## 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.