Maths for Design Optimisation: Gradient-Based Methods

State-of-the-Art Optimisation Algorithms for Engineering Design

What you’ll learn
Intuitive understanding of optimality conditions and constraints
Core gradient-based optimisation algorithms used in engineering
Convergence behaviour, local vs global search, and algorithm trade-offs
Hands-on Python optimisation exercises with Plotly, Sympy, and Scipy

Requirements
Some basic knowledge of mathematical optimisation required

Description
Master State-of-the-Art Optimisation Algorithms for Engineering DesignGradient-based methods form the backbone of most high-performance optimisation tools used in engineering today. This course focuses on understanding how these algorithms work, how they differ, and how to apply them effectively to real design problems.In this hands-on course, you’ll learn how gradient-based optimisation algorithms are constructed and used in practice, building directly on the numerical analysis and derivative concepts developed earlier in the series. You’ll explore how optimisation algorithms choose search directions, determine step size, and handle constraints to converge toward an optimum.We begin with unconstrained optimisation problems, introducing the core building blocks shared by many algorithms. You’ll develop intuition for line-search and trust-region approaches, and study widely used methods such as steepest descent, conjugate gradient, Newton’s method. Rather than treating these algorithms as plain formulas, you’ll learn how and why they behave differently on real optimisation landscapes.The course then moves on to constrained optimisation, where most real engineering problems live. You’ll learn how equality and inequality constraints are handled, how optimality conditions extended to constrained settings, and how practical algorithms such as Sequential Quadratic Programming (SQP) solve constrained problems iteratively. Concepts like active sets and KKT conditions are introduced intuitively, with a focus on how they influence algorithm behaviour rather than long and cumbersome proofs.As throughout the series, the emphasis is on intuition, structure, and application. You’ll work through hands-on Python coding exercises to solve both unconstrained and constrained optimisation problems, visualise algorithm behaviour, and apply gradient-based methods to realistic engineering scenarios — including a final case study on aircraft fuel tank optimisation.By the end of this course, you’ll:Understand how gradient-based optimisation algorithms work in practiceBe able to solve unconstrained and constrained optimisation problemsRecognise the strengths and limitations of different gradient-based methodsDevelop intuition for optimality conditions, convergence criteria, and optimisation strategiesGain hands-on experience implementing optimisation algorithms using Sympy and ScipyBe well prepared to choose and apply appropriate optimisation methods in engineering designThis course is designed for engineers, students and technical professionals who want to move beyond solver settings and black-box tools, and instead understand how modern optimisation algorithms actually operate.A basic familiarity with mathematical optimisation is recommended, as this course builds directly on earlier modules in the Maths for Design Optimisation series.If you’re ready to apply powerful, industry-standard optimisation algorithms with confidence — and understand what’s happening under the hood — this course is for you.

Who this course is for
System designers or engineers interested in MDO
Technical leaders curious about engineering design optimisation
Anyone looking for a more robust, rigorous way to optimise their products

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