The document introduces concepts from search and optimization theory and how they can be applied to understand design processes. It discusses how to model a design problem as a search space with states, actions, and an objective function. Several search algorithms are then presented, such as blind search, hill climbing, and simulated annealing, to guide the exploration of the design space. The document argues that understanding design through computational models of search and optimization can provide insights into creative behaviors.

1. An Introduction to
Search and Optimisation
for DesignTheorists
Akın Kazakçı
MINES ParisTech
PSL Research University, CGS-I3 UMR 9217
akin.kazakci@mines-paristech.fr

2. Everyone designs who devises courses of action aimed at
changing existing situations into preferred ones.
- Herbert Simon

3. Everyone designs who devises courses of action aimed at
changing existing situations into preferred ones.
- States (situations)
- Actions (transformations)
-Value function
Assumes the following ontology: Can be represented with:
-Vectors: x=(x1,…, xn) ∈ S
- Functions: y = f(x), S→S
- Function, v(x), S→ℝ or
S→S°⊂S

4. Maze
States?
Actions?
Goal: find the exit
What changes if goal = find shortest path?
Positions on the board (x, y); here discrete
values for the sake of the example
Current state: (4,4), Goal state: (7,8)
Go North: (0, +1)
Go South: (0, -1)
Go West: (-1, 0)
Go East: (+1,0)
= Agent
= Exit
Value?
Distance to exit

5. Maze
= Agent
= Exit
(4,4)
(5,4)
Right
Up
(5,5)
Right
(5,6)
(6,6)
Right
(6,7)
(7,7)
(7,8)
Up
Up
Up
The path(s) to the solution
cannot be determined in
advance (otherwise why
search?)
So, how to “explore”?

6. - States
- Actions
-Value function
Are state variables discreet or continuous?
What is the search space size?
Which actions are allowed/forbidden in a given states?
What is the cost of applying an action?
Explicit (subset of states) or based on value?
Is the value function known analytically?
Is it costly to evaluate (call) the value function?
Several important factors to consider
when dealing with a search space

7. Generating the search space
The search space is usually not explicitly stated.
That means, it needs to be generated.
Conceptually, this generative process can be
represented as a tree.
Given a “Node” (state), we can
“expand” it with its
“neighbours” (states accessible
through actions).

8. Maze
= Agent
= Exit
(4,4)
(4,5)
Right
Up
= Obstacle
Down
(4,3) (5,4)
Moving left is not allowed at state
(4,4)
Thus, we cannot transform the state
to (3,4)
The (average) number of neighbours
we can generate in a given state is
called (effective) branching factor.

9. Maze
= Agent
= Exit
(3,4)
Left
Down
(3,3)
(4,4)
(5,4)
Right
Up
(5,5)
Right
(5,6)
(6,6)
Right
(6,7)
(7,7)
(7,8)
Up
Up
Up
What if we expand like this?
We loose time (and possibly resources)

10. Blind search
- Two basic strategies for generating the search tree are
breadth-first and deep-first search.
- Neither strategy offers an ideal approach: no guidance of
search, very costly in time, memory or both.
Breadth-first exploration Deep-first exploration
Images from Wikipedia
Numbers represent the order in which nodes are visited
(If branching factor is infinite (i.e. continuous state representations), no guarantee of completeness.)

11. How to guide the search?
• That’s the million dollar question.
• A starting point is to use the “value” function (when it is
available) for evaluating the “promise” of a path.
• This kind of search strategy is called heuristic or informed
search.
• The archetypical example is Hill-Climbing (greedy) search.

12. Greedy local search
(Hill-Climbing)
function Hill-Climbing(problem)
variables: current, neighbour
current⟵ MAKE_NODE(INIT(problem))
loop
neighbour = ARGMAX(VALUE (GET_NEIGHBOURS(current)))
if VALUE(neighbour) < VALUE(current) then return current
current ⟵ neighbour
return current
Initialise with starting state
Generate all neighbours and
select the max valued one
If not better than current
state, return the current state
(if not) change the current
state to the best neighbour

13. Maze
= Agent
= Exit
(4,4)
(5,4)
Right
Up
(5,5)
Up
(5,6)
(4,5)
Up
Up
(4,6)
Up
(5,6)
This is not the (optimal) solution
This is a “local” optima
Very much dreaded in all search and
optimisation literature
= Barrier

14. An illustration of local optima
V(x)
distance
to exit
states
Assume the red dot is the
value of the current state.
What’s the next move?
xk xl
All neighbouring states have higher values!
Remember we are minimising the distance
global optima

15. An illustration of local optima
V(x)
distance
to exit
states
Assume the red dot is the
value of the current state.
What’s the next move?
xk xl
Greedy search easily gets stuck within local optima.
global optima

16. Rastrigin function
Image from MathWorks
You think this is complicated? - What if the number of variables increases?

17. function Simulated_annealing(problem)
variables: current, neighbour, T
current⟵ MAKE_NODE(INIT(problem))
while T > 0
neighbour = PICK_ONE(GET_NEIGHBOURS(current)))
current⟵ neighbour, if probability P (VALUE(neighbour),
VALUE(current), T) > RANDOM(0,1)
DECREASE_TEMPERATURE(T)
return current
Initialise with starting state
Pick one neighbour
Move to neighbour with
some probability
Decrease system temperature
Simulated annealing

18. Simulated annealing
Gif from Wikipedia

19. Demo
http://qiao.github.io/PathFinding.js/visual/

20. Optimisation
Given a search problem, find the optimum value for v(x).
max v(x), s.t. x ∈ S
“subject to”, x is constrained
to take values only in S
Remark: max v(x) = min -v(x)
Same algorithms can be used to solve either of them.

21. Optimisation
Given a search problem, find the optimum value for v(x).
max v(x), s.t. x ∈ S
What changes compared to the previous search setup?
The ideal (desired) state is not known now.
If we are lucky, an analytical form of the value function is known.
In optimization, it is the value function that is being explored
We don’t know which state we want to end up in, but we want it
to have the best possible value.

22. Why is this even a model for
design?
What would be the points?
What does the objective
function represent?
Consider the following, in the
context of a design activity
How would the designer
generate the search tree?
What would be a new move?

23. How would you model Wooten&Ulrich’s notion of feedback?
Bonus question:What kind of “design space”underlies logo design?

24. Can Lehman&Stanley’s approach be seen as a model of creative
behaviour?
Is this different than search & optimisation we have seen?

25. The effect of adding “one”
variable to the “design space”

26. Questions
• What does the model become, if we assume that the
designer can add/remove variables during the course of
the design?
• Where would the designer find new variables?
• How can a designer decide which variable to add?
• How would you model this process?

27. Akın Kazakçı
akin.kazakci@mines-paristech.fr