We are still missing a few more apparatus before we can define an “optimisation process”.
Let an optimisation space be a n-tuple (X,F,B,A,F,P) where:
X is our configuration space
F={f1:X→T,…,fi:X→T,…} is a non-empty collection of objective functions
Where T is some totally ordered set
B⊂X is the “basin of attraction”
A⊂X is the “target configuration space”/”attractor”
F⊂2X is the σ-algebra
2B∪A⊂F
P:F→[0,1] is a probability measure
P(B)≥P(A)
(X,F,P) form a probability space
I.e. an optimisation space is a probability space with some added apparatus
Many optimisation spaces may be defined on the configuration space of the same underlying system.
I expect that for most practical purposes, we’ll take the underlying probability space as fixed/given, and different optimisation spaces would correspond to different choices for objective functions, basins or attractors.
Optimisation Process
We can then define an optimisation process with respect to a particular optimisation space.
Intuitively, an optimisation process transitions a system from a source configuration state to a destination configuration state that obtains a lower (or equal) value for the objective function(s).
Formal Definition
For a given optimisation space (X,T,F,B,A,F,P), an optimisation process is a function o:B→A satisfying:
Optimisation Space
We are still missing a few more apparatus before we can define an “optimisation process”.
Let an optimisation space be a n-tuple (X,F,B,A,F,P) where:
X is our configuration space
F={f1:X→T,…,fi:X→T,…} is a non-empty collection of objective functions
Where T is some totally ordered set
B⊂X is the “basin of attraction”
A⊂X is the “target configuration space”/”attractor”
F⊂2X is the σ-algebra
2B∪A⊂F
P:F→[0,1] is a probability measure
P(B)≥P(A)
(X,F,P) form a probability space
I.e. an optimisation space is a probability space with some added apparatus
Many optimisation spaces may be defined on the configuration space of the same underlying system.
I expect that for most practical purposes, we’ll take the underlying probability space as fixed/given, and different optimisation spaces would correspond to different choices for objective functions, basins or attractors.
Optimisation Process
We can then define an optimisation process with respect to a particular optimisation space.
Intuitively, an optimisation process transitions a system from a source configuration state to a destination configuration state that obtains a lower (or equal) value for the objective function(s).
Formal Definition
For a given optimisation space (X,T,F,B,A,F,P), an optimisation process is a function o:B→A satisfying:
∀x∈B∀f∈F(f(o(x))≤f(x))Next: Motivations for the Definition of an Optimisation Space
Catalogue of changes I want to make:
✔️ Remove the set that provides the total ordering
✔️ Remove “basin of attraction” and “attractor” from the definition of an optimisation space
Degenerate spaces
No configuration is a pareto improvement over another configuration
Redundant objective functions
A collection of functions that would make the optimisation space degenerate
Any redundancies should be eliminated
Convey no information
All quantification evaluates the same when considering only the relative complement of the redundant subset and when considering the entire collection
Removing “basin of attraction” and “attractor” from definition of optimisation space.
Reasons:
Might be different for distinct optimisation processes on the same space
Can be inferred from configuration space given the objective function(s) and optimisation process