In the area of Motion Planning (my own area of research), the most that can be said is that practical solutions exist for a tiny subset of cases, workable methods exist for a larger subset, expensive methods exist for a still larger subset, and everthing else might as well be impossible. - If you've got a low-dimensional problem, say 2D or 3D, without uncertainty (or at least bounded enough to pad obstacles and ignor…
sorry for sidetracking your answer but what actually does convex mean in the context of optimization. I remember looking at a book called convex optimization. Your statement that > Motion planning is very much non-convex, suggests to me that you are very much talking about the same thing. I understand convexity as in a shape. Why is convex good and concave bad in terms of optimization? I don't want you to dumb down t…
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Global optimum
In non-convex optimisation we can't make any assumption about the shape of the
error function: ,--. .--. ε
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`--------- Local optima ----------'
"Optimisation" means that we're trying to find an optimum of a function - a
maximum or a minimum. We're usually interested in the minimum of a function,
particularly a function that represents the error of an approximator on a set
of training data. Generally we prefer to find a _global_ minimum of the error
function because then we can expect the resulting approximator to generalise better
to data that was not available during training.If the error function has a convex shape we're basically guaranteed to find its global minimum. In non-convex optimisation, we're guaranteed to get stuck to local minima.
(Ok, the above is a bit tongue in cheek, there's no _guarantee_ of getting stuck to local minima, but it's very likely).