I'm studying Trust-region method and there is proof in the book using the follow inequality:

$$ \dfrac{1\left\|{g_k}\right\|^4}{2g_k^TB_kg_k}\geq\dfrac{1\left\|{g_k}\right\|^2}{2\left\|B_k\right\|}$$

where $B_k$ is a positive definite matrix, i.e, $g_k^TB_kg_k > 0$.

The book says "Using the Cauchy Schwarz inequality we have this". I've tried use the fact, $\|Ax\| \leq \|A\|\cdot\|x\|$, but I couldn't reach the inequality.

Thanks for help


You have

$$g_k^TB_k g_k\leq \|g_k\|\|B_kg_k\|\leq \|g_k\|\|B_k\|\|g_k\|= \|B_k\|\|g_k\|^2.$$ Taking the reciprocal you obtain

$$\frac{1}{g_k^TB_k g_k}\geq \frac{1}{\|B_k\|\|g_k\|^2}.$$ Now just mutiply this inequality by $\frac{\|g_k\|^4}{2}$ and you are done.

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