Algorithm 1.

Proximal Distance Iteration

Require: An objective $h_{\rho }(\mathit{x})$, gradient $\nabla h_{\rho }(\mathit{x})$, and algorithm map ${𝓜}_{\rho }(\mathit{x})$.
1:	Set tolerances ${\mathit{ϵ}}_d$ and ${\mathit{ϵ}}_g$; fix hyperparameters λ and k.
2:	Set maximum number of $\rho$ values, $t_{\text{max}}$; maximum inner iterations $m_{\text{max}}$.
3:	Initialize the estimate X and define an increasing sequence ${\{\rho (t)\}}_{t\ge 0}$.
4:	for $t\leftarrow 0,1,\dots ,t_{\max }$ do
5:	Set ${\mathit{x}}_0\leftarrow \mathit{x}$ using the current estimate and take $\rho \leftarrow \rho (t)$.
6:	for $m\leftarrow 1,\dots ,m_{\max }$ do
7:	Iterate the algorithm map,$x_{m+1}\leftarrow {𝓜}_{\rho }\left({\mathit{x}}_m\right)$.
8:	if ${\Vert \nabla h_{\rho }\left({\mathit{x}}_{m+1}\right)\Vert }_2\le {ϵ}_g$ then
9:	Break.
10:	end if
11:	end for
12:	Update $\mathit{x}\leftarrow {\mathit{x}}_{m+1}$
13:	if $\text{dist}(\mathit{x},S){ϵ}_d$ then
14:	Break.
15:	end if
16:	end for
17:	Project the final estimate $\mathit{x}\leftarrow P_{S_k}(\mathit{x})$.