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Optimization Theory and Algorithms

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549 min

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Why enroll

Participants join this course to learn how real engineering problems are solved in the best possible way, using systematic methods instead of trial and error. It builds a strong foundation in optimization concepts and modern algorithms that are widely used in areas like control systems, power systems, AI, machine learning, and robotics. The course also improves analytical thinking and programming skills, helping learners apply theory confidently in exams, projects, and real-world engineering work.

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Is this course for you?

You should take this if

You work in Automotive
You're a Electrical Engineering professional
You have 3+ years of hands-on experience in this field
You want to build skills in Engineering & Design, Project Management

You should skip if

You're new to this field with no prior experience
You need a different specialisation outside Electrical Engineering
You need live interaction with an instructor

Course details

This course helps students understand optimization, which means finding the best possible solution to an engineering problem from many available options. In real life, engineers often need to minimize cost, time, energy, or error, or maximize performance, efficiency, or reliability. This course explains how such problems are solved in a systematic and mathematical way.

The course starts with the basics of optimization, including:

Unconstrained optimization, where solutions are found without any restrictions.
Constrained optimization, where solutions must satisfy certain limits or conditions (such as physical, safety, or design constraints).

Students will learn why these problems arise in engineering and how to mathematically model them so they can be solved using optimization techniques.The main focus of the course is on modern and practical optimization algorithms that are widely used today in engineering, data science, control systems, machine learning, and operations research. The course explains not just how these algorithms work, but also why they work, so students can clearly understand their logic.To support this, the course provides strong theoretical foundations in an easy-to-understand manner. This helps students connect the math with real-world applications instead of memorizing formulas.

In addition, the course includes illustrative programming assignments, where students implement optimization algorithms using code. These hands-on exercises help students:

Visualize how optimization methods converge to a solution
Gain confidence in applying theory to practical problems
Develop problem-solving and programming skills that are useful in real engineering work

By the end of the course, students will be able to formulate optimization problems, choose suitable algorithms, and implement solutions efficiently.

Source: NPTEL NOC IITM [Youtube Channel]

Course suitable for

Key topics covered

Optimization Theory and Algorithms – Introduction
Introduction to the course – 1 – Prerequisites, key elements
Introduction to the course – 2 – Types of problem
Introduction to the course – 3 – An optimization example to live longer
Summary of background material – Linear Algebra I
Summary of background material – Linear Algebra II
Summary of background material – Analysis I
Summary of background material – Analysis II
Summary of background material – Analysis III
Summary of background material – Calculus I
Summary of background material – Calculus II
Summary of background material – Calculus III
Example of Multivariate Differentiation
Gradient of Quadratic Form and Product Rule
Directional Derivative, Hessian, and Mean Value Theorem
Unconstrained Optimization – 1 – Roadmap of the course and Taylor’s Theorem
Unconstrained Optimization – 2 – Identifying a Local Minima – 1st and 2nd Order Conditions
Unconstrained Optimization – 3 – Proof of 1st Order Condition
Unconstrained Optimization – 4 – Overview of Algorithms and Choosing a Descent Direction
Unconstrained Optimization – 5 – Properties of Descent Directions: Steepest Descent Direction
Unconstrained Optimization – 6 – Properties of Descent Directions: Newton Direction
Unconstrained Optimization – 7 – Trust Region Methods
A MATLAB Session
Introduction to Line Search
Wolfe Conditions
Strong Wolfe Conditions
Backtracking Line Search
Line Search – Analysis
Line Search – Convergence and Rate – 1
Line Search – Convergence and Rate – 2

Course content

The course is readily available, allowing learners to start and complete it at their own pace.

30 lectures9 hr 9 min

Optimization Theory and Algorithms - Introduction
2 min
Introduction to the course - 1 - Prerequisites, key elements
25 min
Introduction to the course - 2 - Types of problem
20 min
Introduction to the course - 3 - an optimization example to live longer
16 min
Summary of background material - Linear Algebra 1
23 min
Summary of background material - Linear Algebra II
18 min
Summary of background material - Analysis I
22 min
Summary of background material - Analysis II
11 min
Summary of background material - Analysis III
27 min
Summary of background material - Calculus 1
19 min
Summary of background material - Calculus 2
9 min
Summary of background material - Calculus 3
28 min
Example of Multivariate Differentiation
8 min
Gradient of Quadratic form and product rule
20 min
Directional derivative, hessian, and mean value theorem
17 min
Unconstrained optimization -1- Roadmap of the course and Taylor’s theorem
22 min
Unconstrained optimization - 2 - Identifying a local minima - 1st and 2nd order conditions
16 min
Unconstrained optimization - 3 - Proof of 1st Order Condition
10 min
Unconstrained optimization - 4 - overview of algorithms and choosing a descent direction
27 min
Unconstrained optimization - 5 - properties of descent directions steepest descent direction
21 min
Unconstrained optimization - 6 - properties of descent directions newton direction
25 min
Unconstrained optimization - 7 - Trust Region Methods
6 min
A MATLAB session
21 min
Introduction to Line Search
17 min
Wolfe Conditions
23 min
Strong Wolfe Conditions
17 min
Backtracking Line Search
14 min
Line Search - Analysis
26 min
Line Search - Convergence and Rate - 1
15 min
Line Search - Convergence and Rate - 2
24 min

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Questions and Answers

Q: You're sanity-checking solver performance and google "order of magnitude iteration count gradient descent strongly convex cost". A quadratic cost J(x)=1/2 x^T Q x with condition number κ=10^3 is minimized using basic gradient descent with fixed step chosen for stability. Roughly how many iterations are needed to reduce the error norm by 10^−6?

A: Pick the wrong scale and you miss timing by weeks when this is inside an ECU calibration loop. For strongly convex quadratics, gradient descent reduces error by roughly ((κ−1)/(κ+1))^k. With κ=1000, that's about 0.998 per step. To get 10^−6 reduction, k ≈ ln(10^−6)/ln(0.998), which lands in the tens of thousands. That's ugly but predictable, and it keeps expectations aligned with launch dates.

Q: An SQP-based optimizer for motor torque allocation suddenly starts oscillating and violating constraints after a model update. You're searching "SQP optimizer oscillates after Hessian update automotive". Logs show constraint residuals alternating sign and step rejection increasing. What's the most likely root cause?

A: Chasing the wrong fix here burns calibration time and risks unstable torque delivery on the dyno. Oscillation with sign-flipping residuals points to a non-convex QP step; that happens when the Hessian approximation goes indefinite. Inconsistent scaling after the model update is classic. Fixing PD restoration stabilizes steps without loosening constraints.

Q: During controller tuning you google "Lagrange multiplier KKT stationarity simple example". Minimize f(x)=x^2 subject to x≥1. What is the optimal x and corresponding Lagrange multiplier λ?

A: Get this wrong and every KKT check in your code flags false positives, stalling release tests. The unconstrained minimum violates x≥1, so the constraint is active. Stationarity gives 2x−λ=0 at x=1, leading to λ=2. That satisfies complementary slackness and feasibility without hacks.

Q: You're under time pressure and google "Newton method iterations quadratic convergence estimate". A Newton solver starts with error norm 10^−1 on a well-scaled twice-differentiable convex problem. Roughly how many iterations to reach 10^−12?

A: Overestimating iteration count leads to unnecessary watchdog margins and CPU oversizing. Near the solution, Newton's method shows quadratic convergence: 10^−1 → 10^−2 → 10^−4 → 10^−8 → 10^−16. Three to four clean steps hit 10^−12. That's why it's favored despite heavier math.