Computational Engineering / module 16
Optimisation algorithms
How a search finds a good design, why the answer depends on where it started, and the result that says no method wins everywhere.
What you will understand
- Describe a design space and say what makes one hard to search
- Distinguish a local optimum from a global one, and say why that distinction is expensive
- Compare deterministic and stochastic methods on equal evaluation budgets
- Explain exploration and exploitation as a trade-off rather than a preference
- Choose a stopping criterion and say what it gives up
- State the No Free Lunch result and what follows from it in practice
Lessons
The design space
What you are searching, why it is bigger than it looks, and the difference between a local and a global optimum.
Deterministic and stochastic methods
Downhill methods, methods that sometimes go uphill, and why the second kind exists at all.
No free lunch, and knowing when to stop
Why there is no best algorithm, and why every stopping criterion is a decision about what you are willing to miss.
Module summary
- Exhaustive search is honest and does not scale: 21 values in five variables is over four million runs
- A gradient method cannot leave the basin it started in, and does not know there is anything outside
- Where a downhill method ends up on a rugged landscape depends entirely on the starting point
- Stochastic methods accept a worse answer sometimes, which is how they cross a ridge
- Compare methods on evaluations spent, not on the answer alone
- No method wins on every landscape — that is a theorem, not an admission
- Stopping is a decision: every criterion trades computation against the chance of missing something
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Check yourself
A gradient method finds a better answer than a genetic algorithm on a smooth landscape, using a fifth of the evaluations. What follows?
Practice
A design has 6 variables with 12 candidate values each. How many evaluations would a full grid search require?
Practice
Simulated annealing proposes a move 1.2 units worse at a temperature of 0.4. What is the acceptance probability, to four decimal places?