How to find extreme points using optimisation methods - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:59:53Z http://mathoverflow.net/feeds/question/87151 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87151/how-to-find-extreme-points-using-optimisation-methods How to find extreme points using optimisation methods Nimza 2012-01-31T17:44:07Z 2012-01-31T17:44:07Z <p>Im interested in description of image of integral operators of some special type $$f(p) = \int\limits_{ \mathbb{R}^n_+ } e^{-g(p,x)} \mu(dx)$$ Now I'm trying to apply some results from Choquet theory. Let's consider a well-known example. Let M be a set of functions given by $$M = \left[ f \colon \mathbb{R}^n_+ \to \mathbb{R} \mid f(p) = \int\limits_{\mathbb{R}^n_+} e^{-p\cdot x} \mu(dx), \;\;\; \mu \in \text{ Borel probability measures on } \mathbb{R}^{n}_+ \right]$$</p> <p>I want to find extreme points of this set. We can reduce this problem to an optimisation problem</p> <p>$$\langle \nu, f \rangle = \int\limits_{\mathbb{R}^{n}_{+}} f(p) \nu(dp) \to \max\limits(\mu)$$</p> <p>Solution of this problem is given by Bernstein's theorem on monotone functions (optimal measures are Dirac's deltas). But this method is not quite general to use in problems of image description of integral operators. So the task is: is there an optimisation method (for instance, some generalisation of Lagrange's rule) that provides us a possibility to find a solution of this optimisation problem and hence the solution of the initial problem?</p>