Projected gradient descent

For a convex set $C$ and a differentiable convex objective $F$, projected gradient descent uses

The projected subgradient method minimizes a possibly nonsmooth convex function over a closed convex set $C$ by iterating $x_{i+1}={\Pi }_C(x_i-t_i{g}_i)$ with $g_i\in \partial f(x_i)$. Nonexpansiveness of Euclidean projection onto a convex set and the subgradient inequality give its basic convergence bound.

If $\Vert \nabla F(x_i)\Vert \le G$ and $\Vert x_1-x_{\ast }\Vert \le D$, thenIt follows by expanding the squared distance after each projected step, using nonexpansiveness of projection, summing the resulting inequalities, and applying convexity.