Maximising a function

e.g. Each value of parameter x constructs a solution with fitness f(x).
Find the x value that gives maximum fitness f(x).

• Optimization methods

Strategy:

1. If we have an equation for the function and it is differentiable:
   • Differentiate. Search for slope = 0. This will give local max and min points if they exist (if not infinite).

2. Interesting case is where no equation known / not differentiable (but can still judge fitness of any given x).
   • Example of an Unknown or postulated function: Input x = All the parameters that control a mobile robot. Fitness f(x) = How well the robot played soccer. Find x that maximises f(x).
   General approach:
   • Learn from exemplars (many samples of x and f(x)). Build up a map of the function, ever increasing in accuracy and detail.
   • If only interested in max fitness, the map will end up in more detail in uplands (keep exploring) than in lowlands (which we abandon).

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The idea of Maximising a function from exemplars is that "nearby" Input should generate "nearby" Output.

But some functions defeat this simple idea:

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Chaotic functions

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Chaos Theory demo

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Non-chaotic functions

We do not expect in general to be able to maximise a chaotic (or discontinuous) function from exemplars.
The global maximum must be surrounded by some continuous zone of uplands, otherwise how can we find it.
It cannot be a single, isolated point or else the odds of finding that precise x go to zero.