Algorithm 1 Non-linear primal-dual proximal splitting presented in [17] (Algorithm 2.1)

$\mathrm{Choose}\omega \ge 0,\mathrm{and}\tau ,\sigma$ $\mathrm{s}.\mathrm{t}.\tau \sigma \left({sup}_{k=1,...,i}{\Vert \nabla H\left(x^k\right)\Vert }^2\right)<1.$ while Not reached stopping criterion do     $x^{i+1}:={(I+\tau \partial G)}^{-1}(x^i-\tau {[\nabla H\left(x^i\right)]}^{*}{y}^i)$     ${\overline{x}}^{i+1}:=x^{i+1}+\omega (x^{i+1}-x^i)$     $y^{i+1}:={(I+\sigma \partial F^{*})}^{-1}(y^i+\sigma H\left({\overline{x}}^{i+1}\right))$ end while