Probability.

Approx. 14 hours to complete

English

Subtitles: English

What you will 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 will gain

TheoryBusiness AnalyticsMathematical Optimization

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Advanced Level

This course is appropriate for students with experience Calculus, Linear Algebra, and Probability.

Approx. 14 hours to complete

English

Subtitles: English

Instructor

孔令傑 (Ling-Chieh Kung)

副教授 (Associate Professor)

資訊管理學系 (Department of Information Management)

34,881 Learners

8 Courses

Offered by

National Taiwan University

We firmly believe that open access to learning is a powerful socioeconomic equalizer. NTU is especially delighted to join other world-class universities on Coursera and to offer quality university courses to the Chinese-speaking population. We hope to transform the rich rewards of learning from a limited commodity to an experience available to all.

Syllabus - What you will learn from this course

Week

Week 1

1 hour to complete

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.

1 hour to complete

5 videos (Total 65 min), 1 reading, 1 quiz

Week

Week 2

2 hours to complete

Duality

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.

2 hours to complete

14 videos (Total 120 min)

14 videos

2-0: Opening.4m

2-1: Introduction.1m

2-2: Primal-dual pairs – The first example.8m

2-3: Primal-dual pairs – More examples.7m

2-4: Primal-dual pairs – General rule.12m

2-5: Weak duality and sufficiency of optimality.7m

2-6: Dual optimal solution and strong duality.11m

2-7: An example for the theorems.5m

2-8: Complementary slackness.12m

2-9: Motivating examples for shadow prices.9m

2-10: Shadow prices.9m

2-11: Shadow prices and duality.12m

2-12: Computers – Gurobi and Python for shadow prices.13m

2-13: Closing remarks.4m

1 practice exercise

Quiz for Week 220m

Week

Week 3

1 hour to complete

Sensitivity Analysis and Dual Simplex Method

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.

1 hour to complete

8 videos (Total 52 min)

Week

Week 4

2 hours to complete

Network Flow

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.

2 hours to complete

10 videos (Total 80 min)

Week

Week 5

2 hours to complete

Convex Analysis

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.

2 hours to complete

13 videos (Total 125 min)

13 videos

5-0: Opening.4m

5-1: Motivating examples.7m

5-2: Convex sets and functions.12m

5-3: Global optimality and extreme point.11m

5-4: Convex programming.13m

5-5: Convexity of twice differentiable functions.6m

5-6: Example – EOQ12m

5-7: Second-order derivatives.8m

5-8: Positive semi-definiteness.13m

5-9: Analytically solving multi-variate NLPs.6m

5-10: Example – Two-product pricing.10m

5-11: Computers – Implementation of gradient descent.13m

5-12: Closing remarks.3m

1 practice exercise

Quiz for Week 520m

Week

Week 6

2 hours to complete

Lagrangian Duality and the KKT condition

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.

2 hours to complete

15 videos (Total 120 min)

15 videos

6-0: Opening.5m

6-1: Motivation.8m

6-2: Lagrange relaxation.7m

6-3: An example of Lagrange relaxation.4m

6-4: Weak duality of Lagrange relaxation.6m

6-5: The KKT condition.9m

6-6: Visualizing the KKT condition.11m

6-7: Example 1 of applying the KKT condition.10m

6-8: Example 2 of applying the KKT condition.14m

6-9: The KKT condition in general.4m

6-10: More about Lagrange duality.3m

6-11: Convexity and strong duality of Lagrange relaxation.9m

6-12: An example of Lagrange duality.9m

6-13: Lagrange duality vs. LP duality.10m

6-14: Closing remarks.4m

1 practice exercise

Quiz for Week 620m

Week

Week 7

2 hours to complete

Case Study

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.

2 hours to complete

14 videos (Total 90 min)

14 videos

7-0: Opening.6m

7-1: Introduction.3m

7-2: Simple linear regression.6m

7-3: Solving the simple linear regression problem.8m

7-4: Remarks and other regression models.8m

7-5: Support vector machine.8m

7-6: Formulating the SVM model.9m

7-7: Simplifying the objective function.4m

7-8: SVM for imperfect separation.6m

7-9: Dualization for the SVM problem (1).7m

7-10: Dualization for the SVM problem (2).5m

7-11: Convexity of the dual program.5m

7-12: Final remarks.3m

7-13: Closing remarks.5m

1 practice exercise

Quiz for Week 720m

Week

Week 8

1 hour to complete

Course Summary and Future Learning Directions

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.

1 hour to complete

3 videos (Total 29 min)

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Operations Research (3): Theory

Operations Research (3): Theory