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Isn't the problem that in most real-world problems, the optimization function is never convex? At least in many Machine Learning problems (e.g. optimizing a neu

• fmap 13y ago
  For many machine learning problems you can find an equivalent problem which is convex. This seems to be the method of choice for dealing with things like Support Vector Machines. Many academics seem to dislike neural networks because of how easy it is to get stuck in local minima and how hard it is to say anything useful about the result. This won't happen with techniques based on convex optimization. Outside of machine learning solving a convex relaxation of a discrete optimization problem and then trying to round the result is one of the standard techniques in optimization. This is used all over the place, from straightforward linear relaxations for ILPs through the semidefinite relaxations of problems such as max cut. You start with an intractable problem, embed it into a tractable problem and use the result to gain information about the solutions to your original problem.
• rawland 13y ago
  Yes! For instance, a very simple example: When you travel from A to B you sometimes can reach that destination in the same time using different paths. "Same", in the sense of a certain threshold or uncertainty considering Real World (TM) applications. Furthermore, there can be constructed academic problems, where you have an infinite amount of solutions. Say, traveling from the north to the south pole in shortest time over the surface of a perfect globe.