Theory
The following statements are true about the convex minimization problem:
- if a local minimum exists, then it is a global minimum.
- the set of all (global) minima is convex.
- for each strictly convex function, if the function has a minimum, then the minimum is unique.
These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.
Read more about this topic: Convex Optimization
Famous quotes containing the word theory:
“Could Shakespeare give a theory of Shakespeare?”
—Ralph Waldo Emerson (1803–1882)
“There could be no fairer destiny for any physical theory than that it should point the way to a more comprehensive theory in which it lives on as a limiting case.”
—Albert Einstein (1879–1955)
“If my theory of relativity is proven correct, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.”
—Albert Einstein (1879–1955)