Continuous Optimisation HT26, Useful miscellany
Flashcards
Rayleigh quotient eigenvalue bound
@State the Rayleigh quotient bound for eigenvalues.
Suppose:
- $s \in \mathbb R^n \setminus {0}$
- $\mathbf M \in \mathbb R^{n \times n}$ symmetric
Then:
\[\lambda _ \min(\mathbf M) \le \frac{s^\top \mathbf M s}{ \vert \vert s \vert \vert ^2} \le \lambda _ \max(\mathbf M)\]@Prove the ∆rayleigh-ritz-theorem, i.e. that if
- $s \in \mathbb R^n \setminus {0}$
- $\mathbf M \in \mathbb R^{n \times n}$ symmetric
then:
\[\lambda _ \min(\mathbf M) \le \frac{s^\top \mathbf M s}{ \vert \vert s \vert \vert ^2} \le \lambda _ \max(\mathbf M)\]
@todo, once completed sheet 2.
Characterisation of Lipschitz continuity in terms of bounded Hessian
@State an exact characterisation of when $\nabla f$ is Lipschitz continuous in terms of the Hessian.
Suppose $f \in \mathcal C^2$, then $\nabla f$ is Lipschitz continuous iff $\nabla^2 f$ is uniformly bounded above.
@Prove that for $f \in \mathcal C^2$, $\nabla f$ is Lipschitz continuous iff $\nabla^2 f$ is uniformly bounded above.
@todo, once completed sheet 2.