Mathematical Optimization for Engineers from edX

Today, for almost every product on the market and almost every service offered, some form of optimization has played a role in their design.

However, optimization is not a button-press technology. To apply it successfully, one needs expertise in formulating the problem, selecting and tuning the solution algorithm and finally, checking the results. We have designed this course to make you such an expert.

This course is useful to students of all engineering fields. The mathematical and computational concepts that you will learn here have application in machine learning, operations research, signal and image processing, control, robotics and design to name a few.

We will start with the standard unconstrained problems, linear problems and general nonlinear constrained problems. We will then move to more specialized topics including mixed-integer problems; global optimization for non-convex problems; optimal control problems; machine learning for optimization and optimization under uncertainty. Students will learn to implement and solve optimization problems in Python through the practical exercises.

What you'll learn

Mathematical definitions of objective function, degrees of freedom, constraints and optimal solution
Mathematical as well as intuitive understanding of optimality conditions
Different optimization formulations (unconstrained v/s constrained; linear v/s nonlinear; mixed-integer v/s continuous; time-continuous or dynamic; optimization under uncertainty)
Fundamentals of the solution methods for each these formulations
Optimization with machine learning embedded
Hands-on training in implementing and solving optimization problems in Python, as exercises

What's inside

Syllabus

Week 1: Introduction and math review

Mathematical definitions of objective function, degrees of freedom, constraints and optimal solution with real-world examples

Review of some mathematical basics needed to take us through the course

Week 2: Unconstrained optimization

Basics of iterative descent: step direction and step length

Common algorithms like steepest descent, Newton’s method and its variants and trust-region methods.

Week 3: Linear optimization

KKT conditions of optimality for constrained problems

Simplex method

Interior point methods

Week 4: Nonlinear optimization

Penalty, log-barrier and SQP methods

Mixed-integer optimization

Branch and bound method for mixed-integer linear problems

Week 5: Global optimization

Branch and bound method for nonlinear non-convex problems

Constructing relaxations

Different formulations and their numerical performance

Stochastic methods, genetic algorithm and derivative free methods

Week 6: Dynamic optimization

Full discretization, single-shooting and multi-shooting methods

Nonlinear model predictive control

Week 7: Machine learning for optimization

Mechanistic, data-driven and hybrid modelling

Basics of training machine learning models

Optimization with machine learning embedded

Week 8: Optimization under uncertainty

Parametric optimization

Two stage stochastic problems

Robust optimization via semi-infinite problems

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Teaches standard mathematical concepts and their intuitive understanding for solving optimization problems, preparing you for real-world applications

Covers a wide range of formulations, including unconstrained, constrained, linear, nonlinear, mixed-integer, and optimization under uncertainty, giving you a comprehensive understanding of optimization techniques

Provides hands-on training through exercises in Python, allowing you to apply your knowledge and solve optimization problems practically

Involves renowned instructors with expertise in optimization, ensuring the quality and relevance of the course content

Builds a strong foundation in optimization principles and methods, making it suitable for students from various engineering fields

Emphasizes the importance of optimization in designing products and services, highlighting its relevance to industry practices

Reviews summary

Comprehensive mathematical optimization

Learners say this course offers a thorough introduction to the wide-ranging topic of Mathematical Optimization. Its clear lectures and labs deliver foundational concepts for beginners who are interested in digging deeper into the subject.

Provides ample references for further study.

"The literature references provided are important for one who wishes to further dwelve into Mathematical Optimization."

"The course offers various resources, such as references and links, which allow students to explore the topics further."

"There are supplementary materials provided that point to further readings if someone wants to delve deeper."

Clear and in-depth explanations of course material.

"The lectures and labs deliver fundamental concepts in a clear and unambiguous fashion."

"The explanations are very clear and the examples are helpful."

"The course materials make difficult topics easy to understand."

Introduces a broad range of topics in Mathematical Optimization.

"This course covers Mathematical Optimization in its entirety."

"The lectures, readings, and labs touch upon all the important and interesting Mathematical Optimization subjects."

"It offers a general overview of Mathematical Optimization, which is a broad field in itself."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Mathematical Optimization for Engineers with these activities:

Brush Up on Calculus

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Review the fundamentals of calculus to strengthen your foundation for optimization.

Browse courses on Calculus

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Review your notes or a textbook on basic calculus concepts.
Solve practice problems to refresh your skills.
Optional: Seek help from a tutor or online resources if needed.

Crash Course in Dynamic Optimization

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Expand your knowledge of dynamic optimization through guided tutorials, preparing you for the upcoming course content.

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Enroll in an online tutorial or course on dynamic optimization.
Follow the tutorials and complete the associated exercises.
Experiment with different dynamic optimization techniques.

Deep Dive into Nonlinear Optimization

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Build a solid foundation in nonlinear optimization to enhance your understanding of the course materials.

View Practical Optimization on Amazon

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Read the book's introduction and first chapter.
Work through the solved examples in the book.
Attempt the practice problems at the end of each chapter.

Five other activities

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Linear Programming Practice

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Improve your understanding of Linear Programming and prepare for the upcoming course materials.

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Review the concept of linear programming.
Solve a set of practice problems on linear programming.
Compare your solutions with the provided answer key.

Python Script for Optimization

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Apply your Python skills to implement and solve optimization problems, reinforcing your understanding of the course content.

Browse courses on Optimization

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Choose an optimization problem from the course materials.
Develop a Python script to solve the problem.
Test and refine your script to ensure accurate results.
Write a brief report summarizing your approach and findings.

Attend an Optimization Workshop

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Immerse yourself in the world of optimization by attending a workshop led by industry experts.

Browse courses on Optimization

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Find an optimization workshop that aligns with your interests.
Register for the workshop and actively participate in the sessions.
Network with other attendees and experts in the field.

Group Discussion on Optimization Techniques

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Engage with your peers to share insights, clarify concepts, and enhance your comprehension of optimization techniques.

Browse courses on Optimization

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Join a study group or online forum dedicated to the course.
Discuss different optimization techniques and their applications.
Work together to solve optimization problems.
Reflect on the discussion and identify areas for improvement.

Infographic on Optimization Applications

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Create a visual representation to demonstrate the practical applications of optimization, solidifying your understanding of its real-world relevance.

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Research various applications of optimization in different industries.
Design an infographic that illustrates these applications.
Share your infographic with others to educate them about the impact of optimization.

Career center

Learners who complete Mathematical Optimization for Engineers will develop knowledge and skills that may be useful to these careers:

Operations Research Analyst

Operations research analysts apply analytical methods to help organizations improve their efficiency and make better decisions. They identify and solve problems, such as how to optimize production schedules, allocate resources, and design efficient distribution networks. This course provides a strong foundation for operations research analysts by teaching them the mathematical principles and algorithms needed to solve optimization problems. The course covers topics such as linear programming, nonlinear programming, and mixed-integer programming, which are all essential techniques for operations research analysts.

See salaries and explore the career path for Operations Research Analyst

Industrial Engineer

Industrial engineers design, improve, and install integrated systems of people, materials, information, equipment, and energy. They use optimization techniques to improve the efficiency of production processes, logistics systems, and other industrial operations. This course provides industrial engineers with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as simulation and data analysis, which are essential for industrial engineers.

See salaries and explore the career path for Industrial Engineer

Data Scientist

Data scientists use scientific methods, processes, algorithms, and systems to extract knowledge and insights from data in various forms, both structured and unstructured. They use optimization techniques to develop machine learning models, which can be used to solve a wide variety of problems, such as predicting customer churn, recommending products, and detecting fraud. This course provides data scientists with the mathematical foundation they need to understand and apply optimization techniques to machine learning problems. The course covers topics such as linear programming, nonlinear programming, and mixed-integer programming, as well as machine learning algorithms and data analysis techniques.

See salaries and explore the career path for Data Scientist

Management Consultant

Management consultants help organizations improve their performance by identifying and solving problems. They use optimization techniques to develop solutions to problems such as how to improve customer satisfaction, increase sales, and reduce costs. This course provides management consultants with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as decision analysis and risk management, which are essential for management consultants.

See salaries and explore the career path for Management Consultant

Financial Analyst

Financial analysts provide advice to individuals and organizations on investment decisions. They use optimization techniques to develop investment strategies and to value assets. This course provides financial analysts with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as portfolio theory and risk management, which are essential for financial analysts.

See salaries and explore the career path for Financial Analyst

Software Engineer

Software engineers design, develop, and maintain software systems. They use optimization techniques to develop efficient algorithms and to optimize the performance of software systems. This course provides software engineers with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as data structures and algorithms, which are essential for software engineers.

See salaries and explore the career path for Software Engineer

Statistician

Statisticians collect, analyze, interpret, and present data. They use optimization techniques to develop statistical models and to estimate parameters. This course provides statisticians with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as probability theory and statistical inference, which are essential for statisticians.

See salaries and explore the career path for Statistician

Actuary

Actuaries use mathematical and statistical methods to assess risk and uncertainty. They use optimization techniques to develop insurance products and to calculate insurance premiums. This course provides actuaries with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as probability theory and risk management, which are essential for actuaries.

See salaries and explore the career path for Actuary

Quantitative Analyst

Quantitative analysts use mathematical and statistical methods to analyze financial data. They use optimization techniques to develop trading strategies and to manage risk. This course provides quantitative analysts with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as financial econometrics and risk management, which are essential for quantitative analysts.

See salaries and explore the career path for Quantitative Analyst

Investment Analyst

Investment analysts provide advice to individuals and organizations on investment decisions. They use optimization techniques to develop investment strategies and to value assets. This course provides investment analysts with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as portfolio theory and risk management, which are essential for investment analysts.

See salaries and explore the career path for Investment Analyst

Risk Manager

Risk managers identify, assess, and manage risks. They use optimization techniques to develop risk management strategies and to quantify risk. This course provides risk managers with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as probability theory and risk management, which are essential for risk managers.

See salaries and explore the career path for Risk Manager

Business Analyst

Business analysts use data analysis and optimization techniques to improve the efficiency and effectiveness of businesses. They use optimization techniques to develop business strategies and to make decisions. This course provides business analysts with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as data analysis and decision analysis, which are essential for business analysts.

See salaries and explore the career path for Business Analyst

Project Manager

Project managers plan, organize, and execute projects. They use optimization techniques to develop project plans and to allocate resources. This course provides project managers with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as project management and risk management, which are essential for project managers.

See salaries and explore the career path for Project Manager

Supply Chain Manager

Supply chain managers plan, organize, and execute the flow of goods and services from suppliers to customers. They use optimization techniques to develop supply chain strategies and to manage inventory. This course provides supply chain managers with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as supply chain management and logistics, which are essential for supply chain managers.

See salaries and explore the career path for Supply Chain Manager

Transportation Analyst

Transportation analysts plan, organize, and execute the movement of people and goods. They use optimization techniques to develop transportation plans and to manage traffic. This course provides transportation analysts with the mathematical tools they need to solve optimization problems, including linear programming, nonlinear programming, and mixed-integer programming. The course also covers topics such as transportation planning and traffic engineering, which are essential for transportation analysts.

See salaries and explore the career path for Transportation Analyst