An Implementable First-Order Primal-Dual Algorithm for Structured Convex Optimization

Algorithm 1

The proposed algorithm for (1)

Step  0. Input 𝑥⁰,𝑦⁰,𝜆⁰∈𝒳×𝒴.

Step  1. Set

     𝑥^𝑘+1=argmin_𝑥∈𝒳𝑓(𝑥)+𝑟2𝑥−𝑥^𝑘−1𝑟𝐴^𝑇𝜆^𝑘2,                 (∗)

     𝑦^𝑘+1=argmin_𝑦∈𝒴𝑔𝑦+𝑠2𝑦−𝑦^𝑘−1𝑠𝐵^𝑇𝜆^𝑘−𝛽𝐴2𝑥^𝑘+1−𝑥^𝑘+𝐵𝑦^𝑘−𝑏²,    (∗∗)

     𝜆^𝑘+1=𝜆^𝑘−𝛽𝐴2𝑥^𝑘+1−𝑥^𝑘+𝐵𝑦^𝑘+1−𝑏,                  (∗∗∗)

where 𝛽>0, and 𝑟>𝛽𝐴^𝑇𝐴,𝑠>𝛽𝐵^𝑇𝐵.

Step  2. If a termination criterion is not met, go to Step  1.