A question on optimization

Let Y be a normed space, D\subset Y be a proper closed convex cone with nonempty interior. Define the positive polar cone D^+ by D^+=\{\lambda\in Y^*:\lambda(d)\geq 0\text{ for all }d\in D\}. Let \Lambda be a weak*-compact convex base of D^+, i.e, \Lambda is weak*-compact convex, and for any nonzero f\in D^+ there exists unique s>0 and \lambda\in\Lambda such that f=s\lambda.

Given \lambda\in D^+, e\in\text{int}(D), and define \displaystyle\alpha=\inf_{\lambda\in\Lambda}\lambda(e). By the weak*-compactness of \Lambda, the infimum is attained.

Question: Show that it is attained at an extreme point of \Lambda, i.e. \lambda\in\Lambda is an extreme point if there does not exist two distinct \lambda_1, \lambda_2\in\Lambda and t\in (0,1) such that \lambda=t\lambda_1+(1-t)\lambda_2.

It seems this can be done by applying the Krein Milman theorem.

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3 Responses to A question on optimization

  1. KKK says:

    I don’t know anything about optimization and Krein Milman, but I’d like to try ^_^.
    Let S=\{\lambda\in \Lambda: \lambda(e)=\alpha \}. (Btw, your \alpha does not depend on your “given \lambda \in D^+” right? )
    Then S is closed and convex in Y^* in weak* topology, thus by Krein Milman theorem S is the closed convex hull of its extreme points. Take an extreme point \lambda of S. We claim that it is also an extreme point of \Lambda. To see this, let \lambda_i\in \Lambda such that (1-t)\lambda_1+t \lambda_2=\lambda, t\in [0,1] . Then (1-t)\lambda_1(e)+t \lambda_2(e)=\lambda(e)=\alpha forces \lambda_1(e)=\lambda_2(e)=\alpha, i.e. \lambda_i\in S and so \lambda_1=\lambda_2=\lambda, as \lambda is an extreme point of S. Therefore \lambda is an extreme point of \Lambda.

  2. honleungdydx says:

    Yes, given \lambda\in D^+ is a typo.

    The proof is nearly the same as what I think last night, except the topology on Y^* should be the weak* topology. Since your S is weak*-compact and convex, the Krein Milman theorem can still be applied. [Corrected, thanks! -KKK]

    Let me state the general Krein Milman theorem:

    Let C be a nonempty compact convex set of a Hausdorff LCS (locally convex space, in our case, Y^* with weak* topology is an example). Then C is the closed convex hull of its extreme points.

  3. honleungdydx says:

    Perhaps I can add some small remarks.

    If the weak topology and the weak* topology are concerned, we usually add weakly and weak* (not weak*ly, don’t care about grammar) to the related property, for example, weakly closed, weak* open etc.

    In fact, the above problem is just a small part of a proof, which concerns characterizing a certain concept of minimality to the solution of some other optimization problem that the objective function is an extreme point of \Lambda. This result at least guarantees there exists some such point.

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