Module 16 · Lesson 16.1
The design space
What you are searching, why it is bigger than it looks, and the difference between a local and a global optimum.
Why 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)
What the space is
One axis per design variable, and one point per possible design. The objective is a surface over it: a height at every point. Optimisation is finding the lowest point on that surface without being able to see it.
Two properties decide how hard that is.
Dimension. Not size — dimension. A grid of 21 values along each axis is 441 points in two variables, 9 261 in three, and 4 084 101 in five. Each extra variable multiplies the work by 21. This is the curse of dimensionality, and it is why the obvious method stops being available almost immediately.
Ruggedness. A surface with one basin is easy: walk downhill and you arrive. A surface with several basins of different depth is not, because walking downhill takes you to the bottom of whichever basin you started in — and nothing about that bottom tells you there is a deeper one elsewhere.
Local and global
A local optimum is a point with nothing better nearby. A global optimum is the best point anywhere in the space.
Every method finds local optima. The difference between methods is entirely in how they behave about the possibility that a better basin exists somewhere they have not looked.
And here is the uncomfortable part: you almost never know which one you have. Confirming a global optimum requires searching everywhere, which is the thing you could not afford. So an optimisation result is properly reported as the best design found, by this method, from this start, in this many evaluations — and any of those four qualifiers can change the answer.
What makes a structural design space rugged
The smooth-looking problems in textbooks are not typical. Structural design spaces are rugged for specific reasons worth recognising:
- Discrete catalogues. Rolled sections come in steps, so the surface is a staircase rather than a slope, and a gradient is undefined at every step.
- Constraint boundaries. A deflection limit or a fire requirement puts a cliff in the surface. Just inside it the design is fine; just outside it does not exist.
- Changing governing criteria. In one region strength governs and in another deflection does. The surface has a crease where they swap, and a gradient method reads the crease as a direction to follow.
- Indeterminacy. Resizing a member changes the forces everywhere, so the objective is not separable — you cannot optimise members one at a time.
All four are ordinary features of ordinary structures. A design space that is smooth and single-basined is the exception.
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
- Every method finds local optima; they differ in how they treat the possibility of a better basin
- 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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint