Algorithm A1 Embedded T${}_{1\rho }$ with NL-PDPS, adapted from [17] [Algorithm 2.1]

$\mathrm{Choose}\omega \ge 0,\mathrm{and}{\tau }_{\ell },\sigma$ $\mathrm{s}.\mathrm{t}.{\tau }_{\ell }\sigma ({sup}_{k=1,...,i}\Vert {{[\nabla H\left(x^k\right)]}_{\ell }\Vert }^2)<1.$ $\mathrm{T}=\mathrm{diag}({\tau }_1{I}_N,{\tau }_2{I}_N,{\tau }_3{I}_N).$ while Not reached stopping criterion do     ${\tilde{u}}^{i+1}\leftarrow u^i-\mathrm{T}{[\nabla H\left(u^i\right)]}^{*}{\left[v_1^{i^{\mathrm{T}}}{v}_2^{i^{\mathrm{T}}}{v}_3^{i^{\mathrm{T}}}{v}_4^{i^{\mathrm{T}}}\right]}^{\mathrm{T}}$     $u^{i+1}\leftarrow P_{2,a_2}\left(P_{1,a_1}\left({\tilde{u}}^{i+1}\right)\right)$     ${\overline{u}}^{i+1}\leftarrow u^{i+1}+\omega (u^{i+1}-u^i)$     $v_1^{i+1}\leftarrow {(I+\sigma \partial F_1^{*})}^{-1}(v_1^i+\sigma K\left({\overline{u}}_1^{i+1}\right))$     $v_2^{i+1}\leftarrow {(I+\sigma \partial F_2^{*})}^{-1}(v_2^i+\sigma {\nabla }_{\mathrm{S}}{\overline{u}}_2^{i+1})$     $v_3^{i+1}\leftarrow {(I+\sigma \partial F_3^{*})}^{-1}(v_3^i+\sigma {\nabla }_{\mathrm{S}}{\overline{u}}_3^{i+1})$     $v_4^{i+1}\leftarrow {(I+\sigma \partial F_4^{*})}^{-1}(v_4^i+\sigma {\nabla }_{\mathrm{S}}{\overline{u}}_4^{i+1})$ end while