Proof: Condition for a function being strongly convex

Question

Let $\lambda \in \mathbb{R}_{> 0}$. A function $f : \mathbb{R}^d \to \mathbb{R}$ is $\lambda$-strongly convex, if for all $\alpha \in (0, 1)$ and all $u, v \in \mathbb{R}^d$ $$ f(\alpha u + (1 - \alpha) v) \leq \alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha (1 - \alpha) \lVert u - v\rVert^2 $$ holds. In this blog post (ii -> iv) I found a condition for $f$ being $\lambda$-strongly convex:

If $g(x) = f(x) - \frac{\lambda}{2} \lVert x \rVert^2$ is convex, then $f$ is $\lambda$-strongly convex.

The author claims that this condition follows directly from definition of $g$ and convexity, however I'm having problems of proofing it for myself. This is basically how far I got:

$$\begin{align*} &\alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha (1 - \alpha) \lVert{u - v}\rVert^2 \\ &= \alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha (1 - \alpha) \left({\lVert{u}\rVert^2 + \lVert{v}\rVert^2 - 2\langle u, v\rangle}\right) \\ &\geq \alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha (1 - \alpha) \left({\lVert{u}\rVert^2 + \lVert{v}\rVert^2}\right) \\ &= \alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha (1 - \alpha) \lVert{u}\rVert^2 - \frac{\lambda}{2} \alpha (1 - \alpha) \lVert{v}\rVert^2 \\ &> \alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha \lVert{u}\rVert^2 - \frac{\lambda}{2} (1 - \alpha) \lVert{v}\rVert^2 \\ &= \alpha \left({ f(u) - \frac{\lambda}{2} \lVert{u}\rVert^2 }\right) + (1 - \alpha) \left({ f(v) - \frac{\lambda}{2} \lVert{v}\rVert^2 }\right) \\ &= \alpha g(u) + (1 - \alpha) g(v) \\ &\geq g(\alpha u + (1 - \alpha) v) \\ &= f(\alpha u + (1 - \alpha) v) - \frac{\lambda}{2} \lVert{\alpha u + (1 - \alpha) v}\rVert^2 \\ &= ? \\ &= f(\alpha u + (1 - \alpha) v)\end{align*}$$

What am I missing?

Answer 1

Thanks to @whuber I noticed that especially the $-2 \langle u, v \rangle$-term is too valueable to discard. I restarted the calculation from the other side and was able to show the required bound by using $\leq$ exactly one time (applying the premise).

In case anybody else is interested in a sketch of the solution: Start by showing $$ \begin{align*} &f(\alpha u + (1 - \alpha) v) \\ &= f(\alpha u + (1 - \alpha) v) - \frac{\lambda}{2} \lVert{\alpha u + (1 - \alpha) v}\rVert^2 + \frac{\lambda}{2} \lVert{\alpha u + (1 - \alpha) v}\rVert^2 \\ &= \dots \\ &\leq \alpha f(u) + (1 - \alpha) f(v) - \frac{\lambda}{2} \alpha \lVert{u}\rVert^2 - \frac{\lambda}{2} (1 - \alpha) \lVert{v}\rVert^2 + \frac{\lambda}{2} \lVert{\alpha u + (1 - \alpha) v}\rVert^2, \end{align*} $$ and then conclude using only basic linear algebra that $$ - \frac{\lambda}{2} \alpha \lVert{u}\rVert^2 - \frac{\lambda}{2} (1 - \alpha) \lVert{v}\rVert^2 + \frac{\lambda}{2} \lVert{\alpha u + (1 - \alpha) v}\rVert^2 = -\frac{\lambda}{2} \alpha (1 - \alpha) \lVert{u - v}\rVert^2. $$