Hello again,

Another question about vector and potential fields I'm afraid.

Really its just a quick clear up of terms is anyone can help me please.
What is the difference between local minima and local optima. Are these just the same thing? Because I have read up quite a bit on the subject and have become slightly confused again. The places I have seen them refer to local optima it seems more to do with objects getting stuck behind objects due to the repulsive forces of the objects counter-acting the attractive force of the goal and where I have read about them discussing the local minimum seems to refer more to objects getting stuck in concave or U-shaped objects. Is this right or are they just the same thing?

Any clearing up for me would be much appreciated.

Cheers

What is the difference between local minima and local optima. Are these just the same thing?

They are almost exactly the same thing.

The word "optimum" is defined with respect to a particular optimality criterion -- usually either making a certain function's value as big as possible, or as small as possible. If the optimality criterion is that a function be minimized, then "local optimum" and "local minimum" mean the same thing. If the optimality criterion is that a function be maximized, then obviously they are not. Then "local optimum" means "local maximum."

In the case of potential fields, one is trying to minimize the function, so in that context "local optimum" and "local minimum" mean the same thing.

You should probably also know the phrase "critical point." A critical point is any place where a function's gradient is zero. Local minima and maxima are critical points, but so are saddle points, as well as points in flat regions.

They are basically the same thing.

If you think of a wave, minima is the trough of the wave, and the maxima is the crest.

The world 'local' means an area close to the space you are currently searching. Global means the entire space. So local minima is a trough close to the point you are examinimg, the global minima is the lowest point in the whole space.

In the graph below, if you were examining the area around 0.8, you would find a local minima at 0.65 and a local maxima at 0.9ish. The global maxima is at 0.3, and the global minima is at 0.0.




Most functions have complex responses. Like the function above, they're not simple lines or nice big humps like a Gaussian or Sigmoid.

For most functions there is no perfect solution to finding a global minimum or maximum. Short of exploring every single possible variation in parameter the best you (or anyone else) can do is guess. Gradient descent, hillclimbing, backpropagation, polynomial regression, are all different ways of trying to guess where the global maxima/minima of a function is.

They're similar but opposite concepts. A local minimum is where the value bottoms out at a given point (relative to the local area) whereas a local optimum/maximum is where the value peaks at a given point. Of course there are also global variants of each wherein the point has a minimum/maximum value across all points in the domain space.


[edit] Holy triple ninja batman! Heh.

Wow,

Cheers people, 3 beautiful replies all at once.

That's nicely cleared up some things for me.

Thanks!