Table 1 The comparison between Condat ( \(\pmb{\rho_{k}=1}\) ) and PDFP

Form

\(\overline{v}^{k+1}=\operatorname{prox}_{\sigma {f_{2}^{*}}}(\sigma Bx^{k}+\overline{v}^{k})\),

\(x^{k+1}=\operatorname{prox}_{{\tau}{f_{3}}}(x^{k}-\tau \nabla {f_{1}}(x^{k})-{\tau} B^{T} (2\overline{v}^{k+1}-\overline{v}^{k}))\)

\(y^{k+1}=\operatorname{prox}_{{\gamma}{f_{3}}}(x^{k}-\gamma \nabla{f_{1}}(x^{k})-{\gamma} B^{T} \overline{v}^{k})\),

\(\overline{v}^{k+1}=\operatorname{prox}_{\frac{\lambda }{\gamma }{f_{2}^{*}}}(\frac{\lambda}{\gamma}By^{k+1}+\overline{v}^{k})\),

\(x^{k+1}=\operatorname{prox}_{{\gamma}{f_{3}}}(x^{k}-\gamma \nabla {f_{1}}(x^{k})-{\gamma} B^{T} \overline{v}^{k+1})\)

\(f_{1}\neq0\)

\(\sigma\tau\lambda_{\mathrm{max} }(BB^{T})+\tau/(2\beta)\leq1\)

\(0<\lambda< 1/\lambda _{\mathrm{max}}(BB^{T})\), 0<γ<2β

\(f_{1}=0\)

\(0<\sigma\tau\leq1/\lambda_{\mathrm{max} }(BB^{T})\)

\(0<\lambda< 1/\lambda_{\mathrm{max}}(BB^{T})\), 0<γ< + ∞

Relation