Computational Engineering / Optimisation algorithms
The design space
What you are searching, why it is bigger than it looks, and the difference between a local and a global optimum.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
- Reproduce quoted algorithm evaluation counts and reference minima from the actual seeded lab before treating them as verified benchmark results.
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 notesWhy this matters
Module 15 ended with a number: a 23-member truss choosing from an eight-value catalogue has about 5.9 × 10²⁰ possible designs. At a millisecond an evaluation that is nineteen billion years.
So every practical optimisation is a search — a strategy for learning about a space you can never enumerate, from a few thousand samples. This module is about those strategies, what each one buys, and what each one gives up.
By the end of this lesson you should be able to
- Define the design space and say what its dimension is
- Explain why exhaustive search fails on dimension rather than on size
- Distinguish a local from a global optimum
- Say what makes a landscape hard
What you should already know
- Objective, variables and constraints (Module 15)
- One-at-a-time sensitivity and its limits (Module 14)
Design space
Each design variable contributes a dimension, and the feasible points satisfy the stated constraints. A grid with 21 values per variable has 441 points in two dimensions, 9261 in three and 4084101 in five. Enumeration cost grows rapidly with both resolution and dimension.
A local optimum is no worse than feasible neighbours in a specified neighbourhood. A global optimum is no worse than any feasible point. Heuristic searches do not generally establish either simply by stopping. For convex problems, a local minimum is global; other methods can certify bounds or a gap without enumerating every design.
Discrete choices, nonconvex constraints, competing criteria and stiffness-dependent redistribution can make structural optimisation difficult. Constraints define a feasible region; they do not necessarily make the objective discontinuous. Record the formulation, search method, stopping conditions and any actual optimality certificate. Otherwise report the best feasible design found.
Try it
Search algorithm comparison
Six methods on the same landscape, with the evaluation count for each. Everything is seeded, so the numbers here are the numbers the lesson quotes.
Landscape
One basin. Any downhill method reaches the bottom, and the only question is how many evaluations it spends.
Show the path of
Dark blue is the lowest objective value (1.015); orange the highest (14.348). The shading is a secondary cue — every value is in the table below.
| Method | Best found | Gap | Evaluations | Trapped? |
|---|---|---|---|---|
| Grid search | 1.0150 | 0.0150 | 441 | no |
| Gradient descent | 1.0000 | 0.0000 | 79 | no |
| Random search | 1.0432 | 0.0432 | 441 | no |
| Simulated annealing | 1.0019 | 0.0019 | 441 | no |
| Genetic algorithm | 1.0000 | 0.0000 | 401 | no |
| Particle swarm | 1.0001 | 0.0001 | 441 | no |
Gradient descent reached within 0.0000 of the global optimum in 79 evaluations.
What this method is doing
- Follows the slope downhill from where it started. It cannot leave the basin it began in, and it does not know there is anything outside it.
Things worth trying
- On the smooth landscape, compare gradient descent's evaluation count with everything else. It wins outright — right answer, a fifth of the budget.
- Switch to the rugged landscape without changing anything. The same method is now trapped.
- Move the start to (6, 3) on the rugged landscape and gradient descent finds the global optimum. Nothing about the method changed.
- Grid search always evaluates 441 times regardless. In five variables at the same resolution it would be 4 084 101.
What this shows: No method wins on every landscape — and on a rugged one the answer a downhill method gives is decided by where it started.
Worked example
What exhaustive search costs
Given
- A design with 21 candidate values along each variable
- Considered in two, three, five and eight variables
- One evaluation takes 1 second — a small frame analysis
Find
The cost of a complete grid search in each case
Practice
A grid search uses 15 values along each of 4 variables. How many evaluations does it require?
Check yourself
An optimisation reports 'the optimum design has a mass of 84 tonnes'. What is wrong with that statement?
Worked example
When a grid search stops being affordable
Given
- A design has 4 variables, each to be examined at 12 values
- One evaluation takes 3 seconds
Find
How long an exhaustive search takes, and at how many variables it becomes impossible
Check yourself
Why does reducing the number of variables help more than a faster computer?
Check yourself
What is a local optimum?
Summary
- Dimension, not size, is what makes exhaustive search fail
- 21 values in five variables is over four million runs; in eight it is 3.8 × 10¹⁰
- Distinguish a search result from a local or global optimality guarantee
- Discrete catalogues, constraint cliffs, changing governing criteria and indeterminacy all make structural spaces rugged
- Report the best design found, by which method, from which start, in how many evaluations