The problem is to find $$\min_x f(x),$$ where $$ f(x) = g(x) + h(x),$$ and $g$ is convex and $\beta$-smooth, and $h$ is $\alpha$-strongly convex but non-smooth.
I have a modified projected gradient method for composite optimization: $$xt+1 ← \arg \min_x \left\|x-\left(xt - \tfrac{1}{\beta}\nabla g(xt)\right)\right\|^2 + \tfrac{2}{\beta} h(x) $$
I need to prove that this method converges with rate $O \left(\exp \left(-\Theta \left(\tfrac \alpha \beta t \right) \right) \right)$.
I know the definitions of $\beta$-smooth and $\alpha$-strongly convex but I just don't know how to apply them to get to this result.
I would appreciate some help on how to prove this.