 Projected gradient descent

Let $C$ be a nonempty closed convex set in a real finite-dimensional Euclidean space, and let $F$ be a differentiable convex function. A projected gradient descent step isThe Euclidean projection onto a convex set is single-valued under these assumptions and restores feasibility after the unconstrained gradient descent step. Closedness matters: the nearest point to zero in the convex set $(0,1)$ does not exist.

This is the proximal gradient method with nonsmooth term equal to the indicator functional of a constraint set, whose proximal operator is ${\Pi }_C$. If $\nabla F$ is $L$-Lipschitz continuous, a standard choice $0<\eta \le 1/L$ gives the usual convex objective-error bound relative to an existing minimizer.^[ref]

For $C=[0,1]$, ${\Pi }_C(z)=\max (0,\min (z,1))$. Minimizing $F(x)=(x-2{)}^2/2$ from $x_0=0$ with $\eta =1$ gives $x_1={\Pi }_C(2)=1$, the constrained optimum.