## How to find extreme points using optimisation methods

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 $$M = \left[ f \colon \mathbb{R}^n_+ \to \mathbb{R} \mid f(p) = \int\limits_{\mathbb{R}^n_+} e^{-p\cdot x} \mu(dx), \;\;\; \mu \in \text{ Borel probability measures on } \mathbb{R}^{n}_+ \right]$$

I want to find extreme points of this set. We can reduce this problem to an optimisation problem

$$\langle \nu, f \rangle = \int\limits_{\mathbb{R}^{n}_{+}} f(p) \nu(dp) \to \max\limits(\mu)$$

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?

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 Well, Lagrange multiplies also work in Banach spaces, see en.wikipedia.org/wiki/…. – Dirk Jan 31 2012 at 18:21