Table 6 Basic GPSR algorithm

1. Initialize β∈(0,1), $\mu \in (0,\frac{1}{2})$, α₀and z⁽⁰⁾. Also set i=0.

2. Choose α⁽ⁱ⁾to be the largest number of the form α₀β^j, j≥0, such that

$\begin{array}{c}F\left({\left({\mathbf{z}}^{\left(i\right)}-{\alpha }^{\left(i\right)}\nabla F\left({\mathbf{z}}^{\left(i\right)}\right)\right)}_{+}\right)\le F\left({\mathbf{z}}^{\left(i\right)}\right)-\\ \mu \nabla F{\left({\mathbf{z}}^{\left(i\right)}\right)}^T\left({\mathbf{z}}^{\left(i\right)}-{\left({\mathbf{z}}^{\left(i\right)}-{\alpha }^{\left(i\right)}\nabla F\left({\mathbf{z}}^{\left(i\right)}\right)\right)}_{+}\right)\end{array}$

3. Set z^{(i + 1)}=(z⁽ⁱ⁾−α⁽ⁱ⁾∇F(z⁽ⁱ⁾))₊.

4. Check the termination criterion. If neither the maximum number of iterations has passed nor a given stopping condition is fulfilled, increase i and return to the 2nd step.