Proving inequality that arises in projected gradient descent

Question

I am reading a paper in which the author uses projected gradient descent to produce iterates of the form:

$$\pi_{t + 1} = \underset{\pi \in \Pi}{\arg \max}\left\{ \left \langle \pi, Q \right \rangle - \frac{1}{2 \eta} \left \lVert \pi - \pi_t \right \rVert_2^2 \right\}$$

where $\Pi$ is a set of finite-dimensional vectors. The author then claims for all $\pi' \in \Pi$ it holds that

$$\left \langle \pi' - \pi_{t + 1}, \eta Q - \pi_{t + 1} + \pi_t \right \rangle \leq 0$$

By optimality of $\pi_{t + 1}$ I can deduce

$$\left \langle \pi_{t + 1}, Q \right \rangle - \frac{1}{2 \eta} \left \lVert \pi_{t + 1} - \pi_t \right \rVert_2^2 \geq \left \langle \pi', Q \right \rangle - \frac{1}{2 \eta} \left \lVert \pi' - \pi_t \right \rVert_2^2$$

and therefore

$$\left \langle \pi' - \pi_{t + 1}, \eta Q \right \rangle \leq \frac{1}{2} \left \lVert \pi' - \pi_t \right \rVert_2^2 - \frac{1}{2} \left \lVert \pi_{t + 1} - \pi_t \right \rVert_2^2$$

but it remains for me to show

$$\frac{1}{2} \left \lVert \pi' - \pi_t \right \rVert_2^2 - \frac{1}{2} \left \lVert \pi_{t + 1} - \pi_t \right \rVert_2^2 \leq \left\langle \pi' - \pi_{t + 1}, \pi_{t + 1} - \pi_t \right\rangle$$

Any help would be appreciated!

EDIT: Note the update can also be written

$$\pi_{t + 1} = \mathtt{Proj}_{\Pi} \Big \{ \pi_t + \eta Q \Big \}$$

Answer 1

There's a further condition that $\Pi$ is convex, which is key. It is then a direct application of Theorem 2.1 here.