A General Self-Adaptive Relaxed-PPA Method for Convex Programming with Linear Constraints

Algorithm 2

General dual-primal relaxed-PPA method.

Step  1. Initialization. Let 𝛾∈(0,2) and pick (𝑥⁰,𝜆⁰)∈ℜ^𝑛×Λ, set 𝑘=0.

Step  2. Predictor. Let

          𝜆^𝑘=𝑃_Λ𝜆^𝑘−1𝑠(𝐴𝑥^𝑘−𝑏),                 (†)

and

     𝑥^𝑘𝑟=Argmin𝜃(𝑥)+2𝑥−𝑥^𝑘+1𝑟𝐴^𝑇𝜆^𝑘2∣𝑥∈𝒳.        (††)

Step  3. If the stepsize 𝑟 and 𝑠 are chosen satisfying

          𝜑𝑢^𝑘,𝑢^𝑘≥14𝑑𝑢^𝑘,𝑢^𝑘2_𝐺,                    (‡)

then go to Step 4. Otherwise, increase 𝑟,𝑠 and go back to Step 2.

Step  4. Corrector and updating

         𝑢^𝑘+1=𝑢^𝑘−𝛼_𝑘𝐺⁻¹𝑀(𝑢^𝑘−𝑢^𝑘).                    (∧)

Step  5. Adjustment

  𝑟(𝑟,𝑠)=2,𝑠2,if𝜑𝑢^𝑘,𝑢^𝑘≥𝜅∗𝑑𝑢^𝑘,𝑢^𝑘2_𝐺;(𝑟,𝑠),otherwise.Here𝜅>4.

Set 𝑘∶=𝑘+1.