Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am reading Convex Optimization by Boyd and Vandenberghe (free at http://www.stanford.edu/~boyd/cvxbook/) and I am trying to justifying their assertion (p. 53) that if $K$ is a proper cone, $K^*$ is its dual, $v \in \operatorname{int}{K}$, and $\lambda \in K^* \setminus \{0\}$, then $\lambda^T v > 0$. It is obvious that $\lambda^T v \geq 0$. I reworded their statement and got rid of one of the directions of a double implication that was obvious to me. If anyone wants, I can define all the terms I used above.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Let $\epsilon > 0$ be small enough so that $v - \epsilon \lambda \in K$. Then $0 \leq \lambda^T(v - \epsilon\lambda) = \lambda^T v - \epsilon\|\lambda\|^2$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.